Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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168 14. Nonlinear Elliptic Equations

9. Let M be a two-dimensional minimal submanifold of S

, with its standard metric.

Assume M is diffeomorphic to S

. Show that M must be a “great sphere” in S

(Hint:ByExercise7,

II is a symmetric trace free tensor of divergence zero; that is,

II belongs to

V Dfu 2 C

.M; S



/ W div u D 0g;

a space introduced in (10.47) of Chap. 10. As noted there, when M is a Riemann

surface, V  O. ˝ /. By Corollary 9.4 of Chap. 10, O. ˝ / D 0 when M has

genus g D 0:)

10. Prove Lemma 6.11.

6B. Second variation of area

In this appendix to  6, we take up a computation of the second variation of the

area integral, and some implications, for a family of manifolds of dimension k,

immersed in a Riemannian manifold Y .First,wetakeY D R

and suppose the

family is given by X.s; u/ D X

.u/ C s.u/,asin(6.1)–(6.5).

Suppose, as in the computation (6.2)–(6.5), that k@

^^@

kD1

on M , while E

D @

form an orthonormal basis of T

M , for a given point

x 2 M . Then, extending (6.3), we have

.s/ D(6b.1)

j D1

X ^^@

 ^^@

X; @

X ^^@

d u

du

Consequently, A

.0/ will be the integral with respect to du

du

of a sum of

three terms:



i;j

^^@

 ^^@

^^@



^^@

 ^^@

^^@

C 2

i<j

^^@

 ^^@

^^@

i;j

^^@

 ^^@

^^@

 ^^@

(6b.2)

Let us write

(6b.3) A



;

6B. Second variation of area 169

with E

D @

as before. Then, as in (6.4), the ﬁrst sum in (6b.2) is equal to

(6b.4) 

i;j



Let us move to the last sum in (6b.2). We use the Weingarten formula @

 D

  A



, to write this sum as

(6b.5)

i;j



i;j

; r



;

at x. Note that the ﬁrst sum in (6b.5) cancels (6b.4), while the last sum in (6b.5)

can be written as kr

k

. Here, r

is the connection induced on the normal

bundle of M .

Now we look at the middle term in (6b.2), namely,

(6b.6) 2

i<j

`;m



^^E

;

at x,whereE

appears in the ith slot and E

appears in the j th slot in the k-fold

wedge product. This is equal to

(6b.7) 2

i<j





 a





D 2 Tr ƒ



;

at x. Thus we have

(6b.8) A

.0/ D

k

C 2 Tr ƒ



dA.x/:

If M is a hypersurface of R

,andwetake D fN,whereN is a unit normal

ﬁeld, then kr

k

Dkrf k

and (6b.7) is equal to

(6b.9) 2

i<j

R.E

D Sf

;

by the Theorema Egregium,whereS is the scalar curvature of M . Consequently,

if M  R

is a hypersurface (with boundary), and the hypersurfaces M

are given

by (6.6), with area integral (6.2), then

(6b.10) A

.0/ D

krf k

C S.x/f

dA.x/:

170 14. Nonlinear Elliptic Equations

Recall that when dim M D 2,soM  R

(6b.11) S D 2K;

where K is the Gauss curvature, which is  0 whenever M is a minimal surface

in R

If M has general codimension in R

, we can rewrite (6b.8) using the identity

(6b.12) 2 Tr ƒ



D .Tr A



kA



;

where kA



k denotes the Hilbert–Schmidt norm of A



,thatis,



D Tr.A





Recalling (6.13), if k D dim M ,weget

(6b.13) A

.0/ D

k

kA



C k

hH.x/; i

dA.x/:

Of course, the last term in the integrand vanishes for all compactly supported

ﬁelds  normal to M when M is a minimal submanifold of R

We next suppose the family of manifolds M

is contained in a manifold Y 

. Hence, as before, instead of X.s; u/ D X

.u/Cs.u/, we require @

X.s; u/ D

.s;u/ to be tangent to Y .WetakeX.0; u/ D X

.u/.Then(6b.1) holds, and we

need to add to (6b.2) the following term, in order to compute A

.0/:

(6b.14)

ˆ D

j D1

^^@

 ^^@

^^@

;

 D @

 D @

If, as before, @

D E

form an orthonormal basis of T

M ,foragivenx 2 M ,

then

(6b.15) ˆ D

j D1

; E

i; at x:

Now, given the compactly supported ﬁeld .0;u/, tangent to Y and normal to

M , let us suppose that, for each u;

.s/ D X.s; u/ is a constant-speed geodesic

in Y , such that 

.0/ D .0;u/. Thus  D 

.0/ is normal to Y , and, by the

Weingarten formula for M  R

(6b.16) @

 Dr

  A



;

6B. Second variation of area 171

at x,wherer

is the connection on the normal bundle to M  R

and A is as

before the Weingarten map for M  R

. Thus

(6b.17) ˆ D



iDTr A



DkhH.x/; i;

where k D dim M .

