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A uniqueness result in PDE's and parallel meancurvature immersions in euclidean spaceAndrea Ratto a , Marco Rigoli b & Alberto G. Setti ba Dipartimento di Matematica , Universitá della Calabria , Arcavacata di Rende, CS, 87036,Italyb Dipartimento di Matematica , Universitá di milano via saldini 50 , milano, 20133, ItalyPublished online: 29 May 2007.

To cite this article: Andrea Ratto , Marco Rigoli & Alberto G. Setti (1996) A uniqueness result in PDE's and parallel meancurvature immersions in euclidean space, Complex Variables, Theory and Application: An International Journal: AnInternational Journal, 30:3, 221-233

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A Uniqueness Result in PDE's and Parallel Mean Curvature Immersions in Euclidean Space

ANDREA RATTO Dipartimento di Matematica, Universitd della Calabria, 87036 Arcavacata di Rende (CS), Italy

MARC0 RlGOLl and ALBERT0 G. SETTl Dipartimento di Matematica, Universita di Milano via Saldini 50, 20133 Milano, ltaly

It is proven that under suitable curvature assumptions, the image of the spherical Gauss map of a con- stant mean curvature hypersurface in Rn cannot be contained in any closed hemisphere, unless it lies in a great sphere. The proof depends on a uniqueness result for non-negative solutions of the generalised Emden-Fowler equation Au + K(x)u + R(x)uu = 0 on a complete Riemannian manifold (M,g). An application of the latter result to stable minimal hypersurfaces is also given. By applying a variant of the maximum principle developed in [9], it is also shown that the Gauss map of a parallel mean curvature immersion of arbitrary codimension, cannot be obtained in a spherical cap of specified radius.

AMS No. 53C21,53C42,35B05 Communicated: S. Krantz (Received May, 1995)

1. INTRODUCTION

Throughout the paper we will denote by (M,g) a complete, non-compact, con- nected Riemannian manifold of dimension m of scalar curvature s ( ~ ) , and by r(x) the distance function from a fixed point p E M. Given the isometric immersion f : (Mm,g) + R ~ " , let vf : ( M , g ) -+ Sm be its associated spherical Gauss map.

The purpose of this note is to prove the following:

THEOREM 1.1 Assume that M is oriented and that f has parallel mean curvature H, and that for some constants B > 0 and a > -2,

RiccM 2 -(m - I ) B ~ ( I + r)". (1.1)

Suppose that,

where (m - 1 ) 2 ~ 2

4 i f a > - 2

[(m - 1) A - 112 4

if a = - 2 ,

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222 A. RATTO, M. RIGOLI AND A. G. SETTI

and if a>O

A = A , = (1.4)

Then either the image of vf is not contained in any closed hemisphere, or it lies entirely in a great sphere.

Theorem 1.1 is related to a recent result obtained by Leung [7], using different methods. He shows that under the assumption that s(x) is bounded below, vf (M) cannot be contained in a spherical cap of an appropriate radius.

While Leung only assumes a lower bound for s(x), the Gauss equations imply that Ricci is also bounded from below. Moreover, his techniques cannot be pushed to cover the case where the spherical cap is a full hemisphere. Furthermore, we remark that Theorem 1.1 covers the case that the curvature is not bounded below.

The case of a parallel mean curvature immersion f : (M,g) + Rn of arbitrary codimension is treated in Theorem 1.2 below. In order to state our result, we need to introduce some additional notation.

We assume that M is oriented. Let 7f : (M,g) -+ G(n, m) denote the Gauss map of f , where G(n, m) is the Grassmannian of oriented m-planes in Rn. Identifying m planes in G(n, m) with unit multivectors, we can define the cosine of the angle 0 between yf and the m-plane V = Vl A V2 A . . . A Vm as

where {e,) is a local orthonormal frame along f , with ei tangent to M for i = 1,. . ., m, and el A e2 A . . . A em giving the orientation, and e, orthogonal to M for a = m + 1,. . ., n (in short a local Darboux frame along f). Denoting by sG)-' the unit sphere in Am(Rn), we say that yf (p) is contained in the open (resp. closed) spherical cap of radius Oo (0 5 O0 < T ) centered at Vl A V2 A . . . A Vm if and only if cos(0) > cos(Bo) (resp. cos(0) 2 cos(Oo)). Then we have

