GEOMETRIC CAUCHY PROBLEMS FOR SPACELIKE AND TIMELIKE CMC SURFACES IN $\mathbb{R}^{2,1}$ (Differential Geometry of Submanifolds)

GEOMETRIC CAUCHY

PROBLEMS FOR SPACELIKE AND TIMELIKE CMC

SURFACES

IN $\mathbb{R}^{2,1}$ DAVIDBRANDER

ABSTRACT. Wediscussrecentworkoftheauthorandcollaborators ongeneralizations of$Bj\ddot{o}rling$’s

classicalproblemto thecaseofconstantnon-zero meancurvaturesurfaces in$2+1$-dimensional

space-time. The aim istogive anoverview, and to poimoutthe similaritiesanddifferences between the

twocasesoftimelike and spacelikesurfaces. Applications to theconstructionofCMC surfaceswith

prescribed singularitiesarealsodescribed.

1.

INTRODUCTION

This article discusses resultsof work of the author andcollaborators, [2], [5] and[6],

on

the

gener-alizationofBj\"orling’s problem for minimalsurfacestothe

case

ofconstant

non-zero mean

curvature

(CMC) surfaces in the 3-dimensional Minkowski

space

$\mathbb{R}^{2,1}$

.

The classical Bj\"orling

problem is to

findthe minimal surfacein$\mathbb{R}^{3}$

whichcontains

a

prescribed

curve

with the surface tangent distribution

prescribed along the

curve.

It has

a

uniquesolutionviaanalyticextensionbecause thedata is enough

todetermine the(holomorphic)Weierstrass dataalongthe

curve.

The generalization ofthisproblem

toother surfaceclassesbesides minimal is sometimescalled the geometric Cauchy problem.

The problem has also been considered for

zero mean

curvaturesurfacesin $\mathbb{R}^{2,1}$

, both spacelike[1]

andtimelike [7]. Thereis

a

substantial difference between the spacelike and timelike cases,

as

the

formeris

an

ellipticproblem and has

a

Weierstrass representation intermsofholomorphicdata,

as

in

the

case

of minimal surfacesin $\mathbb{R}^{3}$,

whilst thetimelike

case

is hyperbolicandhas

a

d’Alemberttype

solutioninterms of twofunctionsof

one

variable each. Thetimelike

case

iscomplicatedbythe fact

that there is

no

uniquesolutiontothe Bj\"orling problem if the initial

curve

ischaracteristic.

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

Weier-strass/d’Alembert _{typerepresentations}

_are

_now

_{in terms of}

_maps

_into

_a

_loop

_group

_{[9, 8], and}

ob-taining 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 be

approximated numerically, butcannotbewrittendownexplicitly,ingeneral. Therefore

some

workis

necessary

ingoing back and forth between the surface and the corresponding Weierstrass data.

The Bj\"orling problem for non-minimal CMC surfaces in $\mathbb{R}^{3}$,

was

solved by the author and $J$

Dorfmeisterin [3]. Perhaps theessential pointofthe solutionis that, although the Iwasawa

decom-position cannotbe computedexplicitly for generalWeierstrass data,

one

hasconsiderable flexibility

that theIwasawadecomposition is trivialalongtheinitial

curve

in question. This allows

one

towrite

downexplicitly the Weierstrass data corresponding togiven Bj\"orling data along

a curve.

The exact

same

argument

can

be usedto solvetheproblemforspacelike CMC surfacesin$\mathbb{R}^{2,1}$,

so

thatproblem

was

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

case.

Wediscuss thisbelowin Section

2.

Thegeometric Cauchy problemfortimelike CMC surfaces

was

solved bythe author and M

Svens-son

in[5]. Althoughthed’AlembeIt construction isratherdifferent from the Weierstrass construction

of the spacelikecase, the solutionofthe problem againdepends essentially

on

the factthat

one can

choosed‘Alembertdata suchthat the loop

group

decomposition(Birkhoffin thiscase) is trivial along

the

curve

in question. This is discussed in Section

3.

Finally thereisthe singulargeometric Cauchy problem, whicharises naturally forCMC surfaces

in$\mathbb{R}^{2,1}$

.

Both

timelike and spacelike CMC surfaces almost always havegenericsingularities. Most of

thesearise atpoints where the loop

group

decomposition has

a

middle term, andcannotbe avoided

ingeneral. An analogous situation existsforminimal/maximal _{surfaces in}$\mathbb{R}^{2,1}$,and the singularities

have beenstudied recently byvaniousauthors. In particular, the singular Bj\"orling problem, whichis

to findthe 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,

YWKim,S-EKoh and HShinin [11]and[10].

FIGURE 1. The singular geometnic Cauchy

construction

can

be used to produce

CMC surfaces with arbitraly prescribed singular

curves.

The timelike CMC surface shown hasinfinitely

many

swallowtails along

a

cuspidal edge. The first few

can

be

seen

in this plot of

a

finite domain. The geometric Cauchy data

are

$s(v)=\sin(v)$,

Thesingulargeometric Cauchy problemforspacelikeCMC surfaces issolvedbythe authorin [2]. The essential problem here isthat the$SU(1,1)$ framefor thesurface,whichis critical forsolvingthe

problemin the regularsituation of[3], blowsup

as

thesingular

curve

is approached. Theproblemis

solved by multiplying all the

maps

by

a

certainelement of theloop

group,

which

moves

all the data

intothe bigcell,thatistheset

on

whichtheIwasawadecomposition doesnothave

a

middleterm. Itis

then possible to solve the problem with

a

new

“singularframe“ whicharises fromthis idea. An

anal-ogous

device isused to solve the problem for timelikeCMC surfacesinjointwork with M Svensson

in [6]. This will be discussed in Section4.

2. SPACELIKE CMC SURFACES

For explicit details of thefollowingdiscussion,thereader isreferred to[4] and[3].

2.1.

The generalizedWeierstrass representation. The

so

called$DPW$(Dorfmeister/Pedit/Wu [9]),

or

generalized Weierstrass representation, for spacelike CMC surfaces is sketched below. We will

discuss itlocallyforsimplicity.Let$U\subset \mathbb{C}^{2}$be

an open

set.

$\bullet$ A conformalCMCimmersion

$f$

:

$Uarrow \mathbb{R}^{2,1}$ isrepresented by

an

extended

fmme

$\hat{F}$

:

$Uarrow\Lambda G$,

where $G=SU(1,1)$, and$\Lambda G$isthe

group

oftwistedloopsin$G$

.

.

The surface$f$is recoveredfrom $\hat{F}$by

a

simple formula, called the Sym-Bobenkoformula,

whichisessentially of the form$\ovalbox{\tt\small REJECT}(\hat{F})=\lambda(\partial_{\lambda}\hat{F})\hat{F}^{-1}|_{\lambda=1}$

.

.

The extended frameis

characterizedl

among

smooth

maps

into$\Lambda G$by thefact thatits

Maurer-Cartan form$\hat{F}^{-1}d\hat{F}$

can

be written:

$\hat{F}^{-1}d\hat{F}=A_{-1}\lambda^{-1}dz+\infty+\tau(A_{-1})\lambda d\overline{z}$,

where $\tau$ is the involution determiningthe real form$su(1,1)$ in $\epsilon 1(2,\mathbb{C})$, and $\alpha_{0}$ is

a

l-form

which doesnotdependon$\lambda$

.

$\bullet$ Finally (and this is the DPW part of the construction), the extended frame

can

be obtained

from

a

holomorphic

_frame

$\hat{\Phi}$

:

$Uarrow\Lambda G^{\mathbb{C}}$ by

a

pointwiseIwasawadecomposition

(2.1) $\hat{\Phi}=\hat{F}\hat{B}+$, $\hat{F}\in\Lambda G$, $\hat{B}+\in\Lambda^{+}G^{\mathbb{C}}$,

the last

group

denoting those loops in$SL(2,\mathbb{C})$which

can

beholomorphicallyextendedtothe

unitdisc.

$\bullet$ BycompaningtheMaurer-Cartanforms of$\hat{F}$and$\hat{\Phi}$

, andusingthat$\hat{B}+$has

a

Taylor expansion

in $\lambda$, itis

easy

to

deduce that$\hat{\Phi}$

mustbe holomorphic, and is

chamcterized

by its

Maurer-Cartan formbeing ofthe form

$\hat{\Phi}^{-1}d\hat{\Phi}=(\psi_{-1}\lambda^{-1}+\psi_{0}+O(\lambda))dz$

.

The holomorphic functions in the

matrices

$\psi_{i}$

are

arbitrary, apartfrom

a

regularity condition

on one

of them. This

is

the generalized Weierstmss representationfor(spacelike)

CMC

sur-faces.

2.2.

Solution oftheBjorling problem. Now the idea thatisusedin[3]to solvethe Bj\"orling problem is actually

very

simple: The factor$\hat{F}\in\Lambda \mathscr{G}$in the decomposition

2.1

is essentially unique. So

suppose

that $\hat{\Phi}(zo)$ is already AG-valued, i.e. already takes values in the real form. At such

a

point,

the

Iwasawadecomposition(2.1)

is

trivial:

$\mathfrak{H}(zo)=\hat{F}(zo)$

.

We

can use

thistoconstmct

a

special holomorphicframe from the knowledge of$\hat{F}$

along

a

curve

as

follows:

suppose we

know the value of$F$ along

some curve

in $U$,

say

$F(x+0.i)=\hat{F}_{0}(x)$ along

an

interval$J=U\cap \mathbb{R}$

.

Let

us

set

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

,

andthen extendthis holomorphicallyto

a

map

$\hat{\Phi}$

:

$Varrow\Lambda G^{\mathbb{C}}$, from

some open

subset$V$ containing $J$

.

It tums outthat this

can

be done, and

moreover

this

map

$\Phi$ has all the required

properties

of

a

holomorphicextendedRame for

a

CMC surface. Further,the corresponding extendedframe$F$

agrees

with $fi_{0}(x)$ along $J$, because the Iwasawa decomposition is trivial along this

curve.

Thus

we

have

solvedthe Bj\"orlingproblem, providedthat

we

can

construct$\hat{F}_{0}(x)$along$J$fromthe Bj\"orlingdata. It

tums outthatthis

can

be donein

a

fairly straightforward

manner

(see [3]), and

one

finally obtains

a

simple formulafor the holomorphic potential$\hat{\Phi}^{-1}d\hat{\Phi}$interms of the Bj\"orling data.

Using

a

uniquelocalcorrespondencebetweenWeierstrassdataandextendedframes,obtained via

a

normalized Birkhoff splitting, it isalso notdifficulttoshow that the Bj\"orling problemhas

a

unique

solution.

3. TIMELIKE CMC SURFACES

We tum

now

to the timelike

case.

The d’Alembert

representation

given in [8] _differs from the

Weierstrassrepresentation described in Section

2

as

follows:

$\bullet$ The extended frame this time takes values in

a

subgroup of $\Lambda SL(2,\mathbb{C})$ consisting of loops

which

are

in$G=SL(2,\mathbb{R})$ for real values of$\lambda$

.

$\bullet$ TheMaurer-Cartanform oftheextendedframe$\hat{F}(x,y)$,where$x$and$y$

are

lightlike coordinates

for$U\subset \mathbb{R}^{1,1}$,hasthe form

$\hat{F}^{-1}dfi=A_{1}\lambda$dr$+\infty+A_{-1}\lambda^{-1}dy$

.

$\bullet$ The extended frame$fi(x,y)$ is obtained from

a

pair

of

maps

$\hat{X}(x)$ and$\hat{Y}(y)$,where

and thecoefficients $\psi_{i}^{X}$ and$\psi_{i}^{Y}$

are

essentially_{$arbitral\gamma$} functions.

$\bullet$ $\hat{F}$

isobtained from$\hat{X}$ and$\hat{Y}$

byperforming

a

pointwise

_Birkhoff

splitting (3.1) $\hat{X}^{-1}(x)\hat{Y}(y)=\hat{H}_{-}(x,y)\hat{H}_{+}(x,y)$,

where$\hat{H}_{-}(x,y)\in\Lambda^{-}G$_, 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)istrivial along the

curve.

LookingattheirMaurer-Cartanforms,where the leadingtermsmust benon-vanishingfor regular surfaces,

we

can see

that$\hat{X}(x)$ will have

a

pole (in$\lambda$) at $\infty$ while $\hat{Y}(y)$ will have

a

pole at $\lambda=0$

.

Thus, the Birkhoff decomposition is unlikely to

be trivial unless

we

havesomethinglike$\hat{X}^{-1}(x)\hat{Y}(y)=$ constantin$\lambda$, andindeed

we

can

achieve this

along the

curve

$y=x$if

we can

arrange,

for example, that

$\hat{X}(v)=\hat{Y}(v)$

.

