PENGFEI GUAN AND XI SISI SHEN. Dedicated to Professor D. Phong on the occasion of his 60th birthday

Transcription

1 A RIGIDITY THEORE FOR HYPERSURFACES IN HIGHER DIENSIONAL SPACE FORS PENGFEI GUAN AND XI SISI SHEN Dedicated to Professor D. Phong on the occasion of his 60th birthday Abstract. We prove a rigidity theorem for hypersurfaces in space form N n+1 (), generalizing the classical Cohn-Vossen theorem. The classical Cohn-Vossen theorem [2] states that two isometric compact convex surfaces in R 3 are congruent. There is a vast literature devoted to the study of rigidity of hypersurfaces, we refer [10] and references therein. For hypersurfaces R n+1 with n 3, there exist strong local rigidity results when the rank of the second fundamental form is greater than or equal to three, though not all interesting global rigidity questions can be treated this way. Some global rigidity results were established in [9]. In this short note, we follow the ideas of [6, 7, 1, 8] to prove a higher dimensional version of the Cohn-Vossen Theorem for hypersurfaces in space form N n+1 (), n 2, using integral formulas. The original convexity assumption in the Cohn-Vossen Theorem will be replaced by the assumption that hypersurfaces are star-shaped with normalized scalar curvature R >. When = 0 and n = 2, N n+1 () = R 3 and the scalar curvature is the Gauss curvature. Positivity of Gauss curvature of implies the embedding is convex, it in turn is starshaped with respect to any interior point. The integral formulae we establish in Lemma 2 should also be of independent interest. The key ingredient in our proof of this higher dimensional generalization of Cohn- Vossen theorem is the integral formula (10) we will establish. We will make crucial use of a conformal illing field associated with the polar potential of the space form. Let N n+1 () be a simply connected (n+1)-dimensional space form with constant curvature. We will assume = 1, 0 or +1 and n 2. Denote g N := ds 2 the Riemannian metric of N n+1 (). We will use the geodesic polar coordinates. Let S n be the unit sphere in Euclidean space R n+1 with standard induced metric dθ 2, then (1) g N := ds 2 = dρ 2 + φ 2 (ρ)dθ 2. For the Euclidean space R n+1, φ(ρ) = ρ, ρ [0, ); for the elliptic space S n+1, φ(ρ) = sin(ρ), ρ [0, π); and for the hyperbolic space H n+1, φ(ρ) = sinh(ρ), ρ [0, ). We define the corresponding polar potential function Φ and a key vector field V as (2) Φ(ρ) = ρ 0 φ(r)dr, V = φ(ρ) ρ 1991 athematics Subject Classification. 53A05, 53C24. Research of the first author was supported in part by NSERC Discovery Grant. 1

2 2 PENGFEI GUAN AND XI SISI SHEN Therefore, for the cases = 1, 0, +1, the corresponding polar potentials are ρ cos ρ, 2 2, and cosh ρ, respectively. It is well known that V on N n+1 () is a conformal illing field and it satisfies D i V j = φ (ρ)gij N, where D is the covariant derivative with respect to the metric g N. Let N n+1 () be a closed hypersurface with induced metric g and outer normal ν. We define support function as (3) u =< V, ν >. The hypersurface is star-shaped with respect to the origin if and only if u > 0. The following identity (e.g., see [5]) will play an important role in our derivation. (4) i j Φ = φ (ρ)g ij h ij u, where is the covariant derivative with respect to the induced metric g and h = (h ij ) is the second fundamental form of the hypersurface. Denote W = g 1 h the Weingarten tensor and define the 2 nd symmetric function of Weingarten tensor W by (5) σ 2 (W ) = i<j (w ii w jj w ij w ji ) = i<j κ i κ j, where κ = (κ 1,, κ n ) are the eigenvalues of W which are the principal curvatures of. The principal curvatures and the normalized scalar curvature R of the induced metric from ambient space N n+1 () satisfy (6) σ 2 (W ) = n(n 1) (R ). 2 Since R is invariant under isometries, so is σ 2 (W ). From definition (5), σ 2 (W ) = w ll, if i = j; l i σ 2 (W ) = w ji, if i j. Define (7) N+ n+1 () = N n+1 (), = 1 or 0; N+ n+1 () = Sn+1 +, = 1, where S+ n+1 is any open hemisphere. We will restrict ourselves to hypersurfaces in N+ n+1 (). Suppose and are two isometric connected compact hypersurfaces of N n+1 () with the normalized R >. We may identify any point x as a point x through the isometry. This identification will be used in the rest of this article. We may then choose an orthonormal frame on and by the isometry this will correspond to the same orthonormal frame on. We may view as a base manifold since the local orthonormal frames of and can be identified as the same. Denote W and W the corresponding Weingarten tensors of and, respectively. For any fixed local orthonormal frame on, the polarization of σ 2 of two symmetric tensors W = (w ij ) and W = ( w ij ) is defined as (8) σ 1,1 (W, W ) = 1 2 σ 2 (W ) The following lemma is a special case of the Garding inequality [4]. w ij

3 A RIGIDITY THEORE FOR HYPERSURFACES 3 Lemma 1. Given two isometric compact hypersurfaces and in N+ n+1 () with the normalized scalar curvature R >, then σ 2 (W )(x) σ 1,1 (W, W )(x) 0, x. If the equality holds at some point x, then W (x) = W (x). Proof. From identity (6), σ 2 (W ) = σ 2 ( W ) > 0. By the compactness, there are points p and p such that W (p) > 0 and W ( p) > 0. This implies W (x), W (x) Γ 2, x, where Γ 2 = {σ 1 (W ) > 0, σ 2 (W ) > 0} is the Garding cone. The lemma follows from the Garding inequality [4]. Suppose g C 3 and suppose e 1,..., e n is a local orthonormal frame on and W = (w ij ) is a Codazzi tensor on, then for each i, n (9) j=1 ( σ 2 ) j (W ) = σ2 ii (W ) i + j i n = ( = 2 (W ) j w j w ll,i w ii,i ) l=1 j i n w jj,i = 0. l=1 w ll,i w ii,i j i We will establish the integral formulae needed. Lemma 2. Suppose and are two isometric C 2 star-shaped hypersurfaces with respect to origin in the polar coordinates of the ambient space N n+1 () in (1). Identify any local frame on with a local frame on via isometry. Denote Φ and Φ, and u and ũ to be the polar potential functions and support functions of and, respectively. Then σij 2 (W ) Φ j Φ i = [(n 1) φ φ σ 1 (W ) 2 φ uσ 2 (W )], σij 2 ( W ) Φ j Φ i = [(n 1) φ φ σ 1 ( W ) 2 φ uσ 1,1 (W, W )], (10) σij 2 (W )Φ j Φ i = [(n 1)φ φ σ 1 (W ) 2φ ũσ 1,1 ( W, W )], [(n 1)φ φ σ 1 ( W ) 2φ ũσ 2 ( W )]. σij 2 ( W )Φ j Φi = Proof. We first assume and are C 3 star-shaped hypersurfaces. For any local frame {e 1,, e n } on, it is also a local frame on. By the assumption of isometry, g ij = g ij. We note that, with φ and Φ defined in (2), (11) i Φ = i φ, i = 1,, n. Denote φ = φ( ρ), φ = φ ( ρ), Φ = Φ( ρ). It follows from (4) that, φ i j Φ = φ (φ g ij h ij u), φ i j Φ = φ ( φ g ij h ij ũ). Contracting 2 (W ) and σij 2 ( W ) with equations in above, (10) follows from identities (9), (11) and integration by parts. (10) for C 2 hypersurfaces can be verified by approximation. We may approximate them by C 3 hypersurfaces ɛ and ɛ. Note that ɛ and ɛ may not be isometric.

4 4 PENGFEI GUAN AND XI SISI SHEN For any local frame on, the following still holds, φ ɛ i j Φ ɛ = φ ɛ (φ ɛ g ɛ ij h ɛ iju ɛ ), φ ɛ i j Φɛ = φ ɛ ( φ ɛ g ɛ ij h ɛ ijũ ɛ ). Using (g ɛ ) 1 to contract with 2 (W ɛ ) and using ( g ɛ ) 1 to contract with 2 ( W ɛ ), we may perform integration by parts as before at ɛ-level. By (9) and (11), 2 (W ɛ ) Φ ɛ jφ ɛ idv g ɛ = [ φ ɛ φ ɛ σ 1 (W ɛ ) 2 φ ɛ u ɛ σ k (W ɛ )]dv g ɛ, 2 ( W ɛ )Φ ɛ j Φ ɛ idv g ɛ = 2 ( W ɛ ) Φ ɛ jφ ɛ idv g ɛ = 2 (W ɛ )Φ ɛ j Φ ɛ idv g ɛ = [φ ɛ φɛ σ 1 ( W ɛ ) 2φ ