数理解析研究所講究録 1221 巻 2001 年 125-154

125

DISCRETE CONSTANT MEAN CURVATURE SURFACES AND
THEIR INDEXKONRAD POLTHIER AND WAYNE ROSSMAN

ABSTRACT. We define triangulated piece wise linear constant mean curvaturesurfaces using avariational characterization, so that they are critical for area
amongst continuous piece wise linear variations which preserve the boundaryand simplicial structure and also (in the nonminimal case) the volume to one
side of the surface. We then find explicit examples, such as discrete minimal
catenoids and helicoids.
We use these discretized surfaces to study the index of unstable minimal
surfaces, by numerically evaluating the spectrum of their Jacobi operators,
and this approach deviates from other numerical investigations in that we use
avariational characterization to define the discrete approximating surfaces.
Our numerical estimates confirm known results on the index of some smoothminimal surfaces, and provide additional information regarding their areareducing variations.

1. INTRODUCTION

Smooth submanifolds, and surfaces in particular, with constant mean curvature(CMC) have along history of study, and modern work in this field relies heavily on
geometric and analytic machinery which has evolved over hundreds of years. However, nonsmooth surfaces are also natural mathematical objects, even though thereis less machinery available for studying them. (Consider M. Gromov’s approach of
doing geometry using only aset with ameasure and ameasurable distance function[8].)
Here we consider piecewise-linear triangulated surfaces (we call them “discretesurfaces”), which have been brought more to the forefront of geometrical researchby computer graphics. We define
for discrete surfaces inso that they are
critical for volume-preserving variations, just as smoothsurfaces are. Discretesurfaces have both interesting differences from and similarities with smoothones. For example, they are different in that smooth minimal graphs inover abounded domain are stable, whereas discrete minimal graphs can be highly unstable. We will explore properties like this in section 2.
And in section 3we will see some ways in which these two types of surfaces are
similar. We will see that: adiscrete catenoid has an explicit description in termsof the hyperbolic cosine function, just as the smooth catenoid has; and adiscrete
helicoid can be described with the hyperbolic sine function, just as aconformallyparametrized smooth helicoid is; and there are discrete Delaunay surfaces whichhave translational periodicities, just as smooth Delaunay surfaces have.
Pinkall and Polthier [16] used Dirichlet energy and anumerical minimization
procedure to find discrete minimal surfaces. In this work, we rather have the goal todescribe discrete minimal surfaces as explicitly as possible, and thus we are limite$\mathbb{R}^{3}$$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathbb{R}^{3}$$\mathrm{d}$126
KONRAD POLTHIER AND WAYNE ROSSMAN

to the more fundamental examples, for example the discrete minimal catenoid andhelicoid. We note that these explicit descriptions will be useful for implementing aprocedure that we describe in the next paragraphs.
Discrete surfaces have an advantage over smooth ones in the following way:
Function spaces representing smooth variations of smooth surfaces are infinite dimensional, and hence the study of linear operators on these spaces is often very
difficult. In particular, it is often difficult to get explicit information about the
spectra of such operators. However, the function spaces of variations of discretesurfaces contain piece-wise linear functions and are finite dimensional, and linear
operators then reduce to matrices. So the discrete case is much easier to handle.
This suggests that an useful procedure for studying the spectra of the linear Jasurfaces is to consider
cobi operator in the second variation formula of smooththe corresponding spectra of discrete
approximating surfaces. (This is strongly
related to the finite element method in numerical analysis; however, in our case thefinite element approximations will have geometric and variational meaning in theirown right.) As aparticular example of this, consider that aproblem of interest
is to find the index (the number of negative points in the spectrum) of asmoothminimal surface, and that the standard approach to this problem is to replace themetric of the surface with the metric obtained by pulling back the spherical metric
via the Gauss map. This approach can yield the index: for example, the index ofacomplete catenoid is 1([6]), the index of acomplete Enneper surface is 1([6]),the index of acomplete Jorge-Meeks -noid is $2n-3([11], [10])$ and the index ofa-Hoffman-Meeks surface is $2k+3$ for every $k\leq 37([13]$ ,acomplete genus[12] . However, this approach does not yield the eigenvalues and eigenfunctions oncompact portions of the original minimal surfaces, as the metric has been changed.
It would be interesting to know the eigenfunctions associated to negative eigenvalues, since these represent the directions of variations that reduce area, and theabove procedure can provide this information.
In sections 4and 6we establish some tools for studying the spectrum of discretesurfaces, and then we test the above procedure on two simple cases-a(minimal) rectangle, and aportion of asmooth minimal catenoid bounded by two circles.
In these two cases we know the spectra of the smooth surfaces (section 5), and weknow approximating discretesurfaces as well (section 3), so we can check thatthe above procedure produces good approximations for the eigenvalues and smootheigenfunctions (section 7), which indeed must be the case, by the theory of the finite element method [3], [7]. With these successful tests, we go on to consider cases
where we do not apriori know what the smooth eigenfunctions should be, such as
the Jorge-Meeks 3-noid and the genus 1Costa surface (section 7).
We note that the above procedure can also be implemented using discrete approximating surfaces which are found only numerically and not explicitly, such as
surfaces found by the method in [16]. And in fact, we use the method in [16] tofind approximating surfaces for the 3-noid and Enneper surface and Costa surface.
$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$

$n$

$k$

$)$

$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$

2. DISCRETE MINIMAL AND CMC SURFACES
We start with avariational characterization of discrete minimal and discretesurfaces. This characterization will allow us to construct explicit unstable discretesurfaces. (Note that merely finding minima for area with respect to avolume
constraint would not suffice for this, as that would produce only stable examples.
$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$127
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEXFIGURE 1
We will use
discrete surfaces that are unstable for our later numerical spectra
computations.
The following definition for discrete surfaces works equally well for surfaces in, but, as our constructions will all be in, we restrict to this space.
$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathbb{R}^{n}$$\mathbb{R}^{3}$

Definition 2.1.
discrete surface inis a triangular meshwhich has the
topology of an abstract 2-dimensional simplicial complex $K$ combined with a ge0metric-surface realization in . The geornetric realization $|K|$ is later minedby a set of vertices, andcan be identified with the pair
. The simplicial complex $K$ represents the connectivity of the mesh. The 0,
1and 2dimensional simplices of $K$ represent the vertices, edges, and triangles ofthe discrete surface.
Let $T=(p, q, r)$ denote an oriented triangle ofwith vertices , ,. Let$pq$ denote an edge ofwith endpoints , $q\in V$ .
For $p\in V$ , let star(p) denote the triangles of that contain as a vertex. For
an edge , let star(pq) denote the (at most two) triangles of that containasan edge.
$A$$\mathbb{R}^{3}$$C^{0}$$\mathcal{T}$$\mathbb{R}^{3}$$\mathcal{V}=\{p_{1}, \ldots,p_{n}\}\subset \mathbb{R}^{3}$$\mathcal{T}$$(K, \mathcal{V})$$\mathcal{T}$

$T$

$p$

$q$

$r\in \mathcal{V}$$p$$\mathcal{T}$$p$$\mathcal{T}$$\overline{pq}$$\overline{pq}$The area of adiscrete surface isarea(T) $:= \sum$ area$T$,

$T\in \mathcal{T}$where area$T$denotes the area of the triangle$T$

as asubset of$\mathbb{R}^{3}$.

Definition 2.2. Let $V=\{p_{1}, \ldots,p_{n}\}$ be the set of vertices of a discrete surface
variationof is defined as a variation of the vertices .
$A$$\mathcal{T}(t)$$\mathcal{T}$.

$C^{2}$$\mathcal{T}$$p_{*}$$p_{\dot{*}}(t)$:$[0, \epsilon)arrow \mathbb{R}^{3}$

so that Pi (0) $=p:\forall i=1$ ,$\ldots$, .
$n$

The straightness of the edges and the flatness of the triangles are preserved as thevertices move.

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KONRAD POLTHIER AND WAYNE ROSSMAN

In the smooth situation, when the boundary is fixed, the variation space is typically restricted to normal variations, since the tangential parts of the variations
only perform reparametrizations of the surfaces in the variations. However, on discrete surfaces there is an ambiguity in the choice of normal vectors at the vertices,
so we allow arbitrary variations. But we will later see (section 7) that our experimental results can accurately estimate normal variations of asmooth surface when
the discrete surface is aclose approximation to the smooth surface.
In the following we derive the evolution equations for some basic entities undersurface variations.
Letbe avariation of adiscrete surface . At each vertex of , thegradient of area is$\mathcal{T}$$\mathcal{T}(t)$(1)$\nabla_{p}$

area

$p$

$\mathcal{T}=\frac{1}{2}\sum_{T=(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}p}J(r-q)$,

where is rotation of angle in the plane of each oriented triangle
derivative of the surface area is then given by the chain rule$J$$\frac{\pi}{2}$$\frac{d}{dt}$area$\langle p’$$\mathcal{T}=\sum_{p\in \mathcal{V}}$:$\nabla_{p}$area$\mathcal{T}\rangle$$\mathcal{T}$$T$. The first.

The volume of the surface is the oriented volume enclosed by the cone of the surface
over the origin in$\mathbb{R}^{3}$$\mathrm{v}\mathrm{o}\mathrm{l}\mathcal{T}:=\frac{1}{6}\sum_{T=(p,q,r\rangle\in \mathcal{T}}\langle p,q\cross r\rangle=\frac{1}{3}\sum_{T=(p,q,r)\in \mathcal{T}}\langle\overline{N},p\rangle$

where

$p$

is any of the three vertices of the triangle$T$. area

$T$

,

and

$\overline{N}=\frac{(q-p)\cross(r-p)}{|(q-p)\cross(r-p)|}$is the oriented normal of . It follows that(2)$T$

. area $T\cdot\vec{N}+p\cross(r-q)$$\nabla_{p}\mathrm{v}\mathrm{o}\mathrm{l}\mathcal{T}=\frac{1}{6}\sum_{T=(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}p}q\mathrm{x}r=\frac{1}{6}\sum_{T=(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}p}2$and$\frac{d}{dt}\mathrm{v}\mathrm{o}\mathrm{l}\mathcal{T}=\sum_{p\in \mathcal{V}}\langle p’, \nabla_{p}\mathrm{v}\mathrm{o}\mathrm{l}\mathcal{T}\rangle$.

is an interior vertex, then the boundary of star is closed anddisappears from.
In the smooth case, aminimal surface is critical with respect to area for any
variation that fixes the boundary, and asurface is critical with respect to area
for any variation that preserves volume and fixes the boundary. We wish to define
discretesurfaces so that they have the same variational properties for the sametypes of variations. So we will consider variationsof that fix the boundary
and that additionally preserve volume in the nonminimal case, which we callpermissible variations. The condition that makes adiscrete surface area-critical forany permissible variation is expressed in the following definition
Note that if$p$$\sum_{T\in 8\mathrm{t}\mathrm{a}\mathrm{r}p}p\mathrm{x}(r-q)=0$$p$$\nabla_{p}\mathrm{v}\mathrm{o}\mathrm{l}\mathcal{T}$$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathcal{T}(t)$$\partial \mathcal{T}$$\mathcal{T}$129
DISCRETE CONSTANT MEAN CURVATU RE SURFACES AND THEIR INDEXDefinition 2.3. A discrete surface has constant mean curvature (CMC) if there
area
exists a constant $H$ so thatfor all interior vertices . If $H=0$
then it is minimal.
$\nabla_{p}$$=H\nabla_{p}\mathrm{v}\mathrm{o}\mathrm{l}$$p$This definition for discrete minimality has been used in [16]. In contrast, oursurfaces are charsurfaces differs from [14], wheredefinition of discrete
and aconjugation
acterized algorithmically using discrete minimal surfaces intransformation. Compare also [2] for adefinition via discrete integrable systems
which lacks variational properties.
$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$$S^{3}$

2.0.1. Uniqueness of Discrete Minimal Disks. Uniqueness of abounded minimal
surface with agiven boundary ensures that it is stable, and uniqueness can sometimes be decided using the maximum principle of elliptic equations. For example,
the maximum principle ensures that aminimal surface is contained in the convex
hull of its boundary, and, if the boundary has a1-1 projection to aconvex planar
curve, then it is unique for that boundary and is aminimal graph. The maximum
principle also shows that any minimal graph is unique even when the projection ofits boundary is not convex. More generally, stability still holds when the surface
merely has aGauss map image contained in ahemisphere, as shown in [1] (although
their proof employs tools other than the maximum principle).
However, such statements do not hold for discrete minimal surfaces. Consider
the surface shown in the left-hand side of Figure 2, whose height function has alocal maximum at an interior vertex. This example does not lie in the convex
hull of its boundary and thereby disproves existence of adiscrete version of the
maximum principle. Also, the three surfaces on the right-hand side in Figure 3areall minimal graphs over aring-like domain with the same boundary contours andsimplicial structure, and yet they are not the same surfaces, hence graphs with given
simplicial structure are not unique. And the left-hand surface in Figure 3shows
asurface whose Gauss map is contained in ahemisphere but which is unstable(this surface is not agraph) -another example of this property is the first ring-likesurface in Figure 3, which is also unstable. (We define stability of discretesurfaces in section 4).
The influence of the discretization on nonuniqueness, like as in the ring-likeexamples of Figure 3, can also be observed in amore trivial way for adiscrete
minimal graph over asimply connected convex domain. The two surfaces on the, i.e. they are identical as geometricheight-hand side of Figure 2have the same
surfaces, but they are different as discrete surfaces. Interior vertices may be freely
added and moved inside the middle planar square without affecting minimality.
In contrast to existence of these counterexamples we believe that some propertiesof smooth minimal surfaces remain true in the discrete setting, based on numericalexperiments. We say that adiscrete surface is adisk if it is homeomorphic to asimply connected domain.
$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$Conjecture 2.1. Let
be a discrete minimal disk whose boundary projects
injectively to a convex planar polygonal curve, thenis a graph over that plane.
$\mathcal{T}\subset \mathbb{R}^{3}$$\mathcal{T}$The authors were able to prove this conjecture with the extra assumption that all
the triangles of the surface are acute, using the fact that the maximum principle (aheight function cannot attain astrict interior maximum) actually does hold when
all triangles are acute

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KONRAD POLTHIER AND WAYNE ROSSMAN

FIGURE 2. Two views on the left-hand side of asurface that defies the maximum principle, and two discrete minimal surfaces on,
the right-hand side with boundary vertices $(x, 0, z_{1})$ ,$(0, y, z_{2})$ , and. These two surfaces on the rightinhave the sameinbut have different simplicial structures,
and asurprising feature of these examples is that the innermost.
triangles form asquare, regardless of the values of , ,$(\mathrm{x}, 0, z_{1})$$(\mathrm{O}, -y, z_{2})$$\mathbb{R}^{3}$$\mathbb{R}^{3}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$

$x$

$y$

$z_{1}\neq z_{2}$One can ask if adiscrete minimal surface with given simplicial structure andboundary is unique if it has a1-1 perpendicular or central projection to aconvex
polygonal domain in aplane. The placement of the vertices need not be unique,as we saw in the examples on the right-hand side of Figure 2, however, one canconsider if there is uniqueness in the sense that theof inis unique.
$\mathcal{T}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathcal{T}$$\mathbb{R}^{3}$

Conjecture 2.2. Let
be a polygonal curve that either :projects injectively to a convex planar polygonal curve, or :has 1-1 central projection froma point
to a convex planar polygonal curve. Then, for each given simplicial
structure of disk type with boundary compatible to , there exists a discrete minimal
disk with boundary and that simplicial structure, and the trace of is uniquelydetermined. Furthe rmore,is a graph in the case , andis contained in thecone of over in the case .
$A$$\Gamma\subset \mathbb{R}^{3}$

$B$

$a$

$p\in \mathbb{R}^{3}$$\Gamma$$\Gamma$$\mathcal{T}$$\mathcal{T}$$\mathcal{T}$$\Gamma$$p$$A$$\mathcal{T}$$B$

We have the following weaker form of Conjecture 2.2, which follows from Corollary 4.1 of section 4in the case that there is only one interior vertex:

Conjecture 2.3. If a discrete minimal surface is a graph over a convex polygonal
domain, then it is stable.

3. EXPLICIT DISCRETE SURFACES
Here we describe explicit discrete catenoids and helicoids, which seem to be thefirst explicitly known nontrivial complete discrete minimal surfaces (with minimality defined variationally).

3.1. Discrete Minimal Catenoids. To derive an explicit formula for embedded
complete discrete minimal catenoids, we choose the vertices to lie on congruent
planar polygonal meridians, with the meridians placed so that the traces of thesurfaces will have dihedral symmetry. We will find that the vertices of adiscrete
meridian lie equally spaced on asmooth hyperbolic cosine curve. Furthermore,these discrete catenoids will converge uniformly in compact regions to the smoothcatenoid as the mesh is made finer.
We begin with alemma that prepares the construction of the meridian of thediscrete minimal catenoid. We derive an explicit representation of the position of a

131
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEXFIGURE 3. Two unstable and two stable discrete minimal surfacesin. The first figure on the left is unstable, even though it islocally agraph over ahorizontal plane, in the sense that the third
coordinate of the normal vector to the surface is never zero. The
second figure on the left is one of the four congruent pieces ofthe first figure. The middle figure (the leftmost annular graph)is unstable, even though it is agraph over an annular polygonal
region in ahorizontal plane, and it has area-reducing variations
that can deform to either of the last two stable minimal surfaces
on the right, which have the same simplicial structure.
$\mathbb{R}^{3}$vertex surrounded by four triangles in terms of the other four vertex positions. The
center vertex is assumed to be coplanar with each of the two pairs of two opposite
vertices, with those two planes becoming the plane of the vertical meridian and thehorizontal plane containing adihedrally symmetric polygon (consisting of edges ofthe surface).

Lemma 3.1. Consider the vertex $p=(d,0, e)$ surrounded by four vertices $q_{1}=$$(a, 0, b),$ $q_{2}=(d\cos\theta, d\sin\theta, e)$ , $q_{3}=(f, 0,g)$ , and $q_{4}=(d\cos\theta, -d\sin\theta, e)$ ,form$ing$,,, and. Given real numbers
four triangle, ,and angle so that $b\neq e$ , there exists a choice of real numbers andsuch that$(p, q_{1}, q_{2})$

$a$

$b$

$d_{f}e_{f}$$(p, q_{2}, q_{3})$$(p, q_{3}, q_{4})$$(p, q_{4}, q_{1})$$\theta$$f$$\nabla_{p}$$g$area(star ) $=0$$p$

if and only if$2ad> \frac{(e-b)^{2}}{1+\cos\theta}$

Furthermore, when

.

$f$

and

$f$

$=$

$\frac{2(1+\cos\theta)d^{3}+(a+2d)(e-b)^{2}}{2ad(1+\cos\theta)-(e-b)^{2}}$

$g$

$=$

$2e-b$ .

$g$exist, they are unique and must be of the form

Proof First we note that the assumption,

is necessary. If $b=e$ , then one may
choose $g=b$ , and then there is afree 1-parameter family of choices of .
For simplicity we apply avertical translation and ahomothety about the originof
to normalize $d=1$ , $e=0$ , and by doing areflection if necesary, we may
assume $b<0$ . Let $c=\cos\theta$ and $s=\sin\theta$ .
$b\neq e$$f$$\mathbb{R}^{3}$

132
KONRAD POLTHIER AND WAYNE ROSSMAN

area to vanish. The
We derive conditions for the coordinate components ofdefinitions
.
the
star
of
Using
symmetrybyvanishes
component
second$\nabla_{p}$$p$$c_{1}:= \frac{(a-1)s^{2}-b^{2}(1-c)}{\sqrt{2b^{2}(1-c)+(a-1)^{2}s^{2}}}$the first (resp. third) component of(3)$\nabla_{p}$$c_{1}= \frac{g^{2}(1-c)-(f-1)s^{2}}{\sqrt{2g^{2}(1-c)+(f-1)^{2}s^{2}}}$,$c_{2}:= \frac{ab+b}{\sqrt{2b^{2}(1-c)+(a-1)^{2}s^{2}}}$,

area vanishes if, resp.

$c_{2}= \frac{-(f-1)g-2g}{\sqrt{2g^{2}(1-c)+(f-1)^{2}s^{2}}}$.

Dividing one of these equations by the other we obtain(4)$f-1= \frac{c_{2}g(1-c)+2c_{1}}{c_{2}s^{2}-c_{1}g}g$,

is determined by . It now remains to determine if one can find so that$c_{2}s^{2}-c_{1}g\neq 0$ .
If $f-1$ is chosen as in equation 4, then the first minimality
condition of equation 3holds if and only if the second one holds as well. So we only
need to insert this value for $f-1$ into the first minimality condition and check for, we find that the condition becomes,
solutions . When

so

$f$

$g$

$g$

$c_{1}\neq 0$$g$$1= \frac{c_{2}s^{2}-c_{1}g}{|c_{2}s^{2}-c_{1}g|}\frac{g}{|g|}\frac{-(1-c)g^{2}-2s^{2}}{\sqrt{2(1-c)c_{2}^{2}s^{4}+4c_{1}^{2}s^{2}+(2(1-c)c_{1}^{2}+s^{2}(1-c)^{2}c_{2}^{2})g^{2}}}$, note that this equation can hold only ifhave opposite signs, so the equation becomes$\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}-(1-c)g^{2}-2s^{2}<0$$g$$c_{2}s^{2}-c_{1}g$$1= \frac{(1-c)g^{2}+2s^{2}}{\sqrt{2(1-c)c_{2}^{2}s^{4}+4c_{1}^{2}s^{2}+(2(1-c)c_{1}^{2}+s^{2}(1-c)^{2}c_{2}^{2})g^{2}}}$

.
and

,

which simplifies to.

$1= \frac{\sqrt{(1-c)g^{2}+2s^{2}}}{\sqrt{(1-c)c_{2}^{2}s^{2}+2c_{1}^{2}}}$This implies$g^{2}$

is uniquely determined. Inserting the value$g=\pm b$,

one finds that the above equation holds. When $g=b<0$ , we find that $c_{2}s^{2}-c_{1}g<$, which is impossible. When $g=-b>0$ , we find that $c_{2}s^{2}-c_{1}g<0$ if and only
if $2a(1+c)>b^{2}$ . And when $g=-b$ and $2a(1+c)>b^{2}$ , we have the minimality
condition when$0$

$f= \frac{2+2c+ab^{2}+2b^{2}}{2a+2ac-b^{2}}$.

Inverting the transformation we did at the beginning of this proof brings us backto the general case where and are not necessarily 1and 0, and the equations
for and become as stated in the lemma.
When $c_{1}=0$ , we have $(a-1)(1+c)=b^{2}$ and $(f-1)(1+c)=g^{2}$ , so, in particular,
we have $a>1$ and therefore $2a(1+c)>b^{2}$ . The right-hand side of equation (3)implies $g=-b$ and $f=a$ . Again, inverting the transformation from the beginningof this proof, we have that and must be of the form in the lemma for the case$c_{1}=0$ as well.
$d$

$f$

$e$

$g$

$f$

$g$

$\square$133
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEX$y$FIGURE 4. The construction in Lemma 3.1.
Note that the necessary and sufficient condition in the next lemma is identical tothat of the previous lemma. This observation is crucial to the proof of the upcoming
theorem.

Lemma 3.2. Given two points $(a, b)$ and $(d, e)$ in
and an angle , with $b\neq e$ ,there exists an so that these two points lie on some vertical translate of the curve$\mathbb{R}^{2}$$\theta$$r$$(r\cosh\{$if and only if
Proof Define$\frac{1}{e-b}$

arccosh$2ad> \frac{(e-b)^{2}}{1+\cos\theta}$$\{1+\frac{1}{r^{2}}\frac{(e-b)^{2}}{1+\cos\theta}\}t],t)$$t\in \mathbb{R}$,.

. Without loss of generality, we may assume $0<a\leq d$
and $e>0$ , and hence $-e\leq b<e$ . If the points $(a, b)$ and $(d, e)$ both lie on the
curve in the lemma, then$\hat{\delta}=\frac{e-b}{\sqrt{1+\cos\theta}}$

arccosh$(1+ \frac{\hat{\delta}^{2}}{r^{2}})=\mathrm{a}\mathrm{r}\mathrm{c}\cosh$$( \frac{d}{r})-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(6)$. arccosh$( \frac{a}{r})$,where sign(6) $=1$ if $b\geq 0$ and sign(6) $=-1$ if $b<0$ . Note that if $b=0$ , then$=0$ ). This equation is solvable (for either value ofmust equal (and sosign(6) if and only if
$a$

$r$

$\mathrm{a}\mathrm{r}\mathrm{c}\cosh(\frac{a}{r})$$)$$(_{\frac{d}{r}+} \sqrt{\frac{d^{2}}{r^{2}}-1})(_{\frac{a}{r}+}\sqrt{\frac{a^{2}}{r^{2}}-1})=1+\frac{\hat{\delta}^{2}}{r^{2}}+\frac{\hat{\delta}}{r}\sqrt{2+\frac{\hat{\delta}^{2}}{r^{2}}}$

when $b\leq 0$ , or$\frac{\frac{d}{r}+\sqrt{\frac{d^{2}}{r^{2}}-1}}{\frac{a}{r}+\sqrt{\frac{a^{2}}{r^{2}}-1}}=1+\frac{\hat{\delta}^{2}}{r^{2}}+\frac{\hat{\delta}}{r}\sqrt{2+\frac{\hat{\delta}^{2}}{r^{2}}}$

134
KONRAD POLTHIER AND WAYNE ROSSMAN

when $b\geq 0$ , for some $r\in(0, a]$ . The right-hand side of these two equations has thefollowing properties:
1. It is anonincreasing function of $r\in(0, a]$ .
2. It attains some finite positive value at $r=a$ .
.
3. It is greater than the function.
asymptotically as
4. It approaches
The left-hand sides of these two equations have the following properties:
1. They attain the same finite positive value at $r=a$ .
2. The first one is anonincreasing function of $r\in(0, a]$ .
3. The second one is anondecreasing function of $r\in(0, a]$ .
4. The second one attains the value at $r=0$ .
5. The first one is less than the function $4ad/r^{2}$ .
.
6. The first one approaches $4ad/r^{2}$ asymptotically as
So, from these properties it is clear that one of the two equations above has a. This completes the proof.
solution for some if and only if$2\hat{\delta}^{2}/r^{2}$$rarrow \mathrm{O}$$2\hat{\delta}^{2}/r^{2}$$\frac{d}{a}$$rarrow \mathrm{O}$$2ad>\hat{\delta}^{2}$$r$$\square$