If we suppose M is a minimal submanifold of Y ,thenH .x/ is normal to

Y , so, for any compactly supported ﬁeld , normal to M and tangent to Y ,the

computationss (6b.13) supplemented by (6b.14)–(6b.17)gives

(6b.18) A

.0/ D

k

kA



 khH.x/; i

dA.x/:

Recall that A



is the Weingarten map of M  R

We prefer to use B



, the Weingarten map of M  Y . It is readily veriﬁed that

(6b.19) A



D B



2 End T

if  2 T

Y and  ? T

M ; see Exercise 13 in  4 of Appendix C. Thus in (6b.18)

we can simply replace kA



by kB



. Also recall that r

in (6b.18)isthe

connection on the normal bundle to M  R

. We prefer to use the connection

on the normal bundle to M  Y , which we denote by r

. To relate these two

objects, we use the identities

(6b.20)

 Dr

  A



 D

 C II

;/;

 Dr

  B



;

where

r denotes the covariant derivative on Y ,andII

is the second fundamen-

tal form of Y  R

.Inviewof(6b.19), we obtain

(6b.21) r

 Dr

 C II

;/;

a sum of terms tangent to Y and normal to Y , respectively. Hence

(6b.22) kr

k

Dkr

k

kII

;/k

Thus we can rewrite (6b.18)as

(6b.23) A

.0/ D

k

kB



kII

;/k

 Tr A



dA.x/:

172 14. Nonlinear Elliptic Equations

We want to replace the last two terms in this integrand by a quantity deﬁned

intrinsically by M

 Y , not by the way Y is imbedded in R

.NowTrA



hII

/; i,whereII

is the second fundamental form of M  R

On the other hand, it is easily veriﬁed that

(6b.24)  D 

.0/ D II

.; /:

Thus the last two terms in the integrand sum to

(6b.25) ‰ D

kII

;/k

hII

.; /; II

We can replace II

/ by II

/ here, since these two objects have

the same component normal to Y . Then Gauss’ formula implies

(6b.26) ‰ D

.; E

/; E

where R

is the Riemann curvature tensor of Y .WedeﬁneR 2 End N

M ,

where N.M/ is the normal bundle of N  Y ,by

(6b.27) h

R./; iD

.; E

/; E

at x,wherefE

g is an orthonormal basis of T

M . It follows easily that this is

independent of the choice of such an orthonormal basis.

Our calculation of A

.0/ becomes

(6b.28) A

.0/ D

k

kB



R./; 

dA.x/

when M is a minimal submanifold of Y ,wherer

is the connection on the normal

bundle to M  Y , B is the Weingarten map for M  Y ,and

R is deﬁned

by (6b.27). If we deﬁne a second-order differential operator L

and a zero-order

operator B on C



M; N.M /



(6b.29) L

 D .r



; hB./; iDTr.B







respectively, we can write this as

(6b.30) A

.0/ D .L;/

.M /

; L D L

  B./ C R./:

We emphasize that these formulas, and the ones below, for A

.0/ are valid for

immersed minimal submanifolds of Y as well as for imbedded submanifolds.

6B. Second variation of area 173

Suppose that M has codimension 1 in Y and that Y and M are orientable.

Complete the basis fE

g of T

M to an orthonormal basis

W 1  j  k C 1g

of T

Y . In this case, E

kC1

.x/ and .x/ are parallel, so

.; E

kC1

/; E

kC1

iD0:

Thus (6b.27) becomes

(6b.31)

R./ DRic

 if dim Y D dim M C 1;

where Ric

denotes the Ricci tensor of Y . In such a case, taking  D fE

kC1

f,where is a unit normal ﬁeld to M , tangent to Y , we obtain

(6b.32)

.0/ D

krf k







ChRic

; i



jf j

dA.x/

D .Lf; f /

.M /

;

where

(6b.33) Lf Df C 'f; ' DkB



hRic

; i:

We can express ' in a different form, noting that

(6b.34) hRic

; iDS



j D1

hRic

where S

is the scalar curvature of Y . From Gauss’ formula we readily obtain,

for general M  Y of any codimension,

(6b.35)

hRic

iDhR

;/;E

iChRic

kII.E

 k

;II.E

;

where II denotes the second fundamental form of M  Y . Summing over 1 

j  k,whenM has codimension 1 in Y ,and is a unit normal to M ,weget

(6b.36) 2hRic

; iDS

 S

kB



CkH

174 14. Nonlinear Elliptic Equations

If M is a minimal submanifold of Y of codimension 1, this implies that

(6b.37)

' D

 S

/ 



 S

/ C Tr ƒ



We also note that when dim M D 2 and dim Y D 3, then, for x 2 M ,

(6b.38) Tr ƒ



.x/ D K

.x/  K

M/;

where K

D .1=2/S

is the Gauss curvature of M and K

M/ is the

sectional curvature of Y , along the plane T

M .

We consider another special case, where dim M D 1.Wehaveh

R./; iD

jj

.…

M

/,whereK

.…

M

/ is the sectional curvature of Y along the plane

in T

Y spanned by T

M and . In this case, to say M is minimal is to say it is

a geodesic; hence B



D 0 and r

 D

,where

r is the covariant derivative

on Y ,andT is a unit tangent vector to M . Thus (6b.28) becomes the familiar

formula for the second variation of arc length for a geodesic:

(6b.39) `

.0/ D



k

jj

.…



ds;

wherewehaveused instead of M to denote the curve, and also ` instead of A

and ds instead of dA, to denote arc length.

The operators L and L are second-order elliptic operators that are self-adjoint,

with domain H

.M /,ifM is compact and without boundary, and with domain

.M / \ H

.M /,ifM is compact with boundary. In such cases, the spectra of

these operators consist of eigenvalues 

%C1.IfM is not compact, but B and

R are bounded, we can use the Friedrichs method to deﬁne self-adjoint extensions

L and L, which might have continuous spectrum.

We say a minimal submanifold M  Y is stable if A

.0/  0 for all smooth,

compactly supported variations , normal to M (and vanishing on @M ). Thus the

condition that M be stable is that the spectrum of L (equivalently, of L, if codim

M D 1) be contained in Œ0; 1/. In particular, if M is actually area minimizing

with respect to small perturbations, leaving @M ﬁxed (which we will just call

“area minimizing”), then it must be stable, so

(6b.40) M area minimizing H) spec L  Œ0; 1/:

The second variational formulas above provide necessary conditions for a

minimal immersed submanifold to be stable. For example, suppose M is a bound-

aryless, codimension-1 minimal submanifold of Y , and both are orientable. Then

we can take f D 1 in (6b.32), to get

6B. Second variation of area 175

(6b.41) M stable H)





ChRic

; i



dA  0:

If dim M D 2 and dim Y D 3, then, by (6b.37), we have

(6b.42) M stable H)





C S

 2K



dA  0:

In this case, if M has genus g, the Gauss–Bonnet theorem implies that

dA D 4.1  g/,so

(6b.43) M stable H)





C S



dA  8.1  g/:

This implies some nonexistence results.

Proposition 6b.1. Assume that Y is a compact, oriented Riemannian manifold

and that Y and M have no boundary.

If the Ricci tensor Ric

is positive-deﬁnite, then Y cannot contain any com-

pact, oriented, area-minimizing immersed hypersurface M .IfRic

is positive-

semideﬁnite, then any such M would have to be totally geodesic in Y .

Now assume dim Y D 3.IfY has scalar curvature S

>0everywhere, then

Y cannot contain any compact, oriented, area-minimizing immersed surface M

of genus g  1.

More generally, if S

 0 everywhere, and if M is a compact, oriented, im-

mersed hypersurface of genus g  1, then for M to be area minimizing it is

necessary that g D 1 and that M be totally geodesic in Y .

R. Schoen and S.-T. Yau [SY] obtained topological consequences for a com-

pact, oriented 3-manifold Y from this together with the following existence

theorem. Suppose M is a compact, oriented surface of genus g  1, and sup-

pose the fundamental group 

.Y / contains a subgroup isomorphic to 

.M /.

Then, given any Riemannian metric on Y , there is a smooth immersion of M

into Y which is area minimizing with respect to small perturbations, as shown in

[SY]. It follows that if Y is a compact, oriented Riemannian 3-manifold, whose

scalar curvature S

is everywhere positive, then 

.Y / cannot have a subgroup

isomorphic to 

.M /, for any compact Riemann surface M of genus g  1.

We will not prove the result of [SY] on the existence of such minimal immer-

sions. Instead, we demonstrate a topological result, due to Synge, of a similar

ﬂavor but simpler to prove. It makes use of the second variational formula (6b.39)

for arc length.

Proposition 6b.2. If Y is a compact, oriented Riemannian manifold of even

dimension, with positive sectional curvature everywhere, then Y is simply

connected.

176 14. Nonlinear Elliptic Equations

Proof. It is a simple consequence of Ascoli’s theorem that there is a length-

minimizing, closed geodesic in each homotopy class of maps from S

to Y . Thus,

if 

.Y / ¤ 0, there is a nontrivial stable geodesic,  .Pickp 2 ; 

normal to

 at p (i.e., 

2 N

./), and parallel translate  about , obtaining 

2 N

./

after one circuit. This deﬁnes an orientation-preserving, orthogonal, linear trans-

formation  W N

 ! N

.IfY has dimension 2k,thenN

 has dimension

2k  1,so 2 SO.2k  1/. It follows that  must have an eigenvector in N

,

with eigenvalue 1. Thus we get a nontrivial, smooth section  of N./ which is

parallel over ,so(6b.39) implies

(6b.44)



.…



/ ds  0:

If K

.…/ > 0 everywhere, this is impossible.

One might compare these results with Proposition 4.7 of Chap. 10, which states

that if Y is a compact Riemannian manifold and Ric

>0, then the ﬁrst coho-

mology group H

.Y / D 0.

7. The minimal surface equation

We now study a nonlinear PDE for functions whose graphs are minimal surfaces.

We begin with a formula for the mean curvature of a hypersurface M  R

nC1

deﬁned by u.x/ D c,whereru ¤ 0 on M .IfN Dru=jruj,wehavethe

formula

(7.1) hA

X; Y iDjruj

1

u/.X; Y /;

for X; Y 2 T

M ,asshownin(4.26) of Appendix C. To take the trace of the

restriction of D

u to T

M , we merely take Tr.D

u/  D

u.N; N /. Of course,

Tr.D

u/ D u. Thus, for x 2 M ,

(7.2) Tr A

.x/ Djru.x/j

1

u jruj

2

u.ru; ru/

Suppose now that M is given by the equation

nC1

D f.x

/; x

D .x

;:::;x

Thus we take u.x/ D x

nC1

 f.x

/, with ru D .rf; 1/. We obtain for the

mean curvature the formula

(7.3) nH.x/ D

hrf i

f  D

f.rf; rf/

D M.f /;

7. The minimal surface equation 177

where hrf i

D 1 Cjrf.x

. Written out more fully, the quantity in brackets

above is

(7.4)



1 Cjrf j



f 

i;j

M.f /:

Thus the equation stating that a hypersurface x

nC1

D f.x

/ be a minimal sub-

manifold of R

nC1

(7.5)

M.f / D 0:

In case n D 2, we have the minimal surface equation, which can also be written as

(7.6)



1 Cj@

f j



f  2



f  @



f C



1 Cj@

f j



f D 0:

It can be veriﬁed that this PDE also holds for a minimal surface in R

described

by x

D f.x

/,wherex

D .x

;:::;x

/,if(7.6)isregardedasasystemofk

equations in k unknowns, k D n  2,and.@

f  @

f/ is the dot product of

-valued functions. We continue to denote the left side of (7.6)by

M.f /.

Proposition 6.12 can be translated immediately into the following existence

theorem for the minimal surface equation:

Proposition 7.1. Let O be a bounded, convex domain in R

with smooth bound-

ary. Let g 2 C

.@O; R

/ be given. Then there is a solution

(7.7) u 2 C

.O; R

/ \ C.O; R

to the boundary problem

(7.8)

M.u/ D 0; u

D g:

When k D 1, we also have uniqueness, as a consequence of the following:

Proposition 7.2. Let O be any bounded domain in R

.Letu

2 C

.O/ \C.O/

be real-valued solutions to

(7.9)

M.u

/ D 0; u

D g

on @O;

for j D 1; 2.Then

(7.10) g

 g

on @O H) u

 u

on O:

Proof. We prove this by deriving a linear PDE for the difference v D u

u

and

applying the maximum principle. In general,

(7.11) ˆ.u

/  ˆ.u

/ D Lv; L D

Dˆ



u

C .1  /u



d: