数理解析研究所講究録 1416 巻 2005 年 161-174

161

Stationary isothermic surfaces and anew
characterization of the sphere
Shigeru Sakaguchi(坂口茂)

Faculty of Science, Ehime University(愛媛大学理学部)

1Introduction
This is based on the author’s recent work with R. Magnanini and J. Prajapat [MPS].
We establish arelationship between stationary isothermic surfaces and uniformly dense
domains. Astationary isothermic surface is alevel surface of temperature which does
not evolve with time. Adomain in the $N$-dimensional Euclidean space
is said to be uniformly dense in asurfaceof codimension 1if, for every small$r>0$ , the volume of the intersection of with aball of radius and centered atdoes not depend on for
We prove that the boundary of every uniformly dense
domain which is bounded (or whose complement is bounded) must be asphere; thisis anew characterization of the sphere. We then examine auniformly dense domain
with unbounded boundary
and we show that the principal curvatures ofsatisfycertain identities. The case in which the surface coincides with an is particularlyinteresting. In fact, we show that, if the boundary of auniformly dense domain isconnected, then ,if $N=3$ , it must be either asphere, aspherical cylinder or aminimal
surface. We conclude with adiscussion on uniformly dense domains whose boundaryis aminimal surface.
In \S 2, we consider stationary isothermic surfaces in the Cauchy problem for the heat
equation with characteristic functions of domains as initial data. The purpose of \S 3 is
to give the problem we will consider as well as to give the definition of uniformly dense
domains, its relation to the Cauchy problem, and important examples of uniformly
dense domains. In Q4, we show that, if is uniformly dense in , then is smoot$\Omega$$\mathbb{R}^{N}(N\geqq 3)$$\Gamma\subset \mathbb{R}^{N}$$\Omega$

$r$

$x$

$x$

$x\in\Gamma$an$\partial\Omega$$\Gamma$$\Omega$$\Gamma$$\Gamma$$\mathrm{h}$162under some conditions. \S 5 gives the main theorems concerning the symmetry of
where is uniformly dense in . In \S 6, we give very rough outline of proofs.
$\Omega$2$\Gamma$$\Gamma$Stationary isothermic surfaces

Consider the following Cauchy problem for the heat equation:in$\partial_{t}u=\triangle u$$\mathbb{R}^{N}\cross(0, +\infty)$

,

and

on

$u=\mathcal{X}_{\Omega}$$\mathbb{R}^{N}\cross\{0\}$

(2.1)

,

be
. Let
where
is the characteristic function of a domain in
an open subset of a hypersurface. is said to be a stationary isothermic surface of$\Gamma\subset \mathbb{R}^{N}$$\mathbb{R}^{N}(N\geqq 3)$$\Omega$$\mathcal{X}_{\Omega}$$\Gamma$$u$

if there exists a positive function $a=a(t)$ satisfying$u(x, t)=a(t)$for all$(x, t)\in\Gamma\cross(0, +\infty)$

In [CK], I. Chavel and L. Karp have shown that, ifof (2.1) has a stationary isothermic surface F. thencentered atof (see$\Omega$, where,with a little more argument, we can conclude that3$x_{0}$

is bounded and the solution
must extend to a whole sphere$\Omega$$\Gamma$$|\Omega|$$-x_{j} \frac{\partial u}{\partial x_{i}}+x_{i}\frac{\partial u}{\partial x_{j}}$

to

(2.2)

denotes the $N$ -dimensional Lebesgue measure, Theorem 2, p. 275]). Moreover, by using functions$x_{0}= \frac{1}{|\Omega|}\int_{\Omega}xdx$$[\mathrm{C}\mathrm{K}$.

$i\neq j$$\Omega$,is radially symmetric with respect.

Uniformly dense domains

Let $B(x, r)$ be the open ball with radius $r>0$ and centered at$r>0$ , we define the (spherical) average$r$-density ofat$\Omega$$\rho(x, r)=\frac{|\Omega\cap B(x,r)|}{|B(x,r)|}$$x$$x\in \mathbb{R}^{N}$. If$x\in \mathbb{R}^{N}$andas the ratio
(3.1)

.