We now derive an explicit formula for discrete minimal catenoids, which is given
by specifying the vertices along planar polygonal meridians. Then the traces of thesurfaces will have dihedral symmetry of order $k\geq 3$ . The surfaces are tessellated
grid, and each trapezoid can be triangulated
by planar isosceles trapezoids like a
into two triangles by choosing adiagonal of the trapeziod as the interior edge. Either
diagonal can be chosen, as this does not affect the minimality of the catenoid.
The discrete catenoid has two surprising features. First, the vertices of ameridian lie on asmooth hyperbolic cosine curve (which is the profile curve of smoothcatenoids), and there is no apriori reason to have expected this. Secondly, thevertical spacing of the vertices along the meridians is constant.
$\mathbb{Z}^{2}$

Theorem 3.1. There exists a $fou$ ’-parameter family of embedded and completewith dihedral rotational symmetry anddiscrete minimal catenoids
planar meridians. If we assume that the dihedral symmetry axis is the axis anda meridian lies in the $xz$ -plane, then, up to vertical translation, the catenoid iscompletely described by the following properties:$k\geq 3$ , is the dihedral angle.
,
1.
2. The vertices of the meridian in the $xz$ -plane interpolate the smooth $\cosh$ curve$C=\mathrm{C}(0, \delta, r, z_{\mathrm{O}})$$z$$\theta=\frac{2\pi}{k}$$k\in \mathrm{N},$$x(z)=r \cosh(\frac{1}{r}az)$,with$a= \frac{r}{\delta}$

aarrccccoosshh$(1+ \frac{1}{r^{2}}\frac{\delta^{2}}{1+\cos\theta})$,where the parameter $r>0$ is the waist radius of the interpolated $\cosh$ curve,
and $\delta>0$ .
, the profile curve has vertices of
3. For any given arbitrary initial value
the forrn$z_{\mathrm{O}}\in \mathbb{R}$

where
san.

$\delta$$z_{j}$$=$$z_{0}+j\delta$$x_{j}$$=$$x(z_{j})$is the constant vertical distance between adjacent vertices of the merid-

135

FIGURE 5. Adiscrete minimal catenoid and helicoid. (For thishelicoid we have chosen $x_{0}=0.$ )4. The planar trapezoids of the catenoid may be triangulated independently ofeach other.

Proof. By Lemma 3.1, if we have three consequtive vertices $(x_{n-1}, z_{n-1}),(x_{n}, z_{n})$ ,and $(x_{n+1}, z_{n+1})$ along the meridian (the profile curve in the $xz$ -plane), they satisfy
the recursion formula(5)$x_{n+1}= \frac{(x_{n-1}+2x_{n})\hat{\delta}^{2}+2x_{n}^{3}}{2x_{n}x_{n-1}-\hat{\delta}^{2}}$,$z_{n+1}=z_{n}+\delta$,

where $=z_{n}-z_{n-1}$ and. As seen in Lemma 3.1, the verticaldistance between $(x_{n-1}, z_{n-1})$ andis the same as the vertical distance
betweenand $(x_{n+1}, z_{n+1})$ , so we may consider and to be constants
independent of .
In order for the surface to exist, Lemma 3.1 requires that$\hat{\delta}=\delta/\sqrt{1+\cos\theta}$$\delta$$(x_{n}, z_{n})$$(x_{n}, z_{n})$$\hat{\delta}$$\delta$$n$$2x_{n}x_{n-1}>\hat{\delta}^{2}$This implies that all
have the same sign, and we may assume $x_{n}>0$ for all .
Therefore the surface is embedded. Also, as the condition
implies$x_{n}$$n$$2x_{n}x_{n-1}>\hat{\delta}^{2}$$2x_{n+1}x_{n}= \frac{2x_{n}(x_{n-1}+2x_{n})\hat{\delta}^{2}+4x_{n}^{4}}{2x_{n}x_{n-1}-\hat{\delta}^{2}}>\frac{2x_{n}x_{n-1}\hat{\delta}^{2}}{2x_{n}x_{n-1}-\hat{\delta}^{2}}>\hat{\delta}^{2}$,

we see, inductively, that $xj$ is defined for all. Hence the surface is complete.
$x(z)$
One can easily check that the functionin the theorem also satisfies therecursion formula (5), in the sense that if $xj:=x(zj)$ , then these $xj$ satisfy thisrecursion formula. It only remains to note that, given two initial points $(x_{n-1}, z_{n-1})$
andwith $z_{n}>z_{n-1}$ , there exists an so that these two points lie on
the curve $x(z)$ with our given and 0(up to vertical translation) if and only if, as shown in Lemma 3.2.
$j\in \mathrm{Z}$$(x_{n}, z_{n})$$r$$\delta$$2x_{n}x_{n-1}>\hat{\delta}^{2}$$\square$

136
KONRAD POLTHIER AND WAYNE ROSSMAN

Remark 3.1. If we consider the example where $(x_{1}, z_{1})=(1,$ 0) and $(x_{2}, z_{2})=$, then the recursion formula implies that$(1+\hat{\delta}^{2}, \delta)$$(x_{n}, z_{n})=(1+ \sum_{j=1}^{n-1}2^{j-1}a_{n-1,j}\hat{\delta}^{2j}, (n-1)\delta)$

where$\mathrm{a}\mathrm{n}-\mathrm{i},\mathrm{j}$$a_{0,0}=1$,

for$n\geq 3$,

is defined recursively by $a_{n,m}=0$ if $m<0$ or $n<0$ or $m>n$ ,if $n\geq m\geq 1$ .
if $n>0$ , and$an,m=2a_{n-1,m}-\mathrm{a}\mathrm{n}-\mathrm{i},\mathrm{j}$$\mathrm{a}\mathrm{o},0=2$

Thus

,

$a_{n,m}=(\begin{array}{l}n+m2m\end{array})$$+$$\mathrm{a}\mathrm{n}-\mathrm{i},\mathrm{m}-\mathrm{i}$$(\begin{array}{ll}n+m -12m \end{array})$.

are closely related to the recently solved refined alternating sign matrix
These
conjecture [4].
$a_{n,m}$Corollary 3.1. There exists a twO-parameter family of discrete catenoids
whose vertices interpolate the smooth minimal catenoid with meridian $x=\cosh z$ .

$C_{1}(\theta, z_{0})$Proof The waist radius of the discrete meridian must be $r=1$ . Further, we mustchoose the parameter $a=1$ which is fulfilled if and 5are related by $1+\cos\theta+\delta^{2}=$$(1+\cos\theta)\cosh\delta$ . The offset parameter
may be chosen arbitrarily leading to avertical shift of the vertices along the smooth catenoid.
$\theta$$z_{0}$$\square$Corollary 3.2. For each fixed and , the profile curves of the discrete catenoids
approach the profile curve $x=r \cosh\frac{z}{r}$ of a smooth catenoid uniformly.
as 6,
in compact sets of$r$$z_{0}$$C(\theta,\delta,r, \mathrm{z}\mathrm{o})$$\mathbb{R}^{3}$$\thetaarrow 0$

Proof Since,$\lim_{\deltaarrow 0}\frac{1}{\delta}\mathrm{a}\mathrm{r}\mathrm{c}\cosh(1+\frac{1}{r^{2}}\frac{\delta^{2}}{1+\cos\theta})=\frac{\sqrt{2}}{r\sqrt{1+\cos\theta}}$

it follows that the profile curve of the discrete catenoid converges uniformly (insense) to the curve$C^{\mathrm{O}}$$x=r \cosh\frac{\sqrt{2}z}{r\sqrt{1+\cos\theta}}$as$\deltaarrow 0$. Then, as$\thetaarrow 0$

we approach the profile curve $x=r \cosh\frac{z}{r}$ .

$\square$3.2. Discrete Minimal Helicoids. We continue on to the derivation of explicit
discrete helicoids, which are anatural second example of complete, embedded discrete minimal surfaces.
In the smooth setting, there exists an isometric deformation through conjugate
surfaces ffom the catenoid to the helicoid (see, for example, [15]). So, one might
first try to make asimilar deformation from the discrete catenoids in Theorem 3.1
to discrete minimal helicoids. But such adeformation appears to be impossiblein fact, in order to make an associate family of discrete minimal surfaces, one mustallow non-continuous triangle nets having greater flexibility, as described in [17].
Therefore, we adopt adifferent approach for finding discrete minimal helicoids.
The helicoids will be comprised of planar quadrilaterals, each triangulated by four
coplanar triangles, see Figure 5. Each quadrilateral is the star of aunique vertex,
and none of its four boundary edges are vertical or horizontal, and one pair of
opposite vertices in its boundary have the same -coordinate, and the four boundary
$z$

137
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEX

$z$

$x$

FIGURE 6. The construction in Lemma 3.3.

edges consist of two adjacent pairs of edges of equal length. First we derive an
explicit representation of the center vertex of atypical vertex star of the helicoid:
Lemma 3.3. Let
be a point with a vertex star consisting of four vertices ,,and four triangles $\triangle_{i}=(p, q_{i}, q:+1)$ , $i\in\{1,2,3,4\}$ (mod 4). We assurne that $p=(u, 0,0)$ , $q_{1}=(b\cos\theta, b\sin\theta, 1),$ $q_{2}=(b\cos\theta, -b\sin\theta, -1)$ , $q_{3}=$$(t\cos\theta, -t\sin\theta, -1),$$q_{4}=(t\cos\theta,t\sin\theta, 1)$ with real numbers $b<u<t$ and.
eitherIf$p$$q_{2}$$q_{3},$$q_{1}$$q_{4}$$\theta\in(0, \frac{\pi}{2})$t $=-b(1+2u^{2}\sin^{2}\theta)+2u\sqrt{1+b^{2}\sin^{2}\theta}\sqrt{1+u^{2}\sin^{2}\theta}$ orb $=-t(1+2u^{2}\sin^{2}\theta)+2u\sqrt{1+t^{2}\sin^{2}\theta}\sqrt{1+u^{2}\sin^{2}\theta}$ ,then$\nabla_{p}$area vanishes.

Proof. Consider the conormals $J_{1}=J(q_{2}-q_{1})$ , $J_{2}=J(q_{3}-q_{2})$ , $J_{3}=J(q_{4}-q_{3})$ ,, where denotes oriented rotation by angle
containing the edge being rotated. Then
$J_{4}=J(q_{1}-q_{4})$$J$$J_{1}=(2\sqrt{1+b^{2}\sin^{2}\theta}, \circ, 0)$

Since
$0$

and

$\frac{\pi}{2}$in the triangle$J_{3}=(-2\sqrt{1+t^{2}\sin^{2}\theta}, \circ, 0)$$\triangle_{j}$.

$(\mathrm{J}_{4}, (\cos\theta, \sin\theta, 0), (u-b\cos\theta, -b\sin\theta, -1))=$andand $|J_{4}|^{2}=(t-b)^{2}$ , we have that the first component of(and also of ) is$\langle J_{4}, (\cos\theta, \sin\theta, 0)\rangle=0$$J_{4}$$u(t-b)\sin^{2}\theta$$J_{2}$.

$\overline{\sqrt{1+u^{2}\sin^{2}\theta}}$By symmetry, the second and third components of andare equal but opposite
in sign, hence the second and third components of $J_{1}+J_{2}+J_{3}+J_{4}$ are zero. So
for the minimality condition to hold at , we need that the first component of$J_{1}+J_{2}+J_{3}+J_{4}$ is also zero, that is, we need
$J_{2}$$J_{4}$$p$$\frac{u(t-b)\sin^{2}\theta}{\sqrt{1+u^{2}\sin^{2}\theta}}+\sqrt{1+b^{2}\sin^{2}\theta}-\sqrt{1+t^{2}\sin^{2}\theta}=0$

and the solution of this with respect tosolution,area vanishes.
$\nabla_{p}$

$b$

or

$t$

,

is as in the lemma. So, for this$\square$

138

FIGURE 7. Discrete analogues of cylinders and Delaunay surfaces.
Theorem 3.2. There exists a family of complete embedded discrete minimal helicoids, with the connectivity as shown in Figure 5. The vertices, indexed by,
are the points$i,j\in \mathbb{Z}$$\frac{r\sinh(x_{0}+j\delta)}{\sin\theta}(\cos(i\theta),\sin(i\theta)$

for any given reals$\theta\in(0, \frac{\pi}{2})$

and ,
$r$

$\delta\in \mathrm{R}$, $0)+( 0, ir)$ ,.

Note that these surfaces are invariant under the screw motion that combines
vertical upward translation of distance $2r$ with rotation about the -axis by anangle of 20. The term determines the offset of the vertices from the -axis, anddetermines the horizontal spacing of the vertices. The homothety factor is , whichequals the vertical distance between consecutive horizontal lines of edges.
$x_{3}$$\delta$$z$$x_{0}$$r$

Proof. Without loss of generality, we may assume $r=1$ . So for agiven , the$i$vertices are points on the line
, for certain values of .
We chooseand so that thevertex has -value $s_{j-2}=\sinh(x_{0}+(j-$and thevertex has -value $s_{j-1}=\sinh(x_{0}+(j-\mathrm{j}\mathrm{S})/\sin\theta$ .
Lemma 3.3 implies that thevertex has s-value$\{s(\cos(i\theta), \sin(i\theta),i)|s\in \mathbb{R}\}$$\delta$$(j-2)’ \mathrm{t}\mathrm{h}$$x_{0}$$2)\delta)/\sin\theta$$(j-1)’ \mathrm{t}\mathrm{h}$

$s$

$s$

$s$

$j’ \mathrm{t}\mathrm{h}$$s_{j}=-s_{j-2}(1+2s_{j-1}^{2}\sin^{2}\theta)+2\mathrm{s}\mathrm{j}-\mathrm{i}\sqrt{1+s_{j-2}^{2}\sin^{2}\theta}\sqrt{1+s_{j-1}^{2}\sin^{2}\theta}$,

arecursion formula that is satisfied by$s_{j}=\sinh(x_{0}+j\delta)/\sin\theta$.

Lemma 3.3 implies asimilar formula for determining
in terms ofand, with the same solution. Finally, noting that those vertices whose star is aplanar quadrilateral can be freely moved inside that planar quadrilateral without
disturbing minimality of the surface, the theorem is proved.
$\mathrm{S}\mathrm{j}-2$$\mathrm{S}\mathrm{j}-\mathrm{z}$$\mathrm{S}\mathrm{j}-2$$\square$3.3. Discrete Cylinders and Delaunay Surfaces. We now describe some ways
one can find discrete analogues of cylinders and Delaunay surfaces. The simplest
way is to choose positive reals and and an integer $k\geq 3$ , and then choose thevertices to be
$a$

$e$

$pj,\ell=(a\cos(2\pi j/k), a \sin(2\pi j/k)$,$e\ell)$for. We then make agrid of rectangular faces, and cut the faces by
diagonals with endpoints
and $pj+1,\ell+1$ . This is adiscretesurface with
$H=a^{-1}(\cos(\pi/k))^{-1}$ . It is interesting to note that $H$ is independent of the valueof . See the left-hand side of Figure 7.
Another example is to choose positive reals , , , and an integer $k\geq 3$ , and tochoose the vertices to be$j,\ell\in \mathrm{Z}$$pj,\ell$$\mathrm{C}\mathrm{M}\mathrm{C}$

$e$

$a$

$pj,\ell=$$(a\cos(2\pi j/k), a \sin(2\pi j/k)$,$e\ell)$$b$$e$when $jf$ is even, and$\ell$

139

FIGURE 8. Atriply-periodic discrete minimal surface, whose stability properties have been investigated by Karsten
Brauckmann. On the left is afundamental piece, and such pieces areattached as on the right to make an infinite-topology surface withthe same symmetry as the
cubic lattice in .
$\mathrm{G}\mathrm{r}\mathrm{o}\beta \mathrm{e}$$\mathbb{Z}^{3}$$pj,\ell=$$\mathbb{R}^{3}$,$(b\cos(2\pi j/k), b\sin(2\pi j/k)$$e\ell)$when $j+\ell$ is odd,for ,. We then make agrid of quadrilateral faces, and cut the faces by
diagonals with endpoints
and $pj+1,\ell+1$ if $jf$ is even, and by diagonals with
$j+\ell$
endpoints $pj,\ell+1$ and $pj+1,\ell$ if
is odd. By symmetry, it is clear thatarea
andare parallel at each vertex; and for each value of , one can then show
the existence of values of and so that $H$ is the same value at all vertices, using
an intermediate value argument. Thus adiscrete
cylinder is produced. See
the second surface in Figure 7.
Athird example can be produced by taking the vertices to be$j$$\ell\in \mathrm{Z}$$\ell$$pj,\ell$$\nabla_{p_{j,t}}$$\nabla_{p_{j,t}}vol$

$e$

$a$

$b$

$\mathrm{C}\mathrm{M}\mathrm{C}$$pj,\ell=$$(a\cos(2\pi j/k), a \sin(2\pi j/k)$$pj,\ell=$,$e\ell)$$(b\cos(2\pi j/k), b\sin(2\pi j/k)$

,

when

$e\ell)$$\ell$

whenis even, and$\ell$

is odd,for. We then make agrid of isosceles trapezoidal faces, and put an extra
vertex in each of the trapezoidal faces, and connect this extra vertex by edges toeach of the four vertices of the surrounding trapezoid. Keeping the placement of
the vertices of the surface as symmetric as possible, one must move these extra
vertices inso thatand Vvol become parallel at these vertices, and thenone must solve so that $H$ has the same value at all vertices of the surface. This canbe done numerically. See the last two examples in Figure 7.
$j,\ell\in \mathrm{Z}$$\mathbb{R}^{3}$$\nabla area$

Remark 3.2. The 2-dimensional boundaries of the tetrahedron, octahedron, andicosahedron are discretesurfaces. The boundaries of the cube and dodecahedron are not discrete surfaces in our sense, as they are not triangulated. However,by adding a vertex to the center of each face and connecting it by edges to eachvertex in the boundary of the face, we can make discrete surfaces, and then we canmove these face-centered vertices perpendicularly to the faces to adjust the mean
curvature.
$\mathrm{C}\mathrm{M}\mathrm{C}$

4. SECOND VARIATION OF AREA
We now begin to consider the spectra of the second variation for discretesurfaces, which necessarily starts with atechnical and explicit computation of thesecond variation. For notating area and volume, we shall now frequently use “ ”$\mathrm{C}\mathrm{M}\mathrm{C}$

$a$

140
KONRAD POLTHIER AND WAYNE ROSSMAN, for brevity. We will also useand ” ” instead of “area” and “
to signify the area of atriangle $T=(p, q, r)$ .
$V$$\mathrm{v}\mathrm{o}\mathrm{l}"$Lemma 4.1. For a compact discrete$\mathrm{C}\mathrm{M}\mathrm{C}H$

surface$\mathcal{T}$$|T|$

or $|(p, q, r)|$with vertex set ,$\mathcal{V}$$a’(0):= \frac{\partial^{2}}{\partial^{2}t}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}(\mathcal{T})|_{t=0}=\sum_{p\in \mathcal{V}}\langle p’, (\nabla_{p}a)’-H(\nabla_{p}V)’\rangle$

for any permissible variation.
Proof$a’(0)= \sum_{p\in \mathcal{V}}\langle p’, \nabla_{p}a\rangle+\langle p’, (\nabla_{p}a)’\rangle=\sum_{p\in \mathcal{V}}\langle p’, H\nabla_{p}V\rangle+\sum_{p\in \mathcal{V}}\langle p’, (\nabla_{p}a)’\rangle$.

For aminimal discrete surface, the first term on the right hand side is clearly 0.
For adiscretesurface with $H\neq 0$ , the variation $p(t)$ is volume preserving, so$\mathrm{C}\mathrm{M}\mathrm{C}$$\frac{\partial \mathrm{v}\mathrm{o}1(\mathcal{T})}{\partial t}=0\forall t\Rightarrow\sum_{p}\langle p’, \nabla_{p}V\rangle=0\forall t\Rightarrow\sum_{p}\langle p’, \nabla_{p}V\rangle+\langle p’, (\nabla_{p}V)’\rangle=0$

proving the lemma.

,$\square$Definition 4.1. A rninirnal or CMC discrete surface
any pe rmissible variation.

$\mathcal{T}$is stable if $a’(0)\geq 0$ for

We now consider avector-valued functionthat is defined on theinterior verticesof . We may extend this function to theboundary vertices of as well, by assuming
for each boundary vertex. The vectors
are the variation vector field of any boundary-fixing variation of
the form$n$$v_{p_{\mathrm{j}}}\in \mathbb{R}^{3}$$\mathcal{V}:nt=\{p_{1}, \ldots,p_{n}\}$$v_{p}=\overline{0}\in \mathbb{R}^{3}$$\mathcal{T}$$p$$\mathcal{T}$$v_{p_{\mathrm{j}}}$$p_{j}(t)=p_{j}+t\cdot v_{p_{\mathrm{j}}}+O(t^{2})$.

The fact that we have already restricted to boundary-fixing variations is no obstruction, as we will always consider only permissible variations. We define thevectorby$\overline{v}\in \mathbb{R}^{3n}$$v=(\triangleleft v_{p_{1}}^{t}, \ldots, v_{p_{n}}^{t})$.

(also considered as abilinear
We will now find asymmetric $3n\cross 3n$ matrix$a’(0)$
form), so thatis equal tofor any permissible variation with variation
vector field . We define$Q$$\vec{v}^{\ell}Q\overline{v}$$\overline{v}$$\tau(\overline{v}):=\sum_{p\in \mathcal{V}}\langle v_{p}, (\nabla_{p}a)’\rangle=\mathrm{I}\langle v_{p}, \frac{1}{2}\sum_{T=(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(p)}\overline{N}\cross(r’-q’)+\vec{N}’\cross(r-q)\rangle$and,$\mu(\overline{v}):=\sum_{p\in \mathcal{V}}\langle v_{p}, (\nabla_{p}V)’\rangle$

and sofor any permissible variation with variation vector
field . The purpose of the next two propositions is to find matrices
so
and
that
. Thus $Q=Q_{a}-HQv$ .
and$a’(0)=\tau(\overline{v})-H\mu(\overline{v})$$\vec{v}$$\tau(\vec{v})=\vec{v}^{t}Q_{a}\vec{v}$$Q_{a}$$\mu(\vec{v})=\vec{v}^{t}Q_{V}\overline{v}$$Q_{V}$141
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEX

Proposition 4.1. There is a syrnrnetric bilinear form represented by a $3n\cross 3n$
entry
grid with a
can be considered as an, wherematrix$pi,Pj\in
V:nt$
,
so
that
verticespair
eachinteriorofoffor$3\cross 3$$n\cross n$$Q_{a}$$Q_{a}$$Q_{a,p:p_{\mathrm{j}}}$$\mathcal{T}$$\tau(\vec{v})=\vec{v}^{t}Q_{a}\vec{v}$for the variation vector field of any permissible variation. The entry
if the vertices $p:,pj$ are not adjacent, and is$\vec{v}$$Q_{a,p:p_{\mathrm{j}}}$is 0$Q_{a,p:p_{j}}= \frac{1}{2}\sum_{(p:,p_{\mathrm{j}},r)\in star(\overline{p_{}p_{j}})}\frac{1}{|p_{\dot{*}}-p_{j}|^{2}}((p_{i}-p_{j})\cdot \mathcal{J}^{t}(p_{\dot{*}}-p_{j})$$-J(p_{*}.

for-pj)\cdot(p_{\dot{*}}-p_{j})^{t})-\cot\theta_{(p_{\dot{*}},p_{\mathrm{j}},r)}\vec{N}_{(p:,p_{j},r)}\cdot\vec{N}_{(p:,p_{j},r)}^{t}$and $pj$ adjacent and unequal, where
triangle $(p_{i},pj, r)$ at , and is$p_{i}$$\theta(p:,p_{\mathrm{j}},r)$is the interior angle of the$r$$Q_{a,p:p}.

\cdot=\frac{1}{4}\sum_{(p:,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(p:)}\frac{|r-q|^{2}}{|(p_{i},q,r)|}\vec{N}_{(p_{*\prime}q,r)}\vec{N}_{(p:,q,r)}^{t}$when the vertices are both equal to . Here,denotes the oriented unit
$(p,
q,
r)$
no rmal vector of the triangle(which we will subsequently abbreviate to ).
$\vec{N}(p,q,r)$$p_{i}$$\vec{N}$Proof. If

and are variation vector fields for any pair of permissible variations,
we can define abilinear form$\vec{w}$$\vec{v}$$Q_{a}( \vec{v},\vec{w}):=\frac{1}{2}\sum_{T=(p,q,r)\in \mathcal{T}}$$-\langle v_{p}\cross w_{r}-v_{r}\cross w_{p}+v_{q}\cross w_{p}-v_{p}\cross w_{q}+v_{r}\cross w_{q}-v_{q}\cross w_{r},\vec{N}\rangle+$$\frac{1}{2|T|}\langle v_{p}\cross(r-q)+v_{q}\cross$$(p-r)+v_{r}\cross(q-p)$ ,$w_{p}\cross(r-q)+w_{q}\cross(p-r)+w_{r}\cross(q-p)\rangle-$$\frac{1}{2|T|}\langle v_{p}\cross(r-q)+v_{q}\cross(p-r)+v_{r}\cross(q-p),\vec{N}\rangle$$\langle w_{p}\cross(r-q)+w_{q}\cross(p-r)+w_{r}\mathrm{x} (q-p),\vec{N}\rangle$.