THEOREM 1.2 Let f : (M,g) -, Rn be a complete, oriented, isometric immersion with parallel mean curvature H . Assume that for some constants B > 0, d > 0, and a > -2

RiccM > -(m - I ) B ~ ( I + r),, (1.5)

and s(x) 5 m 2 / ~ l 2 - d2(1 + r)(a/2)-1 for r >> 1. (1.6)

Then yf (M) is not contained in any closed spherical cap in SK)-' of radius

J 2(n - m - 1) 0 < arccos

3 n - 3 m - 2 '

In the codimension 1 case (n = m + I), Theorem 1.2 asserts that under the stated curvature hypotheses the image of the Gauss map cannot lie in any closed spherical cap properly contained in an open hemisphere. By contrast, Theorem 1.1 in partic- ular implies that under the assumptions (1.1) and (1.2) the image of the Gauss map is not contained in any open hemisphere. The difference between the conclusions

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A UNIQUENESS RESULT IN PDE'S 223

of the two theorems can be better appreciated by examining the proofs. Let u be the function on M defined by u = (vf, V), where V is a fixed unit vector in R"". Then u satisfies a linear elliptic equation (see (3.2) below). The argument of The- orem 1.1 zmounts to showing that if u is non-negative then it vanishes identically. On the other hand, the proof of Theorem 1.2 only shows that inf u < 0. This is in accordance with the fact that the curvature assumption (1.6) is weaker than the cor- responding hypothesis (1.2) of Theorem 1.1 (Note that if a = -2, (1.2) and (1.6) are comparable, but in the latter the value of the constant d is not specified).

The paper is organised as follows: In $2 we prove a uniqueness result for non- negative solutions of a generalised Emden-Fowler equation on complete manifolds (Theorem 2.1), which may also be of independent interest. In Remark 4.3 below we indicate an example where assumptions (1.2), (1.3), and (1.4) are optimal. In this section we also establish some sharp estimates for the Laplacian of r(x) under the curvature conditions (1.1) which generalise and unify previous results of [I, 4, 91.

Section 3 contains the proof of Theorem 1.1, and another geometrical application of Theorem 2.1 to the problem of stability of minimal hypersurfaces of Rm+l (see Theorem 3.2).

In Section 4 we consider the case of arbitrary codimension. Using ideas of Chern, do Carmo, and Kobayashi [2] and R. Reilly [8] we first show that the cosine of the angle between a fixed plane in G(n,m) and the Gauss map of a parallel mean curvature isometric immersion satisfies a (generally non-linear) partial differential inequality (Lemma 4.1). Using a slight modification of the maximum principle ob- tained in [9] (Lemma 4.2), we then complete the proof of Theorem 1.2.

2. EMDEN-FOWLER EQUATIONS ON COMPLETE MANIFOLDS

In this section we consider the equation

on (M,g). Assuming the non-negativity of R(x), our uniqueness result reduces to showing that the Schrodinger operator A + K(x) has negative first Dirichlet eigen- value on some bounded domain. This is obtained by relating the behaviours of A r and K(x) at infinity. Expressing this relationship in terms of curvature we obtain:

THEOREM 2.1 Suppose that the Ricci temor of (M,g) satisfies (1.1) with a: > -2. For K E cO(M) define

and assume that

K(r ) > 0 on [O,cm), and K(r ) > Cr", for r >> 1, (2.3)

with C satisfying (1.3). If R(x) 2 0 and u is a non-negative solution of (2.1) on M , then u = 0.

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224 A. RATTO. M. RIGOLI AND A. G. SETTI

For the proof we will need some lemmata. The first one is a non-existence result for positive solutions of second order ODE'S. It is actually part of the Hille-Nehari oscillation criterion, and for the proof we refer to Swanson [lo], p. 44 ff.

&MMA 2.2 Let p E c'([o,w)), p > 0, and assume that 00

liminf t 1 p(s)ds > a. t-w

Then 'da > 0, the differential equation

p " + p p = o

has no positive solution in [a, m).

LEMMA 2.3 Assume that ( M , g ) satisfies the curvature condition (1.1), and set

where A is defined in (1.4). Then the inequality

holds pointwise within the cut locus of p , and weak& on M.

