数理解析研究所講究録
第 1775 巻 2012 年 85-93

85

GEOMETRIC CAUCHY PROBLEMS FOR SPACELIKE AND TIMELIKE CMC
SURFACES IN$\mathbb{R}^{2,1}$

DAVID BRANDER
ABSTRACT. We discuss recent work of the author and collaborators on generalizations of’sclassical problem to the case of constant non-zero mean curvature surfaces in $2+1$ -dimensional spacetime. The aim is to give an overview, and to poim out the similarities and differences between thetwo cases of timelike and spacelike surfaces. Applications to the construction of CMC surfaces withprescribed singularities are also described.
$Bj\ddot{o}rling$

1. INTRODUCTION
This article discusses results of work of the author and collaborators, [2], [5] and [6], on the generalization of Bj\"orling’s problem for minimal surfaces to the case of constant non-zero mean curvature(CMC) surfaces in the 3-dimensional Minkowski space. The classical Bj\"orling problem is tofind the minimal surface inwhich contains a prescribed curve with the surface tangent distribution
prescribed along the curve. It has a unique solution via analytic extension because the data is enoughto determine the (holomorphic) Weierstrass data along the curve. The generalization of this problem
to other surface classes besides minimal is sometimes called the geometric Cauchy problem.
$\mathbb{R}^{2,1}$$\mathbb{R}^{3}$

The problem has also been considered for zero mean curvature surfaces in, both spacelike [1]and timelike [7]. There is a substantial difference between the spacelike and timelike cases, as theformer is an elliptic problem and has a Weierstrass representation in terms of holomorphic data, as in
the case of minimal surfaces in , whilst the timelike case is hyperbolic and has a d’Alembert typesolution in terms of two functions of one variable each. The timelike case is complicated by the factthat there is no unique solution to the Bj\"orling problem if the initial curve is characteristic.
$\mathbb{R}^{2,1}$$\mathbb{R}^{3}$

All of the last comments also apply to the corresponding problems for constant non-zero mean
curvature surfaces. Moreover, the non-zero situation has the additional property that the Weierstrass/d’Alembert type representations are now in terms of maps into a loop group [9, 8], and obtaining the solutions from the data entails a loop group decomposition (Iwasawa or Birkhoff). The
loop group decomposition is really an LU decomposition of infinite matrices, which can easily beapproximated numerically, but cannot be written down explicitly, in general. Therefore some work isnecessary in going back and forth between the surface and the corresponding Weierstrass data.

The Bj\"orling problem for non-minimal CMC surfaces in, was solved by the author andDorfmeister in [3]. Perhaps the essential point of the solution is that, although the Iwasawa decomposition cannot be computed explicitly for general Weierstrass data, one has considerable flexibility
in one’s choice of Weierstrass data. It tums out to be possible to choose the Weierstrass data such$\mathbb{R}^{3}$

$J$

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DAVID BRANDER

that the Iwasawa decomposition is trivial along the initial curve in question. This allows one to write
down explicitly the Weierstrass data corresponding to given Bj\"orling data along a curve. The exact, so that problem
same argument can be used to solve the problem for spacelike CMC surfaces in
2.
below
in
SectionthisWe
discuss
was essentially solved by the case.
$\mathbb{R}^{2,1}$$\mathbb{R}^{3}$

The geometric Cauchy problem for timelike CMC surfaces was solved by the author and M Svensson in [5]. Although the d’AlembeIt construction is rather different from the Weierstrass constructionof the spacelike case, the solution of the problem again depends essentially on the fact that one canchoose d‘Alembert data such that the loop group decomposition (Birkhoff in this case) is trivial along
the curve in question. This is discussed in Section 3.
Finally there is the singular geometric Cauchy problem, which arises naturally for CMC surfacesin. Both timelike and spacelike CMC surfaces almost always have generic singularities. Most ofthese arise at points where the loop group decomposition has a middle term, and cannot be avoided, and the singularities
in general. An analogous situation exists for minimal/maximal surfaces inhave been studied recently by vanious authors. In particular, the singular Bj\"orling problem, which is
to find the minimal/maximal surface with a prescribed singular curve, and a further parallel vector
field prescribed along the curve (in analogue with the Gauss map), has been studied by S-D Yang,
YW Kim, S-E Koh and H Shin in [11] and [10].
$\mathbb{R}^{2,1}$$\mathbb{R}^{2,1}$

FIGURE 1. The singular geometnic Cauchy construction can be used to produce
CMC surfaces with arbitraly prescribed singular curves. The timelike CMC surface
shown has infinitely many swallowtails along a cuspidal edge. The first few can beseen in this plot of a finite domain. The geometric Cauchy data are $s(v)=\sin(v)$ ,$t(v)=2,$ $\theta’(v)=0.0001$ . The swallowtails occur when $s=0$ .

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GEOMETRIC CAUCHY PROBLEMS FOR CMC SURFACES IN$\mathbb{R}^{2,1}$

The singular geometric Cauchy problem for spacelike CMC surfaces is solved by the author in [2].
The essential problem here is that the $SU(1,1)$ frame for the surface, which is critical for solving the
problem in the regular situation of [3], blows up as the singular curve is approached. The problem issolved by multiplying all the maps by a certain element of the loop group, which moves all the data
into the big cell, that is the set on which the Iwasawa decomposition does not have a middle term. It isthen possible to solve the problem with a new “singular frame“ which arises from this idea. An analogous device is used to solve the problem for timelike CMC surfaces injoint work with M Svenssonin [6]. This will be discussed in Section 4.

2. SPACELIKE CMC SURFACES
For explicit details of the following discussion, the reader is referred to [4] and [3].

2.1. The generalized Weierstrass representation. The so called $DPW$ (Dorfmeister/Pedit/Wu [9]),or generalized Weierstrass representation, for spacelike CMC surfaces is sketched below. We will
discuss it locally for simplicity. Let
be an open set.
A conformal CMC immersion :,is represented by an extendedfmme :where $G=SU(1,1)$ , andis the group of twisted loops in .
$U\subset \mathbb{C}^{2}$$f$$\bullet$.
.

$Uarrow \mathbb{R}^{2,1}$$\hat{F}$The surface $f$ is recovered fromwhich is essentially of the form$\hat{F}$by a simple formula, called the Sym-Bobenko formula,$\ovalbox{\tt\small REJECT}(\hat{F})=\lambda(\partial_{\lambda}\hat{F})\hat{F}^{-1}|_{\lambda=1}$.

The extended frame is characterizedl among smooth maps into

Cartan form$\hat{F}^{-1}d\hat{F}$$Uarrow\Lambda G$$G$$\Lambda G$$\Lambda G$

by the fact that its Maurer-

can be written:$\hat{F}^{-1}d\hat{F}=A_{-1}\lambda^{-1}dz+\infty+\tau(A_{-1})\lambda d\overline{z}$,where is the involution determining the real form $su(1,1)$ in
which does not depend on .
$\tau$$\epsilon 1(2,\mathbb{C})$, and$\alpha_{0}$is a l-form$\lambda$$\bullet$Finally (and this is the DPW part of the construction), the extended frame can be obtained
from a holomorphic frame :by a pointwise Iwasawa decomposition$\hat{\Phi}$(2.1)$Uarrow\Lambda G^{\mathbb{C}}$$\hat{\Phi}=\hat{F}\hat{B}+$,the last group denoting those loops inunit disc.
$\bullet$$\hat{F}\in\Lambda G$$SL(2,\mathbb{C})$,$\hat{B}+\in\Lambda^{+}G^{\mathbb{C}}$,which can be holomorphically extended to the

By companing the Maurer-Cartan forms of and , and using thathas a Taylor expansionin , it is easy to deduce that must be holomorphic, and is chamcterized by its MaurerCartan form being of the form$\hat{F}$$\lambda$$\hat{\Phi}$$\hat{B}+$$\hat{\Phi}$$\hat{\Phi}^{-1}d\hat{\Phi}=(\psi_{-1}\lambda^{-1}+\psi_{0}+O(\lambda))dz$.

lThis is standard for hannonic maps into symmetric spaces, and the Gauss map is harmonic into$\mathbb{H}^{2}$

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DAVID BRANDER

The holomorphic functions in the matrices are arbitrary, apart from a regularity condition
on one of them. This is the generalized Weierstmss representation for (spacelike) CMC surfaces.
$\psi_{i}$2.2. Solution of the Bjorling problem. Now the idea that is used in [3] to solve the Bj\"orling problem
in the decomposition 2.1 is essentially unique. So suppose
is actually very simple: The factor
is already AG-valued, i.e. already takes values in the real form. At such a point, thethat
Iwasawa decomposition (2.1) is trivial:$\hat{F}\in\Lambda \mathscr{G}$$\hat{\Phi}(zo)$$\mathfrak{H}(zo)=\hat{F}(zo)$.

We can use this to constmct a special holomorphic frame from the knowledge of along a curve as
follows: suppose we know the value of along some curve in , say $F(x+0.i)=\hat{F}_{0}(x)$ along an. Let us set
interval$\hat{F}$

$F$

$U$

$J=U\cap \mathbb{R}$$\hat{\Phi}_{0}(x)=\hat{F}_{0}(x)$,$x\in J$,

containing, from some open subset
and then extend this holomorphically to a map :. It tums out that this can be done, and moreover this map has all the required properties of aholomorphic extended Rame for a CMC surface. Further, the corresponding extended frame agrees
along , because the Iwasawa decomposition is trivial along this curve. Thus we havewithalong from the Bj\"orling data. It
Bj\"orling
solved the
problem, provided that we can constructtums out that this can be done in a fairly straightforward manner (see [3]), and one finally obtains a
in terms of the Bj\"orling data.
simple formula for the holomorphic potential$\hat{\Phi}$$Varrow\Lambda G^{\mathbb{C}}$$V$$\Phi$$J$$F$$fi_{0}(x)$

$J$

$J$

$\hat{F}_{0}(x)$$\hat{\Phi}^{-1}d\hat{\Phi}$

Using a unique local correspondence between Weierstrass data and extended frames, obtained via
a normalized Birkhoff splitting, it is also not difficult to show that the Bj\"orling problem has a uniquesolution.

3. TIMELIKE CMC SURFACES
We tum now to the timelike case. The d’Alembert representation given in [8] differs from theWeierstrass representation described in Section 2 as follows:
consisting of loops
The extended frame this time takes values in a subgroup ofwhich are in $G=SL(2,\mathbb{R})$ for real values of .
$\Lambda SL(2,\mathbb{C})$$\bullet$$\lambda$$\bullet$

The Maurer-Cartan form of the extended frame, has the form
for$\hat{F}(x,y)$, where$x$

and are lightlike coordinates$y$$U\subset \mathbb{R}^{1,1}$$\hat{F}^{-1}dfi=A_{1}\lambda$$\bullet$

dr $+\infty+A_{-1}\lambda^{-1}dy$.

The extended frame $fi(x,y)$ is obtained from a pair of maps,$\hat{X}^{-1}dK=\sum_{\dot{r}=-\infty}^{1}\psi_{i}^{\chi}dx$$\hat{X}(x)$and,$\hat{Y}^{-1}d\hat{Y}=\sum_{j=-1}^{\infty}\psi_{i}^{Y}dy$$\hat{Y}(y)$, where89
GEOMETRIC CAUCHY PROBLEMS FOR CMC SURFACES IN

and the coefficients$\hat{F}$$\bullet$is obtained from$\psi_{i}^{X}$$\hat{X}$

and

and

(3.1)

$\hat{Y}$$\psi_{i}^{Y}$

are essentially$arbitral\gamma$

functions.

by performing a pointwise Birkhoff splitting$\hat{X}^{-1}(x)\hat{Y}(y)=\hat{H}_{-}(x,y)\hat{H}_{+}(x,y)$

where$\hat{H}_{-}(x,y)\in\Lambda^{-}G$$\mathbb{R}^{2,1}$, and $\hat{H}_{+}(x,y)\in\Lambda^{+}G$ .

,

$2$

Then
(3.2)

.

$\hat{F}(x,y)=\hat{Y}(y)\hat{H}_{+}^{-1}(x,y)$3.1. Solution of the geometric Cauchy problem. Here we describe joint work with M Svensson
in [5]. As in the spacelike case, the essential idea is to contrive a situation where the (this time
Birkhoff) decomposition (3.1) is trivial along the curve. Looking at their Maurer-Cartan forms, where
the leading terms must be non-vanishing for regular surfaces, we can see thatwill have a pole(in ) at
while
Thus,
will have a pole at.
the Birkhoff decomposition is unlikely to
be trivial unless we have something likeconstant in , and indeed we can achieve thisalong the curve $y=x$ if we can arrange, for example, that$\hat{X}(x)$$\lambda$$\infty$$\lambda=0$$\hat{Y}(y)$$\hat{X}^{-1}(x)\hat{Y}(y)=$$\hat{X}(v)=\hat{Y}(v)$$\lambda$.

This leads to a solution to the geometric Cauchy problems along non-chamcteristic curves, i.e.
curves which are never tangent to a null curve. For such a curve in, one can always (with apossible change of orientation) locally choose null coordinates $(x,y)$ , such that the curve is given as$u=0$ in the coordinates $u=(x-y)/2,$ $v=(x+y)/2$ .
$\mathbb{R}^{1,1}$

Suppose one can construct, from the geometric Cauchy data, the extended frame
for a timelike CMC surface along the curve $x=y=v$ , i.e. $u=0$ . (It tums out that one can do this).
Let us now set,.
Then it is easy to check that and have precisely the required form for the d’Alembert data of atimelike CMC surface, and thus generate such a surface using the scheme outlined above. Along the
curve $x=y=v$ we have $\hat{X}(x)=\hat{Y}(y)$ , so that, along this curve the Birkhoff decomposition at (3.1)
is just $I=I.I$ , where is the identity matrix. Along this curve, the extended frame for the surface soconstmcted is given by (3.2) as$\hat{F}(0,v)=\hat{F}_{0}(v)$$\hat{X}(x)=\hat{F}_{0}(x)$$\hat{X}$$\hat{Y}(y)=\hat{F}_{0}(y)$$\hat{Y}$$I$$\hat{F}(0,v)=\hat{Y}(v).I=\hat{F}_{0}(v)$.

Thus the constmcted surface solves the geometric Cauchy problem. One can show, again using anormalized Birkhoff decomposition, that the solution is unique.

Finally, the case that the initial curve is characteristic (null), is also treated in [5]. The solution
is not unique, but we describe how to constmct all solutions. We do not treat initial curves whichbecome characteristic at isolated points.
notation for the last two groups is not stnictly correct here, bm in any caseconsists of loops which extend, whileholomorphically toconsists of loops which extend to the unit disc.
$2_{The}$$\Lambda^{-}G$$\{|z|>1\}\subset\hat{C}$$\Lambda^{+}G$90
DAVID BRANDER

4. THE SINGULAR GEOMETRIC CAUCHY PROBLEM
The solution of the singular geometric Cauchy problem, treated in [2] and [6] for, respectively,
spacelike and timelike CMC surfaces is rather complicated to describe. As mentioned in the introduction, the extended frame blows up as such a curve is approached. Therefore, it is clear that thesolutions outlined above cannot be applied for such a curve.
$\hat{F}$We will not attempt to describe the solution here, but only mention a critical idea on which thesolution is built. Let us consider the case of spacelike surfaces. As described in Section 2, theextended frame (and hence the surface) is obtained from the holomorphic frame by an Iwasawa$\mathfrak{H}$$\beta$decomposition$\hat{\Phi}=\hat{F}\hat{B}+\cdot$

When the group is non-compact, the Iwasawa decomposition is only written this way on an open denseofsubset of the loop group, called the big cell. The rest of the loop group is a disjoint unionsubvarieties, of codimension increasing with .
$\bigcup_{i=1}^{\infty}\mathscr{P}_{\pm i}$

$|i|$

On

$\mathscr{P}_{i}$the Iwasawa decomposition reads(4.1)$\hat{\Phi}=\hat{U}0*\hat{B}+$where$\hat{U}\in\Lambda G$

and the middle term

$c\theta$,

is a certain unique special element.

The essential idea, first used in [4], is as follows: suppose that, at some point$\mathfrak{H}(zo)=oe$(The possible terms$\hat{U}(zo)$and$\hat{B}_{+}(zo)$$zo$, we have.

in (4.1) do not affect the following). Now set.
, which is in the big cell. The big cell is an open set, soThentakes values in the big
cell on a neighbourhood of . Thus, around , one has an Iwasawa decomposition$\hat{\Phi}_{\omega}=\hat{\Phi}\text{の_{}i}^{-1}$$\Phi_{\omega}(zo)=I$$\mathfrak{H}_{co}(z)$

$z0$

$z0$

$\hat{\Phi}_{\omega}=\hat{F}_{\omega}\hat{B}+\cdot$

This was used to analyze the surface as
is encountered in [4], and it $mms$ out that finite singu(and,always)and the surface always blows up at $g_{-1}.3$
larities occur onlyat$\mathscr{P}_{\pm 1}$$\mathscr{P}_{1}$Now the important, and not obvious, thing is that it tums out that if we let denote the extended
frame corresponding to , then, on the intersection of the sets where both and are defined, which
is an open dense set in the domain, we have that the Sym-Bobenko formula agrees for both frames:$\hat{F}$$\hat{F}$$\mathfrak{H}$$\ovalbox{\tt\small REJECT}(\hat{F}_{\omega})=\ovalbox{\tt\small REJECT}(\hat{F})$$\hat{F}_{\omega}$.

This means thatcan be regarded as another kind of extended frame for the surface, which, however
is now well defined at the singular set corresponding to.
$\hat{F}_{\omega}$$\hat{\Phi}^{-1}(\mathscr{P}_{1})$$3_{The}$higher order small cells

as they have higher codimension.

$p_{j}$, for $|j|>1$ have not been analyzed, but would not be relevant to generic singularities

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GEOMETRIC CAUCHY PROBLEMS FOR CMC SURFACES IN$\mathbb{R}^{2,1}$

FIGURE 2. Spacelike CMC surfaces with prescribed singularities. Left: swallowtail.
Right: cuspidal cross cap

This singularframe was used by the author in [2] to solve the singular Bj\"orling problem for spacelike CMC surfaces. It is not immediately clear how to define the singular frame from the geometric
Cauchy data, because the definition ofis not geometric but rather comes algebraically from theholomorphic frame via an Iwasawa decomposition. Finding a way to do this was a major issue in thiswork.
$\hat{F}_{\omega}$

The solution of the geometric Cauchy problem for timelike CMC surfaces, also depends on theidea of translating the data into the $(Birkhof0$ big cell and working with a ’singular“ frame. This willappear in a forthcoming article, joint with M Svensson [6].
$|$FIGURE 3. A timelike CMC surface with a cuspidal cross cap singularity. Geometric
Cauchy data: $s(v)=2+0.2v^{2},$ $t(v)=v,$ constant.
$\theta’$92
DAVID BRANDER

For both the timelike and spacelike cases, the singular geometric Cauchy constmction is used tofind the generic singularities of the generalized surfaces define there. In both cases, the generic nondegenerate singularities tum out to be cuspidal edges, swallowtails and cuspidal cross caps.

5. NUMERICAL IMPLEMENTATIONSThe solutions to all of these geometric Cauchy problems can be computed using numerical implefor the spacelike
mentations of the DPW method. The DPW method takes thecase, and the pair
for the timelike case, integrates these, performs a Birkhoff or
Iwasawa decomposition and then applies the Sym-Bobenko formula. All of this can be implemented
numerically. The essential point now is that our solutions for the geometric Cauchy problem give
formulae for these potentials directly from the geometry Cauchy data, and hence can be computed.
$t\prime potentials’’,\hat{\Phi}^{-1}d\hat{\Phi}$$(\hat{X}^{-1}d\hat{X},\hat{Y}^{-1}df)$

FIGURE 4. The singular curve is always a null curve for both timelike and spacelike CMC surfaces. For smooth timelike CMC surfaces, a null curve is characteristic.
However this does not apply to singular curves: the curve is not, in general, characteristic. If the singular curve characteristic, then the singularity is always a straight
line, and the geometric Cauchy problem has infinitely many solutions. Two examples
are shown here.
$is$

The images in this article are from the author’s own implementation, written in Matlab. The noncharacteristic singular geometric Cauchy problem can always be expressed as the problem of finding$f(u,v)$ with$f_{v}=s(-e0+\cos(\theta)e_{1}+\sin(\theta)e_{2})$,$f_{u}=t(-e_{0}+\cos(\theta)e_{1}+\sin(\theta)e_{2})$.

. If all functions are
The DPW potentials are given in [6] in terms of the functions $s(v),$ $t(v)$ and$s\neq\pmt$
non-zero andthen the surface has a cuspidal edge. If vanishes to first order, a swallowtail. If$\theta’(v)$

$s$

93
GEOMETRIC CAUCHY PROBLEMS FOR CMC SURFACES IN$t$$\mathbb{R}^{2,1}$

vanishes to first order, a cuspidal cross cap. See Figures 1 and 3.
Examples of characteristic singular curves are given in Figure 4.

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3. D Brander and JF Dorfmeister, The Bjorling problem for non-minimal constant mean curvature surfaces, Comm. Anal.
Geom. 18 (2010), 171-194.
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Timelike constant mean curvature surfaces with singularities, forthcoming.
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Math. Anal. Appl 377 (2011), 481-494.
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DEPARTMENT OF MATHEMATICS, MATEMATIKTORVET, BUILDING 303 S, TECHNICAL UNIVERSITY OF DENMARK, DK-2800 KGS. LYNGBY, DENMARK

E-mail address: D.Brander@mat.dtu.dk