(We shall use the same symbol –single bars –to denote both the-dimensionalLebesgue measure and the $(N-1)$ -dimensional Hausdorff measure of sets; thus, forinstance,andindicate the $N$ -dimensional Lebesgue measure of and the$|\Omega|$$|\partial\Omega|$$\mathrm{i}\mathrm{V}$$\Omega$163$(N-1)$ -dimensional Hausdorff measure of$\Omega$is unifomly dense in

there exists$\Gamma$$\partial\Omega$, respectively.) If$\Gamma\subset \mathbb{R}^{N}$, we say that

if it satisfies the following property:$r_{0}\in(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\Gamma, \partial\Omega),$$+\infty]$

such that, for each fixed $r\in(0, r_{0})$ .

the function $x\mapsto\rho(x, r)$ is constant on(3.2)$\Gamma$

We notice that (3.2) holds if and only ifthere exists

such that$r_{0}\in(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\Gamma, \partial\Omega))+\infty]$

for almost every fixed $r\in(0, r_{0})$ , the function$x\mapsto \mathrm{a}(\mathrm{x}, r)$(3.3)

is constant on$\Gamma$where$\sigma(x, r)=\frac{|\Omega\cap\partial B(x,r)|}{|\partial B(x,r)|}$

It is clear that, if

distance from$\partial\Omega$$\Omega$is uniformly dense in. In other words,

reason, many of the properties ofTheorem 3.1 Let$\Gamma$$\partial\Omega$$\Gamma$.

(3.4). then any $x\in\Gamma$ must have the same

must be parallel to a portion ofwill be inherited by$\partial\Omega$

and, for this$\Gamma$be a domain in
and letbe the solution of (2.1). Thena stationary isothermic surface forand only if is uniformly dense$r_{0}=+\infty$
with.
$\Omega$$\mathbb{R}^{N}$$u$$\Gamma\subset \mathbb{R}^{N}\iota s$$\Omega$$u\dot{\mathrm{z}}f$$\Gamma$$?.n$

Proof The solution of problem (2.1) is represented by$u(x, t)=(4 \pi t)^{-\frac{N}{2}}\int_{\mathrm{I}\mathrm{R}^{N}}\mathcal{X}_{\Omega}(\xi)e^{-\frac{|x-\xi|^{2}}{4t}}d\xi$

for all$(x, t)\in \mathbb{R}^{N}\cross(0.

+\infty)$

.

(3.5)

We compute that$u(x, t)$

Let ,
$p$

$q\in\Gamma$$=$$(4 \pi t)^{-\frac{N}{2}}\int_{0}^{+\infty}e^{-\frac{r^{2}}{4t}}$$=$$(4 \pi t)^{-\frac{N}{2}}\int_{0}^{+\infty}e^{-\frac{r^{2}}{4t}}|\Omega\cap\partial B(x_{\tau}r)|$$( \int_{\partial B(x,r)}\mathcal{X}_{\Omega}(\xi)dS_{\xi})dr$

dr.

(3.6)

be any pair of points. Since the Laplace transform is injective, (3.6) implies

that $u(p, t)=u(q, t)$ for every $t>0$ if and only if
almost every $r>0$ . This completes the proof.
Now, let us consider the following problem$\square$$|\Omega\cap\partial B(p, r)|=|\Omega\cap\partial B(q, r)|$

for

164
Problem 3.2 When $r_{0}<+\infty$ , classify pairs
dense in$(\Omega, \Gamma)$

satisfying thatis uniformly$\Omega$$\Gamma$

We know several examples with $r_{0}=+\infty$ .
andis called isoparametric if both
Example 3.3 A smooth function :are functions of . The family of the level hypersurfaces of is called an isoparafamily, and each level hypersurface of is called an isoparametric hypersurface.
It was shown in [LC] and [Seg] that any isoparametric family must be either parallel$|\nabla f|^{2}$$\mathbb{R}^{N}arrow \mathbb{R}$

$f$

$f$

$f$

$\triangle f$$f$$metr\dot{\iota}c$

hyperplanes, concentric spheres, or concentric spherical cylinders. See [No, PaTe] forsurveys of isoparametric hypersurfaces. If is either a strip or a half space, and if
If is either a ball,, thenis uniformly dense in
is a hyperplane parallel to$\Gamma$$\Omega$$\Gamma$$\Omega$$\partial\Omega$$\Omega$

is a sphere having the same center as ,Since any spherical cylinder is the Cartesian productan annulus, or the exterior of a ball, and if$\Omega$$\Gamma$then is uniformly dense inof a lower dimensional Euclidean space and a lower dimensional sphere, the similar$\Omega$$\Gamma$

proposition holds if$\Gamma$

is a spherical cylinder.