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 $\mathbb{R}^{1,1}$

,

one

can

always (with

a

possible changeoforientation)locally choose null coordinates $(x,y)$_, such thatthe

curve

is

given

as

$u=0$in_{thecoordinates $u=(x-y)/2,$$v=(x+y)/2$}

_.

Suppose

one

can

construct, from the geometric Cauchydata, the extendedframe $\hat{F}(0,v)=\hat{F}_{0}(v)$

for

a

timelikeCMC surfacealongthe

curve

$x=y=v$, i.e. $u=0$

.

(Ittums outthat

one

can

dothis).

Let

us now

set

$\hat{X}(x)=\hat{F}_{0}(x)$, $\hat{Y}(y)=\hat{F}_{0}(y)$

.

Then it is

easy

tocheck that$\hat{X}$ and$\hat{Y}$

have precisely the required form for the d’Alembertdata of

a

timelikeCMC surface, andthusgenerate such

a

surfaceusing thescheme outlined above. Along the

curve

$x=y=v$

we

have $\hat{X}(x)=\hat{Y}(y)$,

so

that, along this

curve

the Birkhoff decompositionat _(3.1)

is just$I=I.I$,where$I$is theidentity matrix. Along this curve, the extended frame for the surface

so

constmctedisgivenby(3.2)

as

$\hat{F}(0,v)=\hat{Y}(v).I=\hat{F}_{0}(v)$

.

Thus the constmcted surface solves the geometric Cauchy problem. One

can

show, again using

a

normalizedBirkhoffdecomposition,thatthe solutionis unique.

Finally, the

case

that the initial

curve

is characteristic (null), is also treated in [5]. _{The solution} is notunique, but

we

describe how to constmct all solutions. We do not treat initial

curves

which

become characteristicatisolated points.

$2_{The}$_{notation forthe last twogroupsis}

notstnictly correcthere,bm inanycase$\Lambda^{-}G$consists ofloops which extend

4.

THE SINGULAR GEOMETRICCAUCHYPROBLEM

The solution of the singular geometric Cauchy problem, treated in [2] and [6] for, respectively,

spacelike andtimelike CMC surfaces is rathercomplicatedtodescribe. As mentioned in the

intro-duction, the extended frame$\hat{F}$

blows

up

as

such

a

curve

is approached. Therefore, itisclearthatthe

solutions outlinedabovecannotbeapplied for such

a

curve.

We will notattempt to describe the solutionhere, butonly mention

a

critical idea

on

whichthe

solution is built. Let

us

consider the

case

of spacelike surfaces. As described in Section 2, the

extended frame$\beta$ (andhence

the surface)is obtainedfrom theholomorphic frame$\mathfrak{H}$

by

an

Iwasawa

decomposition

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

When the

group is

non-compact,theIwasawadecomposition is onlywritten this

way

on

an open

dense

subset ofthe loop

group,

calledthe bigcell. Therestof theloop

group

is

a

disjointunion$\bigcup_{i=1}^{\infty}\mathscr{P}_{\pm i}$ of

subvarieties,of codimensionincreasingwith $|i|$

.

On $\mathscr{P}_{i}$theIwasawa decompositionreads

(4.1) $\hat{\Phi}=\hat{U}0*\hat{B}+$,

where$\hat{U}\in\Lambda G$andthemiddleterm

$c\theta$is

a

certain

unique specialelement.

The essentialidea, first usedin [4],is

as

follows:

suppose

that,at

some

point$zo$,

we

have

$\mathfrak{H}(zo)=oe$

.

(Thepossibleterms$\hat{U}(zo)$ and$\hat{B}_{+}(zo)$ in(4.1)do not affect thefollowing). Now set

$\hat{\Phi}_{\omega}=\hat{\Phi}\text{の_{}i}^{-1}$

.

Then$\Phi_{\omega}(zo)=I$,whichis inthebigcell. Thebigcellis

an open

set,

so

$\mathfrak{H}_{co}(z)$takes valuesinthebig

cell

on

a

neighbourhoodof$z0$

.

Thus,around$z0$,

one

has

an

Iwasawadecomposition

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

This

was

usedtoanalyze thesurface

as

$\mathscr{P}_{\pm 1}$ isencountered in [4], andit$mms$outthat finite

singu-larities

occur

only(andalways)at $\mathscr{P}_{1}$,and the surface always blows

up

at$g_{-1}.3$

Nowthe

important,

andnotobvious, _thing

is

that ittums outthat if

we

let$\hat{F}$

denote theextended frame correspondingto$\mathfrak{H}$

,then,

on

the intersection of thesetswhere both$\hat{F}$

and$\hat{F}_{\omega}$

are

defined,which

is

an

open

densesetinthedomain,

we

have thatthe Sym-Bobenkoformula

agrees

for bothframes:

$\ovalbox{\tt\small REJECT}(\hat{F}_{\omega})=\ovalbox{\tt\small REJECT}(\hat{F})$

.

This

means

that$\hat{F}_{\omega}$

can

be regarded

as

another kind of extended frame for the surface,which,however

is

now

well

_defined

atthe singularsetcorrespondingto$\hat{\Phi}^{-1}(\mathscr{P}_{1})$

.

$3_{The}$_{higher ordersmallcells$p_{j}$,for$|j|>1$have notbeen analyzed, but wouldnotberelevanttogeneric singularities}

FIGURE2. SpacelikeCMC surfaces with prescribed singularities. Left: swallowtail.

Right: cuspidal

cross

cap

This singularframe

was

usedbytheauthorin [2]tosolvethesingular Bj\"orling problem for

space-likeCMC surfaces. Itis notimmediately clear how to define the singularframe from the geometric Cauchy data, because the definition of$\hat{F}_{\omega}$ is notgeometric but rather

comes

algebraically from the

holomorphic framevia

an

Iwasawadecomposition. Finding

a

way

to dothis

was a

majorissue inthis

work.

The solution ofthe geometric Cauchy problem for timelike CMC surfaces, also depends on the

idea of translatingthedataintothe$(Birkhof0$_{bigcell and}working with

a

$|$

’singular“ frame. This will

appear

in

a

forthcomingarticle,jointwith MSvensson[6].

FIGURE3. Atimelike CMCsurfacewith

a

cuspidal

cross

cap

singularity. Geometric

For both the timelike and spacelike cases, the singular

geometric

Cauchy

constmction is

usedto

find thegeneric singularities of the generalized surfacesdefinethere. In both cases, the

generic

non-degenerate singularities tum out tobecuspidal edges, swallowtailsandcuspidal

cross

caps.

5. NUMERICALIMPLEMENTATIONS

The solutionstoall ofthesegeometric Cauchy problems

can

be computed usingnumerical

imple-mentations ofthe DPW method. The DPW method takes the $t\prime potentials’’,\hat{\Phi}^{-1}d\hat{\Phi}$forthe spacelike

case, and the

pair

$(\hat{X}^{-1}d\hat{X},\hat{Y}^{-1}df)$ 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 formulaeforthesepotentialsdirectly from the geometryCauchydata,and hence

can

becomputed.

FIGURE 4. The singular

curve

is always

a

null

curve

for both timelike and

space-likeCMCsurfaces. For smooth timelikeCMC surfaces,

a

null

curve

is characteristic.

Howeverthis does not apply to singular

curves:

the

curve

isnot, in general, charac-teristic. If thesingular

curve

$is$characteristic, then thesingularityisalways

a

straight

line,and thegeometric Cauchy problemhasinfinitely

many

solutions. Twoexamples

are

shownhere.

Theimages in this article

are

fromthe author’sownimplementation, writtenin Matlab. The

non-characteristic singulargeometricCauchy problem

can

always be expressed

as

the problem of finding

$f(u,v)$ _with

$f_{v}=s(-e0+\cos(\theta)e_{1}+\sin(\theta)e_{2})$

,

$f_{u}=t(-e_{0}+\cos(\theta)e_{1}+\sin(\theta)e_{2})$

.

The DPW potentials

are

given in[6] interms ofthe functions$s(v),$$t(v)$ and$\theta’(v)$

.

If all functions

are

$t$ vanishestofirstorder,

a

cuspidal

cross

cap.

See Figures 1 and3.

Examplesofcharacteristic singular

curves

are

giveninFigure4.

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DEPARTMENT OF MATHEMATICS, MATEMATIKTORVET, BUILDING 303 S, TECHNICAL UNIVERSITY OF

DEN-MARK,DK-2800KGS. LYNGBY,DENMARK