.
,$\mathrm{i}\mathrm{t}\mathrm{f}\mathrm{o}\mathrm{l}1\mathrm{o}\mathrm{w}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{c}1\mathrm{e}\mathrm{a}’ \mathrm{b}\mathrm{i}1\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{t}\mathrm{h}\mathrm{e}1\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{o}\mathrm{f}Q_{a}\mathrm{U}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\vec{N}’=\frac{(q-p)\mathrm{x}(r’-p’)+(q’-p’)\cross(r-p)}{\tau(\overline{v})=Q_{a}(\vec{v},\vec{v}).Q_{a}\mathrm{i}\mathrm{s}2|T|}-\frac{\vec{N}}{2|T|,\mathrm{r}1\mathrm{y}}\langle(q-p)\cross(r’-p’)+(q’-p’)\cross(r-p),\vec{N}\rangle$. The first term is also symmetric in and ,
since
,
,
and
.
It only remains to determine an explicit form for. For agiven interior vertex, suppose and are nonzero only at , that is, thatand
. Thenare obviously symmetric in$\vec{v}$and$\vec{w}$$\overline{v}$$v_{p}\cross w_{r}-v_{r}\cross w_{p}=w_{p}\cross v_{r}-w_{r}\cross v_{p}$$\overline{w}$$v_{q}\cross w_{p}-v_{p}\cross w_{q}=w_{q}\cross v_{p}-w_{p}\cross v_{q}$$v_{r}\cross w_{q}-v_{q}\cross w_{r}=w_{\mathrm{r}}\cross v_{q}-w_{q}\cross v_{r}$$Q_{a}$$\vec{v}$$p$$\vec{w}$$v\triangleleft$$p$$\vec{w}^{t}=$$=(0^{t}, \ldots, 0^{t}, v_{p}^{t}, 0^{t}, \ldots, 0^{t})$$(0^{t}, \ldots, 0^{t}, w_{p}^{t}, 0^{t}, \ldots, 0^{t})$$Q_{a}( \vec{v},\vec{w})=Q_{a,pp}(v_{p},w_{p})=\frac{1}{4}$$\sum$$T=(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(p)$142
KONRAD POLTHIER AND WAYNE ROSSMAN$\frac{1}{|T|}\langle v_{p}\cross(r-q), w_{p}\cross(r-q)\rangle-\frac{1}{|T|}\langle v_{p}\cross(r-q),\overline{N}\rangle\langle w_{p}\cross(r-q),\overline{N}\rangle$$= \frac{1}{4}\sum_{T=(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(p)}$$\frac{1}{|T|}v_{p}^{t}(|r-q|^{2}I-(r-\mathrm{g})(\mathrm{r}-q)^{t}-((r-q)\cross\vec{N})((r-q)\cross\overline{N})^{t})w_{p}$$\sum$$= \frac{1}{4}$$\frac{|r-q|^{2}}{|T|}v_{p}^{t}(\vec{N}\vec{N}^{t})w_{\mathrm{P}}$,$T=(p,q,r)$ Estar(p)is of the form in the proposition.
hence
and
for
Now suppose
some given unequal interior vertices and . If and are not connected by some
, so assume that and are adjacent.
edge of the surface, then clearly Qa
Note that star(pq) then contains two triangles $(p, q,rj)$ for $j=1,2$ and precisely
one of them is properly oriented. Noting also that the normal vector of atriangle
changes sign when the orientation of the triangle is reversed, we have the following
equation:$Q_{a,pp}$$v\vec{t}=$$\overline{w}^{t}=(0^{t}, \ldots, 0^{t}, w_{q}^{t}, 0^{t}, \ldots, 0’)$$(0^{t}, \ldots, 0^{t}, v_{p}^{t}, 0^{t}, \ldots, 0^{t})$$p$$\mathrm{g}$

$p$

$q$

$(\mathrm{v},\overline{w})=0$

$p$

$q$

$\overline{N}$$Q_{a}( \overline{v},\overline{w})=Q_{a,pq}(v_{p}, w_{q})=\frac{1}{2}\sum_{T=(p,q,r_{k}),k=1,2}\langle v_{p}\cross w_{q},\vec{N}\rangle+$$\frac{1}{2|T|}\langle v_{p}\cross (r_{k} -q), w_{q}\cross(p-r_{k})\rangle-\frac{1}{2|T|}\langle v_{p}\mathrm{x}(r_{k} -q),\overline{N}\rangle\langle w_{q}\cross(p-r_{k}),\vec{N}\rangle=$$= \frac{1}{4}\sum_{k=1}^{2}\frac{1}{|T|}v_{p}^{t}$()$(p-r_{k})(q -r_{k})^{t}-(q-r_{k})(p-r_{k})^{t}-\langle p-rk, q-r_{k}\rangle\overline{N}\overline{N}^{t}$$w_{q}$.

For atriangle $(p,q, r)$ , one can check that$(p-r)(q-r)^{t}-(q-r)(p-r)^{t}=$$\frac{2|(p,q,r)|}{|p-q|^{2}}((p-q)(J(p-q))^{t}-J(p-q)(p-q)^{\ell})$so$Q_{a,pq}$,

is as in the proposition.

$\square$Proposition 4.2. There is a symmetric bilinear form represented by a $3n\cross 3n$, where $Qv$ has amatrixentry
for each pair of vertices , $pj\in V_{int}$of , so that$3\cross 3$$Q_{V}$$Q_{V,p:p_{j}}$$p:$$\mathcal{T}$$\mu(\overline{v})=vQ_{V}\triangleleft\overline{v}$

for the variation vector field of any permissible variation. We have $Q_{V,p:p:}=0$ ,$\overline{v}$and$Qv_{p:\mathrm{P}i},=0$

when the vertices :and $pj$ are not adjacent, and$p$$Q_{V,p:p_{\mathrm{j}}}= \frac{1}{6}$$(\begin{array}{lll}0 \mathrm{r}_{2,3}-r_{1,3} r_{1,2}-r_{2,2}\gamma_{1_{\prime}3}-r_{2,3} 0 r_{2,1}-r_{1,1}r_{2,2}-r_{1,2} r_{1,1}-r_{2,1} 0\end{array})$

for adjacent unequal :and , where $(p:,pj, rk)$ are the two triangles in star(pipj)$=(r_{k,1},$
and
for $k=1,2$ , and $(p:,pj,r2)$ is properly oriented and$p$$r_{k}$$\mathrm{r}\mathrm{i},2\mathrm{r}2,3$$(p:,pj,r1)$ is not.

$pj$

143
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEXProof.
$=$$\sum_{p\in \mathcal{V}}\langle p’, (\nabla_{p}V)’\rangle$$=$$\sum_{p\in \mathcal{V}_{nt}}\langle v_{p}, \frac{1}{6}\sum_{(p,q,r)\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(p)}(q\cross r)’\rangle$$\frac{1}{6}\sum_{p\in \mathcal{V}_{nt}}$($\sum_{\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}}$$p,q\neq p\langle v_{p}\cross v_{q},r_{2}-r_{1}\rangle$

to),where $(p, q, r_{2})$ is the properly oriented triangle in star(pg), and $(p, q, r_{1})$ is thenon-properly oriented triangle in star(pg). Thus we have$\mu(\vec{v})=\sum_{p\in \mathcal{V}_{*nt}}.(\sum_{q\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{t}\mathrm{o}p,q\neq p}v_{p}^{t}(Q_{V,pq})v_{q})$

where
is a
the fact that
$Q_{V,pq}$,

matrix defined as in the proposition. Thus $Q_{V,pp}=0$ , andis skew-symmetric in and implies $Qv$ is symmetric.

$3\cross 3$$Q_{V,pq}$$p$Corollary 4.1. If a discrete
is stable.

$\mathrm{C}\mathrm{M}\mathrm{C}$surface$\square$$q$$\mathcal{T}$has only one interior vertex, then it. Then $Q_{a}=Q_{a,pp}$
Proof. Denote the single interior vertex by , so star(p)and $Q_{V}=Q_{V,pp}$ arematrices. By Propositions 4.1 and 4.2, $Q_{V}=0$ and forany vector
at we have$=\mathcal{T}$$p$$3\cross 3$$u_{p}\in \mathbb{R}^{3}$$p$,$u_{p}^{t}Q_{a}u_{p}= \frac{1}{4}\sum_{(p,q,r)\in \mathcal{T}}\frac{|r-q|^{2}}{|(p,q,r)|}u_{p}^{t}\vec{N}\vec{N}^{t}u_{p}=\frac{1}{4}\sum_{(p,q,r)\in \mathcal{T}}\frac{|r-q|^{2}}{|(p,q,r)|}\langle u_{p},\vec{N}\rangle^{2}\geq 0$

so $a’(0)\geq 0$ for all permissible variations.

$\square$

5. JACOBI OPERATOR FOR SMOOTH$\mathrm{C}\mathrm{M}\mathrm{C}$

SURFACES

We now are able to begin the study of the spectra of the second variation of
discretesurfaces, as the second variation is now in an explicit form. However,
we postpone this to the next section, in order to discuss the spectra of the second
variation of smoothsurfaces here. We digress to the smooth case for latercomparison with the discrete case (section 7). In particular, here we explicitly
determine the eigenvalues and eigenfunctions of the Jacobi operator for portions ofsmooth catenoids.
Let:
be an immersion of acompact 2-dimensional surface $M$ . Let
be aunit normal vector field on $\Phi(M)$ (we write
simply as
defined on$M)$ . Let
be asmooth variation of immersions forso that $\Phi(0)=\Phi$andfor all. Let
be the variation vector field on. We can assume, by reparametrizing $(t) for nonzero if necessary, that thecorresponding variation vector field at $t=0$ is $\vec{E}(0)=u\vec{N}$ ,. Let $a(t)$be the area of $\Phi(t)(M)$ and $H$ be the mean curvature of$(M). The first variational
formula is$\mathrm{C}\mathrm{M}\mathrm{C}$$\mathrm{C}\mathrm{M}\mathrm{C}$$Marrow \mathbb{R}^{3}$$\Phi$$\vec{N}$$\Phi^{*}\vec{N}$$\Phi(t)$$t\in(-\epsilon, \epsilon)$$\Phi(t)|_{\partial M}=\Phi(0)|_{\partial M}$$\Phi(t)$$\vec{N}$$t\in(-\epsilon, \epsilon)$$\vec{E}(t)$$t$$u\in C_{0}^{\infty}(\mathrm{A}\mathrm{t})$$a’(0):= \frac{d}{dt}a(t)|_{t=0}=-\int_{M}\langle nH\vec{N}, u\vec{N}\rangle dA$,

where , and dA are the metric and area form on $M$ induced by the immersion. We now assume $H$ is constant, so $a’(0)=-nH \int_{M}udA$ . Let $V(t)$ be thevolume of $\Phi(t)(M)$ , then $V’(0)= \int_{M}udA$ . The variation is volume preserving if$\langle$$\Phi$$\rangle$

144
KONRAD POLTHIER AND WAYNE ROSSMAN. In particular, $\int_{M}udA=0$ when $t=0$ ,for all
md $(M) is critical for area amongst all volume preserving variations.
soisThe second variation formula for volume preserving variations$t\in(-\epsilon, \epsilon)$$\int_{M}\langle\overline{E}(t),\overline{N}(t)\rangle dA(t)=0$$\mathrm{a}’(0)=0$$\Phi(t)$$a’(0):= \frac{d^{2}}{dt^{2}}a(t)|_{t=0}=\int_{M}\{|\nabla u|^{2}-(4H^{2}-2K)u^{2}\}dA=\int_{M}$where $K$ is the Gaussian curvature on $M$ induced by$\Phi$uLudA,, and$L=-\triangle-4H^{2}+2K$

operator .
is the Jacobi operator with Laplace-Bel
surface can be defined:
There are two ways that the index of asmooth
maximum possible dimenis
the
index
for
definition
naturalThe geometricallyfor whichsion of asubspace of volume preserving variation functions$\mathrm{a}’(0)<0$ for all nonzero $u\in S$ , which we call Ind(M). (We are identifying $\Phi(M)$with $M$ so that we can write simply Ind(M), rather than Ind( (At)).)
The analytically natural definition for index is the number of negative eigenvaluesof the operator , which equals the maximum possible dimension of asubspace $Su$
for whichof (not necessarily volume preserving) variation functions$u\in
S_{U}$
thewhereInd(/(M),. We call this index
uLudA $<0$ for all nonzero
subscript stands for Unconstrained index”.
. It is also not difficult to see that lndu(M)-lClearly, indu(M)Ind(M) [9]. As it is geometrically more natural, we want to compute Ind(M). But
lndu(M) is more accessible to computation than Ind(M), and since they differ byat most 1, computing Jndu{M) means that we know Ind(M) is either lndu(M) or$\triangle$$\mathrm{t}$$\mathrm{C}\mathrm{M}\mathrm{C}$$u\in C_{0}^{\infty}(M)$$S$$\Phi$$L$$u\in C_{0}^{\infty}(\mathrm{A}\mathrm{t})$$\int_{M}$

$u$

$U$

$\leq$$\geq \mathrm{I}\mathrm{n}\mathrm{d}(M)$$\mathrm{I}\mathrm{n}\mathrm{d}_{U}(M)-1$.

In the case that we are considering minimal surfaces, as in section 7the volume.
constraint is not necessary, and hence Ind(M)$=\mathrm{I}\mathrm{n}\mathrm{d}u(M)$5.1. Eigenvectors of L for Rectangles. Consider the rectangle
A# $=\{(x,$ y,$0)\in \mathbb{R}^{3}|0\leq x\leq x_{0},0\leq y\leq y_{0}\}$as asmooth minimal immersion (inclusion map) into , and consider functions on
it with Dirichlet boundary conditions. In this case, $L=-\triangle$ , and its eigenvalues
and eigenfunctions are$\mathbb{R}^{3}$$\lambda_{m,n}=\frac{m^{2}\pi^{2}}{x_{0}^{2}}+\frac{n^{2}\pi^{2}}{y_{0}^{2}}$for$(m, n)\in \mathbb{Z}^{+}\cross \mathbb{Z}^{+}$,$\phi_{m,n}=\frac{2}{\sqrt{x_{0}y_{0}}}\mathrm{s}.\mathrm{n}\frac{m\pi x}{x_{0}}\mathrm{s}.\mathrm{n}\frac{n\pi y}{y_{0}}$. hence Ind(M)

$=$

$0$

.

5.2. Eigenvectors of L for Catenoids. We can consider the catenoid as amap
: $(x, y)\in \mathcal{R}arrow(\cos x\cosh y, \sin x\cosh y,y)\in \mathbb{R}^{3}$ ,$\Phi$

where$\mathcal{R}=\{(x,y)\in \mathbb{R}^{2}|0\leq x\leq 2\pi, y\mathrm{o}\leq y\leq y_{1}\}$,and the left and right boundary segments of 7? are identified with each other. This isoperator, and Gauss curvatureaconformal map, and the metric, Laplace-Belare$\mathrm{t}$$ds^{2}=\cosh^{2}y\cdot(dx^{2}+dy^{2})$,$\triangle=\frac{\frac{\partial^{2}}{\partial^{2}x}\dagger\overline{\partial}^{\mathrm{T}}\overline{y}\partial^{2}}{\cosh^{2}(y)}$,$K=-\cosh^{-4}y$ .

145
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND THEIR INDEXFIGURE 9. The function $f(y)$ (computed numerically) in the eigenfunction $\cos(mx)f(y)=f(y)$ when $m=0$ for the catenoidwith $y_{1}=-y_{0}=1.91$ . The corresponding eigenvalue is,and all other eigenvalues are positive. We show this function forlater comparison with the first eigenvector field of the second variation of adiscrete catenoid in section 7.
$\Phi(\mathcal{R})$$\lambda\approx-0.54$

We put Dirichlet boundary conditions on the upper and lower boundary segmentsof 7?.
Lemma 5.1. Anbasis of eigenfienctions of the Jacobi operator $L=-\triangle+2K$ of
can be chosen so the eigenfunctions are of the form $\sin(mx)f(y)$ or $\cos(mx)f(y)$ ,$L^{2}$$\Phi$.
for
Proof It is well known that$m\in \mathrm{N}\cup\{0\}$, with respect to the Dirichlet boundary condition, has
adiscrete spectrum in , and that, for all,
is afinite dimensional
space of smooth functions. Furthermore, an orthonormal basis of thespace over7? (with respect to) can be obtained as aset of smooth eigenfunctions ofsatisfying the Dirichlet boundary condition.
Define the operator. Then $DL=LD$ , so.
For functions and that are -periodic in we have$L$$\mathbb{R}$$\lambda\in \mathbb{R}$$\mathrm{k}\mathrm{e}\mathrm{r}(L-\lambda)$$L^{2}$$ds^{2}$$L$$D=i \frac{\partial}{\partial x}$$u$$D:\mathrm{k}\mathrm{e}\mathrm{r}(L-\lambda)arrow \mathrm{k}\mathrm{e}\mathrm{r}(L-\lambda)$$2\pi$$v$$x$

ydxdy $=0$ ,$\langle\frac{\partial}{\partial x}u, v\rangle_{L^{2}}+\langle u, \frac{\partial}{\partial x}v\rangle_{L^{2}}=\int_{R}(u_{x}\overline{v}+u\overline{v}_{x})\cosh^{2}$

which implies that the operator
is skew symmetric. Therefore $D$ is symmetric.
$D$ is elliptic, so
it has abasis of eigenfunctions in each finite dimensional space. So we can choose aset of functions that is simultaneously anbasis
of eigenfunctions for both $D$ and . Since the eigenfunctions of $D$ must be of theform $e^{mxi}f(y)$ with, the lemma follows.
$\frac{\partial}{\partial x}$$L^{2}$$\mathrm{k}\mathrm{e}\mathrm{r}(L-\lambda)$$L$$m\in \mathbb{Z}$$\square$Now note that an eigenfunction $\sin(mx)f(y)$ of$L(\sin(mx)f(y))$$L$satisfies$=$

A $\sin(mx)f(y)$$=$$\frac{m^{2}\sin(mx)f(y)}{\cosh^{2}y}-\frac{\sin(mx)f_{yy}(y)}{\cosh^{2}y}-\frac{2\sin(mx)f(y)}{\cosh^{4}y}$,and asimilar computation holds for an eigenfunction $\cos(mx)f(y)$ . It follows that$f_{yy}=$ $(m^{2}-\lambda \cosh^{2}y-2\cosh^{-2}y)f$,

and finding the eigenvalues Aamounts to finding solutions of this equation thatsatisfy the boundary conditions $f(y_{0})=f(y_{1})=0$ . Thus we know all of the eigenfunctions, up to solutions of adetermined-order ordinary differential equation$2\mathrm{n}\mathrm{d}$146
KONRAD POLTHIER AND WAYNE ROSSMAN

6. JACOBI OPERATOR FOR DISCRETE

SURFACES$\mathrm{C}\mathrm{M}\mathrm{C}$

Since we know the second variation matrix explicitly (section 4), we are now
able to find the “discrete Jacobi operator” for compact discretesurfaces ,analogous to in the smooth case (section 5). We first convert variation vector
fields into piece-wise linear continuous functions, in order to naturally describe thenorm, we then find thenorm on the space of variation vector fields. With thiscorrect matrix for the discrete Jacobi operator, and this matrix has the eigenvalues
and eigenfunctions of the second variation of .
Consider apermissible variation$Q$$\mathcal{T}$$\mathrm{C}\mathrm{M}\mathrm{C}$$L$$L^{2}$$L^{2}$$\mathcal{T}$,$p_{j}(t)=p_{j}+t\cdot v_{p_{\mathrm{i}}}+\mathcal{O}(t^{2})$where the vector-valued function
defined on the interior verticesof (if is aboundary vertex) comprises its variation vector, as defined in section 4.
field
There is anatural way to extend the function to acontinuous piece-wise linear-valued function defined at every point of . In order to define , we first
define aset of piece-wise linear continuous head functions:$v_{p_{j}}\in \mathbb{R}^{3}$$\{p_{1}, \ldots,p_{n}\}$$v_{p}=\vec{0}$$\mathcal{T}$$\mathcal{V}_{int}=$

$n$

$p$

$v\vec{\ell}_{=}(v_{p_{1}}^{t}, \ldots, v_{p_{\mathfrak{n}}}^{t})$$\vec{v}$$\mathbb{R}^{3}$$\mathcal{T}$$v$$\mathrm{u}$, letDefinition 6.1. For
be the head function onwhich is 1atand is 0at all other vertices of T. We then extend
to every point of (in theunique way) so that it is linear on each edge and each face of .
$p\in \mathcal{V}:nt$$\mathcal{T}$$\psi_{p}$$p$$\mathcal{T}$$\psi_{p}$$\mathcal{T}$There is ahead function for eachsupport ofis star(pj).

$pj\in \mathcal{V}_{\dot{|}nt}$, hence there are$n$of them, and the$\psi_{\mathrm{P}\mathrm{j}}$Definition 6.2. We define$v$

associated to$\overline{v}$by$v|_{T}=v_{p}\psi_{p}+v_{q}\psi_{q}+v_{r}\psi_{f}$,for all triangles $T=(p, q, r)$ in .
$\mathcal{T}$The function has the following four properties:
1. is continuous,
2. is linear on each triangle,,
3. is 0on
4. is the variation vector field for thesurface variation induced by the
associated vertex variation.
We will consider the
to be the -valued coefficients of with respect to thebasis of functions. And, as theform abasis for all functions
with the above properties, they are abasis (with scalars in) for the following$3n$ -dimensional function space:
$v$

$v$

$T\subset \mathcal{T}$$v$$\partial \mathcal{T}$$v$$C^{0}$$v$$p_{j}(t)$$\mathbb{R}^{3}$$v$$v_{\mathrm{P}i}$$\{\psi_{p_{1}}, \ldots, \psi_{p_{n}}\}$$v$$\psi_{p_{j}}$$\mathbb{R}^{3}$

Definition 6.3. Define$S_{h}:=${$S_{h}$of the discrete surface,$v:\mathcal{T}arrow \mathbb{R}^{3}|v\in C^{0}(\mathcal{T})$$v$$\mathcal{T}$to beis linear on each$T\in \mathcal{T}$and$v|\partial \mathcal{T}=0$}.

We have named this space , in keeping with the notational conventions of thetheory of finite elements. Note that the component functions of any function $v\in S_{h}$all have bounded Sobolev
norm.
Now we can find an explicit form for the
inner product onwith respect
to the basis:$S_{h}$$H^{1}$$L^{2}$$\{\psi_{p_{1}}, \ldots, \psi_{\mathrm{P}n}\}$$S_{h}$

147

FIGURE 10. The eigenvectors of the discrete square with $n=15$
associated to the eigenvalues A392, A393,, and. Note thatthese eigenvectors closely resemble the eigenfunctions $\sin(x)\sin(y)$ ,$\sin(x)\sin(2y)-\sin(2x)\sin(y)$ , $\sin(x)\sin(2y)+\sin(2x)\sin(y)$ , and$\sin(2x)\sin(2y)$ from the smooth case.
$\lambda_{394}$$\lambda_{395}$

Proposition 6.1. There is a positive definite $3n\cross 3n$ matrix.

$S=(\langle\psi_{p:}, \psi_{p_{\mathrm{j}}}\rangle_{L^{2}}I_{3\cross 3})_{*,j=1}^{n}$so that$\langle u, v\rangle_{L^{2}}=\vec{u}^{t}S\vec{v}$for all, $v\in S_{h}$ with associated vectors. The matrix consists ofblocksin angrid, with the diagonal (resp. nondiagonal) blocks eachbeing multiples of theidentity matrix,$S$$\vec{u},\vec{v}\in \mathbb{R}^{3n}$$u$$n\cross n$$S_{p:p_{j}}$$3\cross 3$$S_{p\mathrm{j}p_{j}}=( \sum_{T\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(p_{j})}\frac{|T|}{6})I_{3\cross 3}$when$p_{i}$

and$p_{j}$

resp.

,$S_{p.p_{\mathrm{j}}}.=( \sum_{T\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(\overline{pp_{j}})}\frac{|T|}{12})I_{3\cross 3}$

are adjacent, and $S_{p.p_{j}}.=0$ when :and $pj$ are not adjacent.
$p$

Proof We first note that$|v|_{L^{2}}^{2}:= \int_{\mathcal{T}}\langle v, v\rangle dA=\sum_{T\subset \mathcal{T}}\int_{T}\langle v|_{T}, v|\tau\rangle dA$

Acomputation yields that for each triangle$\int_{T}\psi_{p}^{2}dA=\frac{|T|}{6}$,$T\subset \mathcal{T}$,$\int_{T}\psi_{p}\psi_{q}dA=\frac{|T|}{12}$,.

$3\cross$

$3$

148

FIGURE 11. The eigenvectors of the discrete square,,with
15 associated to the eigenvalues, and.
Note that these eigenvectors closely re$\sin(x)\sin(3y)$ ,semble the eigenfunctions $\sin(3x)\sin(y)$$\sin(3x)\sin(y)+\sin(x)\sin(3y)$ , $\sin(3x)\sin(2y)-\sin(2x)\sin(3y)$ ,and $-\sin(3x)\sin(2y)-\sin(2x)\sin(3y)$ fiom the smooth case.
$\lambda_{396}$

$=$

$n$

$\lambda_{397}$$\lambda_{399}$$\lambda_{398}$–

for any vertices

$p$

and

$q$

of . Thus$T$$|v|_{L^{2}}^{2}= \sum_{T=(p,q,r)\in \mathcal{T}}\frac{|T|}{6}\{|v_{p}|^{2}+|v_{q}|^{2}+|v_{r}|^{2}+\langle v_{p}, v_{q}\rangle+\langle v_{p}, v_{r}\rangle+\langle v_{q}, v_{r}\rangle\}$.

Hence, for any two functions $u,v\in S_{h}$ , we have$\langle u,v\rangle_{L^{2}}=$$\sum_{T=(p,q,r\rangle\in \mathcal{T}}\frac{|T|}{12}\{\langle u_{p}+u_{q}+u_{r},v_{p}+v_{q}+v_{r}\rangle+\langle u_{p},v_{p}\rangle+\langle u_{q}, v_{q}\rangle+\langle u_{r},v_{r}\rangle\}$$= \sum_{p_{j}\in \mathcal{V}_{\mathfrak{n}t}}(\langle u_{p_{\mathrm{j}}},v_{p}\rangle(\sum_{T\in\epsilon \mathrm{t}\mathrm{a}\mathrm{r}(p_{j})}\frac{|T|}{6})+$$p_{\mathrm{j}} \langle u_{p_{\mathrm{j}}}, v_{p:}\rangle(\sum_{T\in \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}(\overline{p_{}p_{j}})}\frac{|T|}{12}))$$p: \in \mathcal{V}:nt\sum_{\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}}$toHence the$3\cross 3$

blocks$S_{p:p_{\mathrm{j}}}$are as in the proposition.

149
DISCRETE CONSTANT MEAN CURVATU RE SURFACES AND THEIR INDEX$\mathrm{f}$

$r$

$\

$tr$

$\backslash \cdot‘\backslash [searrow]_{\backslash }^{[searrow]\backslash }\backslash /_{\hat{\gamma}_{\mathrm{k}}^{r}\ovalbox{\tt\small REJECT}-\nearrow_{\int_{\wedge}’}^{f}}’\sim[searrow]\nearrow.-\backslash \mathrm{I}f1\prime_{\backslash }\backslash \backslash \ovalbox{\tt\small REJECT}^{\nwarrow}\mathit{1}\iota_{\mathit{1}}^{\mathrm{t}\backslash ^{\backslash }}\sim\bigvee_{\sqrt}’\swarrow_{\backslash }^{\swarrow_{\sqrt[\prime]{}}}$$\backslash r_{A}\backslash$

”

.

-4

FIGURE 12. The eigenvector associated to the eigenvalue-0.542 of adiscrete catenoid. Here we have also shown thisvector field on ahorizontal planar grid (where each
-vector isvertical with length equal to that of the corresponding -vector
in the-eigenvector field on the discrete catenoid), so that we
can compare it directly with the eigenfunction shown in Figure
9for the smooth case. Notice that the plot on the right closely
resembles the curve in Figure 9.
$\lambda_{0}\approx$$\mathbb{R}^{3n_{-}}$$\mathbb{R}^{3}$$\mathbb{R}^{3}$$\mathbb{R}^{3n}$We now compute the discrete Jacobi operator: $S_{h}arrow S_{h}$ associated to thesecond variation formula for the surface, i.e.
for all $v\in S_{h}$(recall that $Q=Q_{a}-HQv$ ). We need the property
so that we canconsider the eigenvalue problem for. And we also wish
to be linear andsymmetric (for all , $v\in S_{h}$ ). With these properties, thechoice ofis canonical:$L_{h}$$\int_{\mathcal{T}}v^{t}L_{h}vdA=\vec{v}^{t}Q\vec{v}$$L_{h}(S_{h})\subset S_{h}$$L_{h}$$\int_{\mathcal{T}}u^{t}L_{h}v=\int_{\mathcal{T}}v^{t}L_{h}u$$L_{h}$$u$$L_{h}$

Proposition 6.2. There exists a unique linear operator
is symmetric in and and$\int_{\mathcal{T}}u^{t}L_{h}vdA$$u$$L_{h}$: $S_{h}arrow S_{h}$ so that$v$$\int_{\mathcal{T}}v^{t}L_{h}vdA=vQ\triangleleft\vec{v}$for all$\vec{v}_{f}$. Furthermore, if
is the function in$v\in S_{h}$then$L_{h}v$$v$$S_{h}$

is the function inassociated to the$S^{-1}Qvrightarrow$

Proof For

$v= \sum_{j=1}^{n}v_{p_{j}}\psi_{p_{j}}$$S_{h}$

associated to the-vector$S_{h}$-vector.

, we define$L_{h}v:= \sum_{i,j,k=1}^{n}(S^{-1})_{p:p_{k}}((Q_{a,p_{k}p_{j}}-HQ_{V,p_{k}p_{j}})v_{p_{j}})\psi_{p:}$which is the function inand$\mathbb{R}^{3n}$$\mathbb{R}^{3n}$associated to$S^{-1}Q\overline{v}$. This map$L_{h}$,

is clearly linear,$\int_{\mathcal{T}}u^{t}L_{h}vdA=\langle u, L_{h}v\rangle_{L^{2}}=\overline{u}S(S^{-1}Q\vec{v})=\vec{u}^{t}Q\vec{v}$for all , $v\in S_{h}$ . Hence, since
$u$

$v$

$Q$

is symmetric,$\int_{\mathcal{T}}u^{t}L_{h}vdA$

is symmetric in$u$.

Uniqueness of$L_{h}$

with the above properties follows from the following:$\int_{\mathcal{T}}u^{t}L_{h}vdA=\frac{1}{2}(\int_{\mathcal{T}}(u+v)^{t}L_{h}(u+v)dA-\int_{\mathcal{T}}u^{t}L_{h}udA-\int_{\mathcal{T}}v^{t}L_{h}vdA)$

and

150

,.., .’–$\sim\backslash \backslash -$–—–$1_{-\neg-\#^{l}}^{--}--_{l}’$.

$,\sim$.

$”\backslash \backslash _{\mathrm{R}}\grave{\backslash }\backslash _{r}[searrow]^{\backslash _{\mathrm{t}}}\backslash \backslash 1\backslash \backslash /,’//’///\iota\acute{\prime}/’\nearrow$$\mathrm{h}$$\mapsto-\backslash \neg-\nu^{f}$$\backslash \downarrow \mathrm{t}$$-_{-arrow},t$

, ” ,

,

$j$

$\backslash \cdot$$*\cdot---arrow’\backslash$$r$$\backslash ^{\backslash }\backslash$$\sim_{---d}\backslash$$\nearrow_{\sim}^{*---\sim}\prime^{\phi}----\mathrm{c}’\backslash ’\sim\wedge$$\ovalbox{\tt\small REJECT}_{r^{f}}^{/1}\backslash$$\backslash \backslash \backslash \backslash ^{\backslash _{\backslash }}\sim\backslash \backslash \backslash \backslash \backslash \backslash$$\wedge$

FIGURE 13. TwO-thirds of the eigenvectors axe approximately tangential to the surface. For example, here we show the
eigenvector fields associated to the eigenvalues Ai,, and A3
(whose values are just slightly greater than 0).

$\mathbb{R}^{3n_{-}}$$\lambda_{2}$$= \frac{1}{2}((\overline{u}+\overline{v})^{t}Q(\overline{u}+\vec{v})-\overline{u}^{t}Q\overline{u}-vQ\prec\overline{v})$Henceis uniquely determined for all $u\in S_{h}$ , so
mined for each $v\in S_{h}$ .
$\int_{\mathcal{T}}u^{t}L_{h}vdA$.
$L_{h}v$

is uniquely deter$\square$

So the spectrum of the second variation of is the set of eigenvalues of $S^{-1}Q$ .
One can check that $S^{-1}Q$ is self-adjoint with respect to theinner product on, thus all the eigenvalues of $S^{-1}Q$ are real.
$\mathcal{T}$$L^{2}$$S_{h}$

Remark 6.1. Another way to see that $S^{-1}Q$ is the correct discrete Jacobi operator
is to consider the Rayleigh quotient$\frac{\tilde{v}^{t}Q\overline{v}}{\langle v,v\rangle_{L^{2}}}=\frac{\overline{v}^{t}S(S^{-1}Q\overline{v})}{\vec{v}^{t}S\overline{v}}$.

Using the standard procedure for producing eigenvalues from the Rayleigh quotientin this case would produce the eigenvalues of $S^{-1}Q$ .

7. APPROXIMATING SPECTRA OF SMOOTH CMC SURFACES
We can now implement the procedure described in the second half of the intr0duction, since we know $S^{-1}Q$ explicitly.
If asequence of compact
discrete surfaces {converges (in the Sobolev
norm as graphs over the limiting surface) to asmooth compact
surface:, then standard estimates from the theory of finite elements (see, forexample, [3] or [7] imply that the eigenvalues and eigenvectors (piece-wise linearlyextended to functions) of the operatorsof theconverge to the eigenvalues
and eigenfunctions of the Jacobi operator of (convergence is in thenorm
for the eigenfunctions).
For the first two examples here -aplanar square and rotationally symmetricportion of acatenoid –we know the approximating discrete minimal surfaces exactly, and we know the eigenvalues and eigenfunctions of for the smooth minimal
surfaces exactly, so we can check that convergence of the eigenvalues and eigenfunctions does indeed occur.
In the final example -asymmetric portion of atrinoid –the spectrum of thesmooth minimal surface is unknown, so we see estimates for the eigenvalues andeigenfunctions for the first time. Our experiments confirm the known value 3for theindex of this unstable surface, and additionally show us the directions of variation$\mathrm{C}\mathrm{M}\mathrm{C}$$\eta_{\dot{\iota}=1}^{\infty}$$H^{1}$$\Phi$$\mathrm{C}\mathrm{M}\mathrm{C}$$Marrow \mathbb{R}^{3}$$)$$L_{h}$$L$$\mathcal{T}_{j}$$L^{2}$$\Phi$$L$$\mathrm{s}$151’$\mathrm{R};_{/^{0_{\tilde{\mathrm{w}}}[searrow] \mathit{1}_{-}}}\wedge\iota_{\wedge}^{-1}\mathrm{d}_{\mathrm{k}}..-\grave{]_{\backslash }}0_{\backslash \backslash ^{\mathrm{r}\mathrm{e}}}\ovalbox{\tt\small REJECT}\Leftrightarrow\backslash \overline{/}\mathrm{i}r_{\mathrm{x}_{\backslash }}^{\acute{i}}(\backslash ^{\backslash ^{\mathrm{v}_{X}}}J^{-}/J\grave{\wedge^{\mathrm{m}}}\cdot$}$\mathrm{r}_{r/^{\mathrm{v}}w_{\mathrm{q}}\mathrm{i}_{\}_{1}}1_{\vee},\backslash \backslash \backslash \backslash }rl_{*\mathit{1}_{\grave{c}_{\mathrm{f}\}}}\mathfrak{l}.1\backslash \mathrm{c}}"’\grave{\nwarrow}" J_{\backslash }r\acute{\downarrow}\backslash \cdot$$/^{l}\cdot/_{\nearrow\nearrow \mathrm{f}1\nwarrow\backslash }f\dagger.1\backslash \tau_{\zeta\iota^{\iota\infty}}h\prime \mathrm{b}_{/}-_{f}\}^{1\#}-arrow\prime \mathit{4}\backslash \backslash arrow\vee$$\nwarrow*J_{\alpha_{r_{\mathrm{i}}}}/l-/,\mathit{4}.l1.,\iota_{\Lambda_{\mathrm{x}\sqrt}^{P_{*}}}{}_{1_{*\mathrm{t}}}\mathrm{C}t\delta_{\iota}[searrow]^{\mathrm{k}_{\mathrm{r}}\backslash }\mathrm{I}1\mathrm{I}\backslash \backslash J^{-y’}\theta \mathrm{A}\backslash \mathrm{t}\uparrow\grave{\mathrm{e}}_{\mathrm{i}[searrow]\iota_{!^{l_{\backslash }}}}\tilde{\tau}^{\backslash }*\mathrm{e}\backslash \backslash ^{\mathrm{s}^{\backslash }\backslash }\hat{\{}[_{\downarrow l}^{\overline{\{}}\nwarrow\nwarrow$$\downarrow \mathrm{k}^{\mathrm{X}_{1\grave{\mathrm{J}}}^{s\backslash \acute{\mathrm{Y}}^{\grave{\backslash }}l^{\dot{d}_{*}}\}\psi}}\underline{\dot{\mathrm{s}}}^{\backslash }1^{\mathrm{I}}\mathrm{t}\{y\mathrm{c}_{-\vee}-l\acute{X}\wedge$$\}\mathrm{t}^{\mathrm{A}_{\downarrow r_{\wedge}}^{\gamma_{1}^{\tilde{[searrow]}i_{\backslash }}\backslash }}-[searrow]-\backslash ^{\underline{\backslash -r_{r^{\gamma_{\backslash }^{\mathrm{v}_{\mathrm{e}}}}J_{\vee}^{4}arrow}}^{\prime_{\backslash }}}[searrow]\uparrow-\overline{\mathrm{e}}_{\mathrm{R}_{\backslash }^{\backslash }}^{*g_{1\acute{\swarrow}l}}$$\backslash _{11\mathfrak{j}}\backslash \backslash \nwarrow\backslash \mathrm{C}\uparrow\{\prime r^{r_{J’}}|^{\prime \mathrm{x}}l^{\gamma 7^{j}}$$-1-\mathfrak{s}$

$r$

’

,

$/$

FIGURE 14. One-third of the eigenvectors are approximately perpendicular to the surface. For example, here we show the
eigenvector fields associated to the eigenvalues A147,
,
,
and
.