Example 3.4 Let $N=3$ . A right helicoid $?t$ is defined as the set$H$ $=${( ,$x_{1}$$x_{2}$,$x_{3})\in \mathbb{R}^{3}$:$x_{1}=s\cos t$ ,$x_{2}=s\sin$t. $x_{3}=at+b$ , $(s,$$t)\in \mathbb{R}^{2}$},

up into two connected components, and. Furthermore, we havelet be one of them. Then is uniformly dense in
and the symmetry of implies thatthatfor every $x\in H$ and every. It is evident that, when $N\geqq 4$ ,the solution of (2.1) satisfies
on $H$

where $a\neq 0$ , are real constants. 7{ splits$b$$\Omega$$\mathbb{R}^{3}$$\mathcal{H}(=\partial\Omega)$$\Omega$$r>0\backslash$$\rho(x, r)=\frac{1}{2}$$\Omega\cross \mathbb{R}^{N-3}$$\cross(0, +\infty)$$u= \frac{1}{2}$$u$is uniformly dense in $H$$\cross \mathbb{R}^{N-3}$$\mathcal{H}$.

In [Ni] J. Nitsche settled a conjecture of G. Cimmino [Cim]. He showed that thesuch that, forplane and the right helicoid are the only (smooth) hypersurfaces in$\mathbb{R}^{3}$on the surface, every sufficiently small sphere centered at has its areabisected by the surface. This result was derived by computing the Taylor’s formula fornear $r=0$ up to the relevant degrees, where is one part of a neighborhood offor sufficiently small
split up by the surface. Since J. Nitsche assumes that
other than$r>0$ , his result does not rule out the existence of hypersurfaces in
the helicoid, the plane, the sphere, or the spherical cylinder such that is uniformly
dense in for some domain .

each point

$x$

$x$

$x$

$\Omega$$\sigma(x_{7}r)$$\sigma(x, r)\equiv\frac{1}{2}$$\Gamma$$\mathbb{R}^{3}$$\Omega$$\Gamma$$\Omega$165
4Regularity of$\Gamma$

where$\Omega$is uniformly dense in$\Gamma$

The purpose of this section is to show that, if is uniformly dense inthen issmooth under some conditions. We consider the case where $r_{0}<+\infty$ . The first
theorem takes care of the case where, and the second one takes care of thecase where.
$\Omega$$\Gamma$$\Gamma$$\Gamma\subset\partial\Omega$$\Gamma\cap\partial\Omega=\emptyset$

Theorem 4.1 Let be an open set inwith boundaryof class , and let
an open subset of . If is u mformly dense in , then must be smooth.
$\Omega$$\mathbb{R}^{N}$$\partial\Omega$

Proof Choose
convolution$\psi$$\psi$$\Omega$$\Gamma$$\in C_{0}^{\infty}(\mathbb{R}^{N})$$\star \mathcal{X}_{\Omega}$such thatbelongs to$C^{\infty}(\mathbb{R}^{N})$$C^{0}$$\partial\Omega$$\Gamma$be$\Gamma$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\psi)\subset B(0, r_{0})$and $\psi(x)=\eta(|x|)$ . The

and we have that$\psi$$\star \mathcal{X}_{\Omega}(x)=\int_{B(0,r_{\mathrm{O}})}\psi(\xi)\mathcal{X}_{\Omega}(x-\xi)d\xi=\int_{0}^{r_{0}}\eta(r)(\int_{\Omega\cap\partial B(x,r)}dS_{\xi})$

dr.

(4.1)

Since satisfies (3.3), we infer thatmust be constant on and hence is thelevel surface of a smooth function. If we prove that, for every
is afunction such thathas non-vanishing gradient at , then, by the implicit
function theorem, we can conclude that is smooth.
$\Omega$$\psi’\star \mathcal{X}_{\Omega}$$\Gamma$$\Gamma$$x\in\Gamma\backslash \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$\psi$$\psi$$\star \mathcal{X}_{\Omega}$$\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\}_{1}$$x$$\Gamma$

Assume, by contradiction, that there exists a pointsuch thatfor every function with the properties stated above. Since$x_{0}\in\Gamma$$0$$\nabla(\psi\star \mathcal{X}_{\Omega})(x_{0})=$$\psi$$\nabla(\psi\star \mathcal{X}_{\Omega})(x_{0})$$=$$\int_{B(x_{0},r_{0})}\mathcal{X}_{\Omega}(\xi)\eta’(|x_{0}-\xi|)\frac{x_{0}-\xi}{|x_{0}-\xi|}d\xi=$$\int_{0}^{r_{0}}\eta’(r)(\int_{\Omega\cap\partial B(x_{0},r)}\frac{x_{0}-\xi}{|x_{0}-\xi|}dS_{\xi})dr$,

then$\int_{0}^{r0}\eta’(r)M(r)$$dr=0$ ,

(4.2)

where
$M(r)= \int_{\Omega\cap\partial B(x_{0},r)}\frac{x_{0}-\xi}{|x_{0}-\xi|}dS_{\xi}$.