Proof It is enough to prove that (2.6) holds pointwise within the cut locus of p (cf. Yau [I l l , Appendix). By the Laplacian comparison theorem,

a' Ar 5 (m- 1)-

a (2.7)

where a > 0 is the solution of the initial value problem

By the second Sturm Comparison Theorem,

where li/ is a positive subsolution of (2.8), i.e.

First we consider the case a > -2, and we look for a positive solution + of (2.9) of the form (2.5), namely

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A UNIQUENESS RESULT I N PDE'S 225

We need to verify that, if A is as in (1.4), then I) = h is a subsolution of (2.9). It is clear that h(0) = 0 and h'(0) = 1, and by direct computation

1 2 A h l ' ( r ) = [f ( 1 + r)"12-' + A ( 1 + r )@] - { - 1 + erp r ) ( 2 + @ ) / 2 - 1)

A

It follows easily that the condition that h be a subsolution is implied by

or equivalently

(A' - 8') + A?(I + r)-@l2-l 2 0. 2

(2.10)

Thus if a 2 0, we can take A = B , and (2.9) holds. When -2 < a < 0, the left hand side of (2.10) is increasing with r , and its infimum

is - B2 + aAI2 . The smallest A for which that is non-negative is

As for a = -2, we look for solutions of (2.9) of the form

1 h = -[(I + r)*- I ] .

A

The methods used above show that the smallest A for which (2.9) holds is

A = L + L W . 2 2

Remarks 2.4 If a 2 0, we could satisfy (2.9) with the choice

Observe that

LEMMA 2.5 Let li, be a solution of the initial value problem

where h is defined in (2.5), and K ( r ) satisfies (2.3). Then ?C, has a first zero at some Ro > 0.

Proof Assume by contradiction that I) > 0 in [ O , c o ) . We rewrite (2.11) in the form

{a(r>*'(r))' + b(r)*(r) = 0

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226 A. RATTO, M. RIGOLI AND A. G. SETTI

with a(r) = hm-'(r) and b(r) = hm-l(r)K(r).

It follows from the definition of h that l / a E ~ ' ( + m ) , so that we can perform the standard change of variables

Then K : (0, m ) + (0, m ) is strictly increasing, and, setting

a direct computation shows that y satisfies

with

An application of Cauchy's Theorem, together with the definition of K, and (2.5) shows that

so that (1.3) and (1.4) imply that the right hand side of (2.14) is strictly greater than 114. By Lemma 2.2, (2.13) cannot have an everywhere positive solution, thus contradicting the assumed positivity of $, and completing the proof of the Lemma.

rn After this preparation we can proceed to the

Proof of Theorem 2.1 We begin by showing that there exists a geodesic ball BR,(p), and a Lipschitz continuous radial function v on BR,(p) such that

with K defined by (2.2). Since v has the form v = $ o r , it suffices to choose $ appropriately. Observe that

Av = $"(I) + $'(r) Ar, (2.16)

and since by Lemma 2.3,

hf A ( m - 1 ) weakly on M,

h

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A UNIQUENESS RESULT IN PDE'S

with h defined in (2.5), we are led to considering the solution $ of

Note that the existence of $ E c2([0,m)), which does not follow immediately from the standard existence theorems, can be proven by applying usual iteration proce- dures. Our assumptions (1.1), and (2.3), together with Lemmata 2.2, 2.3, and 2.4, imply that $ has a first zero Ro > 0. Then $ > 0 in [0, Ro), and it follows from (2.17) that $' < 0 in (0, Ro]. Combining these with (2.16) and (2.17) yields (2.15).

The next step is to use the radial function v to show that the Schrodinger operator L = -A - K(x) has negative first Dirichlet eigenvalue on sufficiently large geodesic balls.

Observe first of all that by the radial symmetry of v, the co-area formula easily yields

and that we can assume that v has norm 1 in L ~ ( B ~ , ( ~ ) ) . Since v is admissible for the variational characterisation of the first Dirichlet eigenvalue X1(Ro) of L on BRo(p), using (2.15) and (2.18), we have

The strict domain 'monotonicity of X1(R) (which in turn follows from the unique continuation property of L, cf. Kazdan [5 ] ) , implies that for R1 > Ro

To conclude the proof of the Theorem assume by contradiction that u $ 0 is a non- negative solution of (2.1) on M, with R(x) > 0. Take R1 > Ro large enough so that (2.19) holds and u $ 0 on B R , ( ~ ) . Let w > 0 be the normalised eigenfunction be- longing to X1(R1). By the Boundary Point Lemma, denoting by v the outward unit normal to dBR,@), we have

Green's formula and w = 0 on dBRl@), give

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228 A. RATTO, M. RIGOLI AND A. G. SETTI

so that, using the fact that w is an eigenfunction for X1(R1), and the non-negativity of R(x), u, and w, we obtain

Since w > 0 and u 2 0 $ 0 in BRl(p), (2.19) and (2.20) yield the desired contradic- tion.

3. GEOMETRICAL APPLICATIONS: HYPERSURFACES

Proof of Theorem 1.1 Let A be a unit vector in Rm", and set

where the braces denote inner product in Rmf l. Assume that vf(M) is contained in a closed hemisphere centered at A E Sn c Rm+l. It follows that u = UA 2 0 on M.

A straightforward computation using Gauss and Codazzi equations shows that for an isometric immersion into Rmf ', u satisfies

where I1 denotes the second fundamental tensor. Under the assumption that f have parallel mean curvature H , that equation reduces to

Since, again by Gauss equations,

we see that (1.2) implies

1 / I r for r >> 1, vol(aBr(~)) . 8B,(p)

with C satisfying (1.3). By Theorem 2.1 we conclude that (3.2) u must be identically 0, so that vf (M) lies in a great sphere. w

Our second application of Theorem 2.1 concerns stable minimal hypersurfaces into Rmfl . We recall that a minimal immersion f : M 4 N is said to be stable if it minimises area up to second order. In the case of minimal hypersurfaces into R ~ + ' , the condition of stability is

(cf. Lawson [6], p. 43 ff.), and is equivalent to the fact that the Schrodinger operator

a + 1 1 1 1 ~

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A UNIQUENESS RESULT IN PDE'S 229

is negative definite on C,OO(M) c L ~ ( M ) . By virtue of Theorem 1 in Fisher-Colbrie and Schoen [3], (3.3) is in turn equivalent to the existence of a positive solution u of

A u + 1 1 1 1 ~ ~ = 0, on M .

Thus Theorem 2.1 immediately yields the following:

THEOREM 3.2 Let f : ( M , g ) -t R ~ " be a minimal immersion. If

and 1 r

s ( x ) < - C r " , for r > l , VOl(&P)) 18,.,,,

for constants B > 0, cr 2 -2, and C satisfying inequality (1.3), then the immersion f cannot be stable.

4. HIGHER CODIMENSIONAL IMMERSIONS

In this section M is oriented, and f : ( M , g ) -+ Rn is a parallel mean curvature, isometric immersion, with associated Gauss map yf : M -+ G(n , m) .

Maintaining the notation introduced in Section 1, we represent yf as the m-unit vector el A e;! A . . . A em, where {e,)z=l is a local Darboux frame along f .

Given a unit multivector V = Vl A V2 A . . . A Vm, we set

so that u is the cosine of the angle between 7f and the m-plane determined by I/ Then we have

LEMMA 4.1 Iff is a parallel mean curvature immersion, then for every unit multi- vector A, the function u = uv satisfies the differential inequality

Au 5 -11112 { u - 7 - 1 n - m ,

Proof The differential of u is m

du(e i ) = C ( v , e l A . . . A ej-1 A II(ei,ei) A . . . A em) ) , j =1

so that its Hessian is

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230 A. RATTO, M. RIGOLI AND A. G. SETTI

Assuming without loss of generality that MV,iej = 0 at p , the second sum in (4.2) vanishes at p , while a straightforward computation shows that the first one can be written as

Since by Codazzi's equations vi(II(ei,ek)) = v&(II(ei,ej), taking traces and using V H = 0, yield

Let Q be the sum at the right hand side of (4.3). Writing II(ei,ek) = hge, we have

with

Lopti = (V,e, A ep A . . . A Z t A . . . A C i A . . . A em) , (4.4)

where * means "omit". To estimate Q from above we follow Reilly [8]. First, an application of the CauchySchwarz inequality yields

Next note that since V is a unit multivector

On the other hand, Lemma 1 of [2] states that if C and D are symmetric n x n matrices, then their Hilbert-Schmidt norms satisfy the inequality