$\mathbb{R}^{3n_{-}}$$\lambda_{171}$$\lambda_{204}$$\lambda_{210}$

that reduce area. Thus we have approximations for maximal spaces of variation
vector fields on the smooth minimal surfaces for which the associated variations
reduce area. (For the approximating discrete surfaces in this example, we do nothave an explicit form; however, the theory of finite elements applies and we canstill expect convergence of the eigenvalues and eigenfunctions (innorm), if wechoose the discrete aproximations so that they converge (innorm) to the smoothminimal surfaces.)$L^{2}$$H^{1}$

7.1. The flat minimal square. Considering the square $M=\{0\leq x\leq\pi$ , $0\leq y\leq$
included inas asmooth minimal surface, the eigenvalues and eigenfunctions$\mu_{m,n}=m^{2}+n^{2}$of areand $\phi_{m,n}=\frac{2}{\pi}\sin(mx)\sin(ny)$ for $m$ ,(section 5).
Now we consider the discrete minimal surface that is $M$ with aregular squaregrid. In each subsquare of dimension, we draw an edge from the lower
left corner to the upper right corner, producing adiscrete minimal surface withcongruent triangles with angles , , and .
For this , $S^{-1}Q$ has no negative eigenvalues, as expected, since the smoothminimal square is stable. However, we must take tangential motions into account in
the discrete case, and we find that (when writing the eigenvalues in increasing order)
the first tw0-thirds of the eigenvalues are 0and their associated eigenvectors are
entirely tangent to the surface. The final one-third of the eigenvalues are positive,
with eigenvectors that are exactly perpendicular to the surface. Examples of theseperpendicular vector fields are shown in Figures 10 and 11 for $n=15$ . (There are
196 interior vertices, and so there are 588 eigenvalues
of $S^{-1}Q$ andandwhen $j\in[392,587].)$ The
vectors shown in these figures
and their eigenvalues are close to those of the smooth operator of $M$ . We have$\lambda_{392}=2.022$, $\lambda_{393}=5.094$, A394 $=5.148$,,$\lambda_{396}=10.434$, $\lambda_{397}=10.445$, $\lambda_{398}=13.649$, A393 $=14.12$.
$\mathbb{R}^{3}$$\pi\}$$L$$n\in \mathbb{Z}^{+}$$\mathcal{T}$$n\cross n$$\frac{\pi}{n}\cross\frac{\pi}{n}$$2n^{2}$$\frac{\pi}{4}$$\frac{\pi}{4}$$\frac{\pi}{2}$$\mathcal{T}$$\lambda_{\mathrm{O}}=\ldots=$$\lambda j$$\lambda_{391}=0$$\lambda_{j}>0$$\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{n}\backslash$$L$$\approx\mu_{1,1}$$\approx\mu 1,3$$\approx\mu_{1,2}$$\approx\mu 3,1$$\approx\mu_{2,1}$$A\mathit{3}\mathit{9}\mathit{5}=8.347$$\approx/\mathrm{i}2,3$$\approx\mu_{2,2}$$\approx$$\mu_{3,2}$

7.2. Discrete Minimal Catenoids. By Corollary 3.2, we know that the discrete
minimal catenoids converge to smooth catenoids as the meshes are made finer.
Hence the eigenvalues and eigenvectors of the discrete catenoids converge to theeigenvalues and eigenfunctions of the smooth catenoid. For the discrete catenoids
with relatively fine meshes, we find that tw0-thirds of the eigenvectors are approximately tangent to the surface, and the remaining ones are approximately perpendicular. The approximately perpendicular ones (considered as functions which are
multiplied by unit normal vectors) and their eigenvalues converge to the eigenfunctions and eigenvalues of the smooth catenoid (computed in section 5)

152
KONRAD POLTHIER AND WAYNE ROSSMAN

$\{$

$\{$

$.\rfloor\{$$\{$$1.\mathrm{J}\rfloor 1$$\mathrm{t}$$,|\ovalbox{\tt\small REJECT}$$\{$$1|_{1}$$\mathrm{r}\Downarrow\#^{1\#}\mathrm{t}_{\{1\ovalbox{\tt\small REJECT}_{|||\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}||\uparrow||\uparrow||}}|\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\# 1|||_{1}$$.\mathrm{t}_{\mathrm{t}1\#_{\mathrm{t}}^{\mathrm{t}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{1\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}}.$$1|\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$|.|\Downarrow$$\ovalbox{\tt\small REJECT}.\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}^{\mathrm{f}}\ovalbox{\tt\small REJECT}\Downarrow$$|^{1}$$||[\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}^{\vee}?|\mathrm{t}$

$|\#$

.

$||||$$|$$\#$FIGURE 15. Here we show the projected versions of the eigenvectors in the previous figure, for use in comparing with the eigenfunctions of the smooth case considered in section 5. These projectedversions are made the same way as in Figure 12.

Consider the example shown in the Figures 12, 13, 14, and 15. Here the catenoid.
has $9\cross 14=126$ interior vertices, so the matrix $S^{-1}Q$ has dimension 378$j\in[1,377]$,
for all-0.542 and
The first eigenvalue of this matrix isisas expected, since the smooth complete catenoid has index 1([6]). Note thatvery close to the negative eigenvalue for the smooth case, described in the caption of
Figure 9(the closest matching smooth catenoid portion satisfies $y_{1}=-y_{0}=1.91$ ).
The first eigenfunction in the discrete case (Figure 12) is also very close to the first
eigenfunction in the smooth case (Figure 9).
$\cross 378$$\lambda_{0}\approx$$\lambda j>0$$\lambda_{\mathrm{O}}$7.3. Discrete Minimal Trinoids. Since the trinoid has index 3, we find thatapproximating discrete surfaces with relatively fine meshes have 3negative eigenvalues. And we can look at the corresponding eigenvector fields (which estimate
the eigenfunctions in the smooth case), shown in Figure 16. For the approximating discrete trinoid in Figure 16, the first four eigenvalues are approximately-3.79, -1.31, -1.31, 0.014, so we indeed have 3negative eigenvalues and the second
eigenvalue has multiplicity 2.
REFERENCES[1] J. L. Barbosa and M. do Carmo. Stability of minimal surfaces and eigenvalues of the laplacian.
Math. Z., 173:13-28, 1980.
[2] A. Bobenko and U. Pinkall. Discrete isothermic surfaces. J. reine angew. Math., 475: 187-208,

1996.

[3] S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods. SpringerVerlag, 1994.
[4] D. Bressoud and J. Propp. How the alternating sign matrix conjecture was solved. Notices of
the AMS, 46(6):637-646, 1999.
[5] J. Dodziuk and V. K. Patodi. Riemannian structures and triangulations of manifolds. J. Inchan
Math Soc 40:1-52, 1976.
[6] D. Fischer-Colbrie. On complete minimal surfaces with finite Morse index in three manifolds.
Invent. Math. 82:121-132, 1985.
[7] G. Fix and G. Strang. An analysis of the finite element method. Prentice Hall, 1973.
[8] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics, vol. 152, Springer-Verlag, 1999.
[9] L.L. de Lima and W. Rossman. On the index of constant mean curvature 1surfaces in hyperbolic space Indiana Math. J. 47(2):685-723, 1998.
[10] S. Montiel and A. Ros. Schrodinger operators associated to aholomorphic map. Global Dif1481, Berlin 1991, 147-174
ferential Geometry and Global Analysis, Lect. Notes in Math.
$\#$

153
DISCRETE CONSTANT MEAN CURVATURE SURFACES AND$\mathrm{T}\mathrm{H}\mathrm{F}!\mathrm{I}\mathrm{R}$$\mathrm{I}\mathrm{N}\mathrm{D}\mathrm{F}_{\ell}\mathrm{X}$$i^{\prime^{Z}}j’\mathrm{a}$”$\backslash .\sim\sim\backslash \sim’\sim\sim\acute{J,}\swarrow_{/}}\sim’\approx\nearrow.\int_{\backslash }’\acute{\mathrm{I}}\iota_{\vee}\backslash \backslash \prime \mathrm{t}\sim\backslash$$\sim,\sim--j_{\alpha}-’\dot{y}arrow-i.".\backslash \nearrow.\mathrm{i}arrow\swarrow f_{\mathit{1}\backslash }^{f\}\backslash }r_{\acute{\mathit{1}}}\backslash \backslash \backslash$$\alpha..\nearrow’,/’,J_{\mathrm{t}}.l_{1,\backslash }\dagger\wedge/\nearrow\prime\prime-,$$.\backslash .\mathrm{s}^{1}$,”$;_{j}^{\mathrm{s}_{\mathrm{k}^{1}}^{\backslash }}-.’\wedge.-*.\sim.\mathrm{i}\mathrm{L}|\mathit{1}’..’.\cdot$]$..1.\iota-.\sim\backslash 1_{-}\backslash ,,\}\mathit{1}|_{\frac{}{\iota_{5}\{}\xi_{\mathrm{I}\downarrow-1_{\backslash }}^{l_{\mathrm{f}\nu\iota}^{\mathrm{t}_{i,\prime}^{j}}}}1_{\backslash }’\backslash \dagger\backslash \mathrm{t}r_{\ell}l14\mathrm{e}_{1}\downarrow|t^{r}\mathit{1}^{l\}i_{j_{r}’}^{\theta}}\tilde{\}}’\downarrow 1\iota_{4}^{\mathrm{I}l\dot{1}_{\alpha_{1}}\backslash }\mathrm{S}\backslash \mathrm{t}_{\backslash }\backslash \backslash \cdot\backslash \backslash \backslash \backslash _{\mathrm{R}}\backslash \neq f^{\iota^{\mathrm{s}.\prime}}backslash \iota_{\backslash }^{1}11\dot{1}i\nwarrow i_{\mathrm{x}_{\mathrm{b}}}$$\backslash 1\backslash r_{1}^{J}\nearrow’,’l\overline{\overline{\overline{[searrow]_{\backslash }^{\sim_{\backslash }^{\sim}}}}}\backslash ..\cdot\cdot\vee’\iota_{\backslash }^{\backslash ^{1}}|arrowarrow;\backslash ’\prime\prime\prime--\sim\nwarrow\backslash \backslash \vee\backslash ^{\backslash \backslash 1_{\acute{\prime}}}\backslash ^{\iota\iota’\prime}\backslash \mathrm{t}^{\acute{J}}\sim$,$\backslash \nearrow_{/}’\gamma=_{\iota}^{f}arrow’|[searrow]_{\backslash }^{\backslash }\prime_{\nearrow P}1\sim_{\mathrm{h}}\backslash ..arrow.-.\backslash \backslash \backslash \cdot\backslash \backslash \sim\backslash \backslash \backslash \backslash \backslash -\backslash \backslash \backslash \backslash ^{1}\acute{\iota}^{1}\backslash _{\acute{\acute{\mathrm{t}}}_{t}^{;}}^{l}\prime\prime\acute,\backslash \cdot \mathrm{a}$$[searrow]$$\backslash -\cdot.\backslash \cdot.\cdot..d\backslash ’.\simeq\simeq^{-}\simarrowarrow’\vee|_{\eta}\backslash \wedge-.]_{\mathrm{A}^{\backslash }}\vee\cdot.\cdot\backslash \cdot.\cdot.\mathrm{p}..r:_{\vee-\backslash \sim\backslash \backslash \backslash \backslash ^{1}\backslash \mathrm{t}\hat{f}_{\mathrm{Y}}^{\mathrm{t}}}\sim\backslash \cdot$

FIGURE 16. Variation vector fields for three area-reducing variations of adiscrete approximation of acompact portion of atrinoid.
The lower row has overhead views of these variation vector fields,
as well as an overhead view of the variation vector field associatedto the fourth (and first positive) eigenvalue.

:. . . .
$\backslash$[.

’$\vee\prime \mathrm{c}^{\backslash ^{\sim}}$$1^{l}$$–\vee\cdot$$\sim$$.’.\cdot..\ell\backslash\cdot’,$$.’\dot{J}\overline{.}i^{*:}\backslash
l**’\backslash 1.,\cdot.,\cdot,\cdot\backslash ’\wedge$.

.

$\nearrow-$\sim --\sim.

$\nwarrow^{\wedge}$

FIGURE 17. The first eigenvector field (whose corresponding eigenvalue is the first one and is negative) for adiscrete approximationof acompact portion of agenus 1Costa surface. Two other viewsof this discrete surface are shown

154
KONRAD POLTHIER AND WAYNE ROSSMAN

FIGURE 18. The first eigenvector field for adiscrete approximationof acompact portion of an Enneper surface. The associated first
eigenvalue is negative and is the only negative eigenvalue that is
not approximately zero, corresponding to the fact that the smooth
Enneper surface has index 1. Those other negative (approximatelyzero) eigenvalues have corresponding eigenvector fields that appear
roughly tangent to the surface.
[11] S. Nayatani. Lower bounds for the morse index of complete minimal surfaces in Euclidean-space. Osaka J. Math., 27:453-464, 1990.
[12] S. Nayatani. Morse index of complete minimal surfaces. The Problem of Plateau, ed. by Th.
M. Rassias, World Scientific (1992), 181-189.
[13] S. Nayatani. Morse index and Gauss maps of complete minimal surfaces in Euclidean 3-space.
$C$omment. Math. Helv. 68:511-537, 1993.
[14] B. Oberknapp and K. Polthier. An algorithm for discrete constant mean curvature surfaces. In
H.-C. Hege and K. Polthier, editors, Visualization and Mathematics, pages 141-161. Springer
Verlag, Heidelberg, 1997.
[15] R. Osserman. A Survey of Minimal Surfaces. Dover, 1986.
[16] U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experim. Math., $2(1):15-36$ ,1993.
[17] K. Polthier. Conjugate harmonic maps and minimal surfaces. preprint, 2000