Equation (4.2) implies that the distributional derivative of the bounded function$\mathrm{M}\{\mathrm{r})$equals zero on $(0, r_{0})$ . Therefore, by observing that $\lim_{rarrow 0+}M(r)=0$ , we conclude that
$M(r)$ equals

zero for almost every $r\in(0, r_{0})$ , and hence$\int_{\Omega\cap\partial B(x_{0},r)}(\xi-x_{0})dS_{\xi}=0$

for almost every $r\in(0, r_{0})$ .

166
Thus, by integrating this equation in , we see that$r$

Hence,$x_{0}$(4.3)

for every $r\in(0, r_{0})$ .

$\int_{\Omega\cap B(x\mathrm{o},r)}(\xi-x_{0})d\xi=0$

must be the center of mass of the set$\Omega\cap B_{(}^{/}x_{0}$, ) for every $r\in(0, r_{0})$ .
$r$Now, by choosing $r>0$ sufficiently small and by eventually translating and rotating

be represented, in a neighborhood
the axes, we can suppose that $x_{0}=0$ and that
, whereof $x_{0}=0$ , by the graph of a continuous function :is a suitable neighborhood of 0 and $\varphi(0)=0$ . Let $\varphi_{\pm}(y)=\max[\varphi(y), \pm\sqrt{r^{2}-|y|^{2}}]$ for$y\in B’=\{y\in \mathbb{R}^{N-1} : |y|<r\}\subset U(0)$ ; the set $\Omega\cap B(x, r)$ can be represented as$\partial\Omega$$\{(y, y_{N})\in B’\cross \mathbb{R} :\varphi_{-}(y)<y_{N}<\varphi_{+}(y)\}$$U(0)\subset \mathbb{R}^{N-1}$$U(0)arrow \mathbb{R}$$\varphi$. Therefore, we can infer that$\int_{\Omega\cap B(x,r)}(\xi_{N}-x_{N})d\xi=\int_{\Omega\cap B(x,r)}y_{N}dy_{N}dy=$$\int_{B}$$( \int_{\varphi-(y)}^{\varphi+(y)}y_{N}dy_{N})dy=\frac{1}{2}\int_{B}$$[\varphi_{+}(y)^{2}-\varphi_{-}(y)^{2}]dy>0$

,

,

,

which contradicts (4.3).

$\square$satisfying the interior sphere condition
be an open set in
Theorem 4.2 Let
and suppose thatis a domain satisfying the interior cone condition and such that
. Let
. If is uniformly dense in , then there exists an open subset$\mathbb{R}^{N}$$\Omega$$D$$\Gamma=\partial D$$\overline{D}\cap\overline{\Omega}=\emptyset$$\Lambda$of$\partial\Omega$$\Gamma$$\Gamma$satisfying the following properties:(i) both(ii)$\Omega$$\Gamma$andand$\Lambda$$\Lambda$

are smooth;are parallel;(iii) each principal curvature ofis smaller than the number$\Lambda$$\frac{1}{R}$with respect to the exterior normal direction to, where$R=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\Gamma, \partial\Omega)$$\partial‘\Omega$.

Proof We proceed as in the proof of Theorem 4.1 and calculate$v$ )$\star \mathcal{X}_{\Omega}(x)$and$\nabla(\psi\star$forsuch thatevery function with the properties stated in the beginning of the proof of Theorem
4.1, we conclude that (4.3) holds for every $r\in(0, r_{0})$ . Define the function $d=d(x)$ by$\mathcal{X}_{\Omega})(x)$. By supposing that there exists a point$\nabla(\psi\star \mathcal{X}_{\Omega})(x_{0})=0$$x_{0}\in\Gamma$$\psi$$d(x)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, \partial\Omega)$

for every$x\in \mathbb{R}^{N}\backslash \overline{\Omega}$

.

(4.3)

167
Since

$\Omega$is uniformly dense in , as is observed in \S 3 we have$\Gamma$$d(x)=R$

where$R=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\Gamma, \partial\Omega)$

for every$x\in\Gamma$

,

(4.5)

. Since $D$ satisfies the interior cone condition, there exists a

finite right spherical cone $K$ with vertex atsuch thatand.
By translating and rotating if needed, we can suppose that $x_{0}=0$ and that $K$ is the$K\subset\overline{D}$$x_{0}$$\overline{K}\cap\partial D=\{x_{0}\}$

set $\{x\in B(0, \rho) : x_{N}<-|x|\cos\theta\}$ , for some choice of $\rho\in(0, R)$ and

Since$K\subset\overline{D}$and$\overline{K}\cap\partial D=\{0\}$$\theta\in(0, \frac{\pi}{2})$.

, (4.5) implies that$d(x)>R$ for every $x\in K$ .

(4.6)

The set defined by$V=\{x\in\partial B(0, R) : x_{N}\geq R\sin\theta\}$

(4.7)

,

is such that

.

(4.8)

$\partial\Omega\cap\partial B(0, R)\subset V$

because, otherwise, there would be a point in $K$ contradicting (4.6). Thus, from (4.8)
it follows that there exists a number $\delta>0$ such that$d(x)\geq\delta$

for every$x \in \mathrm{d}\mathrm{B}(0, R)\cap\{x_{N}\leq R\sin\frac{\theta}{2}\}$

(4.9)

.

Choose $r \in(R, \min\{R+\delta, r_{0}\})$ . Then (4.9) yields that$\Omega\cap B(0, r)$$\subset B(0, r)\cap\{x_{N}\geq R\sin\frac{\theta}{2}\}$(4.10).

This contradicts the fact that (4.3) holds for $x_{0}=0$ .
Therefore, it follows that must be smooth, and we can complete the proof byfollowing that of [ , Lemma 3.1].
$\Gamma$$\mathrm{M}\mathrm{S}$5$\square$Symmetry of$\Gamma$

where$\Omega$is uniformly dense in$\Gamma$

Under the hypotheses either of Theorem 4.1 or of Theorem 4.2, we know that a part ofcorresponding to is a smooth hypersurface. Let, $j=1$ ,, $N-1$ be theprincipal curvatures ofatwith respect to the exterior normal direction to$\partial\Omega$$\Gamma$$\kappa_{j}(x)$$\partial\Omega$$x\in\partial\Omega$$\cdots$

168$\partial\Omega$. For eachof the surface$\in\{1, \ldots, N-1\}$ ,$g$$\partial\Omega$$K_{j}(x)$denotes the -th symmetric invariant $K_{j}(x)$$j$evaluated at , that is$x$$\kappa_{i_{1}}(x)\cdots\kappa_{i_{j}}(x)$$K_{j}(x)= \sum_{i_{1}<<i_{J}}$, $j=1$ ,$\cdots$, $N-1$ .

.

With this definition, $H=K_{1}/(N-1)$ and $K=K_{N-1}$ are the mean and the Gauss, respectively. We will use the notation $\omega_{N}=|\partial B(0,1)|$ , where
curvature of$\partial\Omega$$B(0,1)\subset \mathbb{R}^{N}$

Let us first consider the case where$\Gamma\subset\partial\Omega$.

, and suppose is an open subset of(N
Theorem 5.1 Let be a domain in. If is uniformly dense in I. thenthe boundary
which is of class$\mathbb{R}^{N}$$\Omega$$C^{0}$$\partial\Omega$$\geqq 3)$$\Gamma$$\Omega$as $rarrow 0$ ,$\sigma(x, r)=\frac{1}{2}+\sigma_{1}(x)r+\sigma_{3}(x)r^{3}+O(r^{5})$$x\in\Gamma$

(5.1)

,

where
(5.2)

$\sigma_{1}(x)=\frac{\omega_{N-1}}{2\omega_{N}}H(x)$andif $N=3$ ,$\frac{1}{256}[K_{1}(x)^{3}-4K_{1}(x)K_{2}(x)]$(5.3)$\sigma_{3}(x)=\{$

if$\frac{\omega_{N-1}}{16\omega_{N}(N^{2}-1)}[K_{1}(x)^{3}-4K_{1}(x)K_{2}(x)+4K_{3}(x)]$are constant on,$\Gamma$Corollary 5.2 Let
be a domain insubset of the boundary
which is of classanalytic and the following hold:$\mathbb{R}^{N}(N\geqq 3)$$\Omega$$\partial\Omega$

(i)

If

(ii)

If $N=3$ and$N\geqq 3$$N\geqq 4$,$\Gamma=\partial\Omega$$\Gamma$, and$\partial\Omega$$C^{0}$$\Gamma$$\Gamma$

is an open

. If . is uniformly dense in . then$\Omega$

is bounded, thenis connected, then, and suppose that$\partial\Omega$$\Gamma$$\Gamma$ismust be a sphere.

must be $eit/ier$ a portion of a $sphere_{j}$ of$a$spherical cylinder or of a minimal surface.

Proof of Corollary 5.2. (i) follows from Theorem 5.1 and Aleksandrov’s Soap Bubble
Theorem (see [Alek]), and Theorem 5.1 implies (ii) directly.

$\square$169
Remark 5.3 In [Ni] J. Nitsche computed$\sigma(x.r)=\frac{1}{2}+\sum_{n=1}^{\infty}\sigma_{n}(x)r^{n}$$\Gamma$$\sigma_{5}(x)$when $N=3$ for the series expansion. He showed that, if $H(x)=\sigma_{5}(x)=0$ for every $x\in\Gamma$ . thenmust be either a portion of a plane or of a right helicoid.

Let us consider the case where$\Gamma\cap\partial\Omega=\emptyset$.

Theorem 5.4 Under the hypotheses and the situation of Theorem 4.2, for every
there exists a unique pointsuch that, and furthermore wehave that, for each. as $rarrow R+0$ ,$x\in\Gamma$$y\in\Lambda$$\overline{B(x,R)}\cap\overline{\Omega}=\{y\}$$x\in\Gamma$$\rho(x, r)=\frac{2^{\frac{N+1}{2}}N\omega_{N-1}}{(N^{2}-1)\omega_{N}R^{N}}\{\prod_{j=1}^{N-1}(\frac{1}{R}-\kappa_{j}(y))\}^{-\frac{1}{2}}(r-R)^{\frac{N+1}{2}}+o((r-R)^{\frac{N+1}{2}})$

In particular, both$\Gamma$and$\Lambda$$\prod_{j=1}^{N-1}$

Moreover, if$\partial\Omega$(5.4)

are analytic, and for some constant $c>0$($\frac{1}{R}-\kappa j$$(y))=c$ for every

is bounded and connected, then$\partial\Omega$$y\in\Lambda$

(5.5)

.

must be a sphere.

Remark 5.5 As in [MS], by Aleksandrov’s uniqueness theorem in [Alek] we see that
equality (5.5) implies that, if$\partial\Omega$

is bounded and connected, then$\partial\Omega$must be a sphere.

Remark 5.6 In Example 3.4, by using Theorem 5.4, we see thatis an isolated
isothermic surface. Indeed, if there exists another isothermic surface sufficiently close
to , then by Theorem 5.4$\mathcal{H}$$Tt$$( \frac{1}{R}-\kappa_{1}(y))(\frac{1}{R}-\kappa_{2}(y))=c$

for every $y\in H$

for some positive constant . Thus, since $\kappa_{1}(y)+\kappa_{2}(y)\equiv 0$ , the Gauss curvaturemust be constant on $H$ . This is a contradiction.
$c$$\kappa_{1}(y)\kappa_{2}(y)$Nitsche’s result [Ni] does not rule out the existence of minimal surfaces (other thanthe helicoid or the plane) which are boundaries of uniformly dense domains. Here, we
consider the case of embedded minimal surfaces of finite total curvature. The theoryof complete embedded minimal surfaces of finite total curvature inhas developed
recently (see [HK],, and [PeRo] for some surveys). In particular, in [Kap] N.
$\mathbb{R}^{3}$$[\mathrm{L}\mathrm{o}\mathrm{M}]$170
Kapouleas constructed large families of such minimal surfaces with symmetries, andmoreover in [T] M. Traizet showed the existence of such minimal surfaces with nosymmetries. Note that the catenoid and the plane are the classical examples of complete

embedded minimal surfaces of finite total curvature, and the helicoid is not of finite
total curvature because of its periodicity. By combining Nitsche’s result [Ni] and thetheory of complete embedded minimal surfaces of finite total curvature inconclude our analysis of uniformly dense domains with the following result.

$\mathbb{R}^{3}$, we

Theorem 5.7 Let be a complete embedded minimal surface of finite total curvature, and letinbe one of the two domains disconnected by from . If is uniformly, then must be a plane.
dense
$S$

$\mathbb{R}^{3}$$S$$\Omega$$\Omega$$S$$\dot{l}nS(=\partial\Omega)$

6$\mathbb{R}^{3}$

On proofs of theorems

In this section we give very rough outline of proofs of Theorem 5.1 and Theorem 5.7.

See [MPS] for their details.

On the proof of Theorem 5.1. For x, denote by Tx(dQ) and the tangent
space and the interior normal unit vector toat , respectively. For fixedwith $|v|=1$ , let
be the plane through spanned by and . We may assume
that, for $r>0$ sufficiently small, each point in $\Omega\cap B(x, r)$ can be parameterized inspherical coordinates as:$\in\partial\Omega$$\nu$$\partial\Omega$$\pi_{x}(v, \nu)$$v\in T_{x}(\partial\Omega)$

$x$

$x$

$v$

$\nu$

$z$

$z=x+\rho\cos\varphi$ $v+\rho\sin\varphi\nu$$v\in T_{x}(\partial\Omega)\cap \mathrm{S}^{N-2}$

where, for fixed$v\in T_{x}(\partial\Omega)\cap \mathrm{S}^{N-2}$

in polar coordinates. Expand,$\theta(r, v)$,$\theta(\rho, v)\leqq\phi$$\phi=0(\mathrm{p}, v)$

in

,

$r$

$\leqq\pi/2,0\leqq\rho\leqq r$,

parameterizes the curve(6.1)$\partial\Omega\cap\pi_{x}(v, \nu)$as$\theta(r, v)=\theta_{1}(v)r+\theta_{2}(v)r^{2}+\theta_{3}(v)r^{3}+\cdot$

The Jacobian of the change of variables (6.1) is$\rho^{N-1}\cos$$| \Omega\cap B(x, r)|=\int_{\mathrm{S}^{N-2}}\int_{0}^{r}\rho^{N-1}\int_{\theta(\rho,v)}^{\pi/2}\cos$$\phi$.