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A UNIQUENESS RESULT IN PDE'S

so that

Using Newton's inequalities, the last sum can be estimated by 2

2: 1 1 1 ~ 1 ~ 1 1 p j ~ 4 ( n - m ) ( n - m - 1)

2 aP

Inserting (4.7) and (4.6) into (4.5) yields

which substituted into (4.3) gives (4.1), and completes the proof of the Lemma. Note that for n = m + 1, all the inequalities become equalities and (4.1) reduces to (3.2). w

Having shown that the function u satisfies a differential inequality, the next step towards the proof of Theorem 1.2 is to obtain a priori bounds for the infimum of solutions of (4.1). This is the content of the next Lemma, which is a modification of a generalised maximum principle recently obtained in [9]. Since the argument is very similar to that of [9] we will only give a brief outline of the proof.

LEMMA 4.2 Suppose that ( M , g ) satisfies the curvature condition (1.5), and let u be a solution of

A u 5 - b (x ) f (u ) , with inf u > -m, (4.8)

where f is a continuous function on R, and b ( x ) is a function on M satisfying

for a constant d > 0 and r ( x ) >> 1. Then

f (inf u ) 5 0.

Sketch of Proof. Set inf u = N and assume by contradiction that f ( N ) > 0. The first step is to take R1 large enough so that (4.9) holds on 'BR, (p) . Then

arguing as in [9] one verifies that there exists a radial function v E C ' , ' ( ~ B ~ , ( ~ ) ) , smooth within the cut locus of p, such that

A v > -b(x) holds weakly on ' BR , (p ) ,

lim v ( x ) = -m, r ( x ) - m

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232 A. RATTO. M. RIGOLI AND A. G. SETTI

The second step is to use the maximum principle and a connectedness argument to show that u cannot achieve its infimum at a point of M . Then, having set NR =

infBR(p) u, take RZ > R1 so that

N 5 u(x) 5 NR2 implies f (u(x)) 2 6 (4.11)

for a fixed 6 > 0 sufficiently small. Next define w(r) = infsB,(p)u, and observe that (4.8), (4.11), and b(x) > 0 in

'BR,(p) imply that w(r) is strictly decreasing in [R2, a ) , and that N = lim,,, w(r). In order to conclude the proof, define a function J = u - qv, (17 > 0), where v

is the function in (4.10). Since u is bounded below and v tends to - a , by taking 17 < 6 sufficiently small we can arrange that J attains an absolute minimum at z E M\BR2(p) , with u(z) < NR2 Applying the maximum principle at z, and using (4.8), and (4.10) yields the desired contradiction. rn

Remark 4.3 As mentioned in the Introduction, in the linear case (f (u) = u) the conclusion of the Lemma is that under the stated hypotheses, inf u is < 0. That is not sufficient to conclude that such an infimum is attained. Indeed, it is easy to check that the function v defined on the hyperbolic space H ~ ( - B ~ ) by v = p o r, with

is a bounded, non-negative solution of

Av = -K(r(x))v with K(r) > 0 V r > 0 and

which tends to zero at ca but never vanishes on H"(-B~). Observe that in this ex- ample a = 0, so that (4.9) certainly holds. This also illustrates a difference between Theorem 2.1 and Lemma 4.2.

Proof of Theorem 1.2 The proof is now immediate. By Lemma (4.1), the function u = u~ satisfies the differential inequality

with 2(n - m - 1)

b(x) = 11112(x), and f (u) = u - GJI-UI. Since by (1.6), 1111~ = n 2 1 ~ 1 2 - s(x) satisfies (4.9), Lemma (4.2) implies that f (inf u) < 0. Solving the inequality one obtains

inf u I. 3n-3m-2

and (1.7) follows. rn

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A UNIQUENESS RESULT IN PDE'S 23 3

Remark 4.4 Using a refined version of the Omori-Yau maximum principle, one can verify that the conclusion of Lemma 4.2 holds assuming that

and b(x) > 6 > 0. Correspondingly the conclusion of Theorem 1.2 holds assum- ing that (4.12) holds and that IIII2 > 6 > 0. Since by Schwarz's inequality 1III2 > (n - r n ) / n l ~ l ~ , and the latter is constant by the assumption of parallel mean cur- vature, (1.7) holds for non-minimal parallel mean curvature immersions provided (4.12) is verified.

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[3] D. Fisher-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211.

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