(6.2), so that we can write:$\phi d\phi d\rho dS_{v}$

,

(6.3)

171
where

denotes the surface element on
respect to and dividing by, we get:$dS_{v}$$\mathrm{S}^{N-2}$. By differentiating this formula with$\omega_{N}r^{N-1}$$r$$\sigma(x, r)=\frac{1}{\omega_{N}}\int_{\mathrm{S}^{N-2}\theta(}\int_{r,v)}^{\pi/2}\cos$$\phi d\phi dS_{v}=\frac{1}{2}-\frac{1}{\omega_{N}}\int_{\mathrm{S}^{N-2}}\int_{0}^{\theta(r,v)}\cos$$\phi d\phi dS_{v}$.

Here we have.

$\int_{0}^{\theta(r,v)}\cos$$\mathrm{C}^{)}d\phi=\theta_{1}(v)r+\theta_{2}(v)r^{2}+[\theta_{3}(v)-\frac{N-2}{6}\theta_{1}(v)^{3}]r^{3}+\cdot$(6.4)

Without loss of generality, we suppose that is the origin inandcoincides with the hyperplane $\{(y, y_{N})\in \mathbb{R}^{N} : y_{N}=0\}$ , where we use the letter fo
denote an element of, that is, $y=$
. Suppose thatisthe graph of a smooth function in the neighborhood of a point $x=0\in\partial\Omega$ andwe compute the coefficients (6.2) in terms of the derivatives of . We may assume
that the function :then parameterizes
in a neighborhood of $x=0$ ,that isis represented by the equation $y_{N}=\varphi(y)$ , where $\varphi(0)=0$ ,$x$$\mathbb{R}^{N}$$T_{x}(\partial\Omega)$$y$$\mathbb{R}^{N-1}$$(y_{1}, \ldots, \mathrm{y}\mathrm{w}-\mathrm{i})\in \mathbb{R}^{N-1}$$\partial\Omega$$\varphi$$\varphi$$\mathbb{R}^{N-1}arrow \mathbb{R}$$\partial\Omega$$\varphi$$\partial\Omega$$\nabla\varphi(0)=0_{\backslash }$and$-\nabla^{2}\varphi(0)=$curvatures of

diag
at

$(\kappa_{1}, \cdot\cdot 1 , \kappa_{N-1})$. Here, $j=1$ , .

, $N-1$ are the principal$\cdot$$\kappa j$with respect to the exterior normal direction to.
also use a standard multi-index notation for the derivatives of : if $i=(i_{1}, \ldots, i_{N-1})$
is a multi-index, we denote $|i|=i_{1}+\cdot\cdot+i_{N-1}$ , $i!=i_{1}$ !
,$\partial\Omega$$0\in\partial\Omega$$\partial\Omega$$\mathrm{W}^{\gamma}\mathrm{e}$$\varphi$$\cdots$$D^{i}\varphi=\partial_{y1}^{i_{1}}$and$\varphi$$\partial_{yN-1}^{i_{N-1}}\varphi$,

for. With these notations and assumptions, the Taylor
in a neighborhood of $y=0$ is$y^{i}=y_{1}^{i_{1}}\cdots y_{N-1}^{i_{N-1}}$

expansion of..

$i_{N-1}!$$y\in \mathbb{R}^{N-1}$

where $P4\{v$ )$\varphi(y)=\sum_{n=2}^{\infty}P_{n}(y)$$= \sum_{|\mathrm{i}|=n}\frac{D^{i}\varphi(0)}{i!}y^{i}$. $n=0,1$ ,(6.5)$\cdots$

Since $r\sin\theta(r, v)=\varphi(r\cos\theta(r, v)v)$ for sufficiently small , we have:$r$$\sin\theta(r_{\backslash }v)=\sum_{n=2}^{\infty}r^{n-1}\cos\theta(r, v)P_{n}(v)$By expanding both sides in$\theta_{1}(v)=P_{2}(v)$

,

$r$

.

(6.6)

and comparing their coefficients, we can get:$\theta_{2}(v)=P_{3}(v)$,and$\theta_{3}(v)=P_{4}(v)-\frac{5}{6}P_{2}(v)^{3}$

.

(6.7)

172
Hence, combining this with (6.4) yields that in the Taylor expansion (5.1)$\sigma_{1}(x)=-\frac{1}{\omega_{N}}\int_{\mathrm{S}^{N-2}}P_{2}(v)dS$,

an (x)$=- \frac{1}{\omega_{N}}\int_{\mathrm{S}^{N}2}P_{3}(v)dS$

, and

(6.8)
$\sigma_{3}(x)=-\frac{1}{\omega_{N}}\int_{\mathrm{S}^{N-2}}[P_{4}(v)-\frac{N+3}{6}P_{2}(v)^{3}]dS_{v}$Lemma 6.1 Let $i=$$(i_{1}, \ldots, i_{N-1})$.

be a multi-index. We have$\int_{\mathrm{S}^{N-2}}v^{i}dS_{v}=0$

if at least one entry of$i$is odd; otherwise,(6.9)$\frac{1}{\omega_{N-1}}\int_{\mathrm{S}^{N-2}}v^{2\mathrm{i}}dS_{v}=\frac{(N-3)!!(2i)!}{(2|\mathrm{z}|+N-3)!!2|i|i!}$$[ \frac{n-1}{2}]$where $n!!= \prod_{k=0}(n-2k)$ .

Consider$\sigma_{1}$. Since$\sigma_{2}$first. Lemma 6.1 and (6.8) directly imply that

$P_{2}(v)=- \frac{1}{2}\sum_{j=1}^{N-1}\kappa_{j}v_{j}^{2}$$\sigma_{2}=0$. Let us consider, we have from Lemma 6.1 and (6.8)$\sigma_{1}(x)=\frac{\omega_{N-1}}{2(N-1)\omega_{N}}\sum_{j=1}^{N-1}\kappa_{j}=\frac{\omega_{N-1}}{2\omega_{N}}H(x)$,which is just (5.2). Therefore, the assumption that is uniformly dense inthat $H(x)\equiv H_{0}$ on for some constant. By using this fact, we get$\Gamma$$\Omega$$\Gamma$

implies$H_{0}$.

$(1+| \nabla\varphi|^{2})\triangle\varphi’-\sum_{k,\ell=1}^{N-1}\frac{\partial\varphi}{\partial y_{k}}\frac{\partial\varphi}{\partial y\ell}\frac{\partial^{2}\varphi}{\partial y_{k}\partial y_{l}}\equiv-(N-1)H_{0}(1+|\nabla\varphi|^{2})^{\frac{3}{2}}$This fact implies that$\varphi$

is analytic in . By differentiating this equation twice and$y$letting $y=0$ , we can get (5.3) from Lemma 6.1 and (6.8).

Remark 6.2 It can be shown that$\sigma(x, r)$admits the series expansions$\sigma(x, r)=\frac{1}{2}+\sum_{n=1}^{\infty}\sigma_{n}(x)r^{n}$Here, for each$n\in \mathrm{N}$is a polynomial, without, $P_{n+1}(v)$ and hence each coefficient, the integrand in the expression forzeroth order coefficient, of the functions$P_{2}(v)$,$\ldots$(6.10).
$\sigma_{n}(x)$is a polynomial, without zeroth order coefficient, of$n+1$ . In particular, we obtain that $\sigma_{2k}(x)=0$ for any $k=1,2$ ,$\sigma_{n}(x)(n\in \mathrm{N})$$D^{\beta}\varphi(0,0)$$\cdots$, $2\leq|\beta|\leq$173
On the proof of Theorem 5.7. We recall from [PeRo, p. 18] thatcompleteembedded minimal surface inwith finite total curvature, outside a big ball in space,
has a nice shape: there are a finite number of parallel ends and each end is asymptotic
to a plane or to a halfcatenoid” (see also [, Proposition 2.5, pp. 36-37] for a more
precise description concerning complete, nonplanar, minimal surfaces with finite total
curvature). On one end of , we can see that, as goes to the end,$‘(\mathrm{a}$$\mathbb{R}^{3}$$\mathrm{H}\mathrm{K}$

$S$

$x$

$\sigma j(x)arrow 0$

Hence, since$\Omega$is uniformly dense in$\sigma j(x)\equiv 0$

which shows that$\sigma(x, r)\equiv\frac{1}{2}$[Ni], we can conclude that

$S$

$S$

for every$\in \mathrm{N}$$\gamma$.

, we must havefor every $x\in S$ and every$j\in \mathrm{N}$,for sufficiently small $r>0$ . Finally, by Nitsche’s resultmust be a plane.

Acknowledgement.
This work was partially supported by a Grant-in-Aid for Scientific Research (B)15340047) of Japan Society for the Promotion of Science.

$(\mathrm{C}$

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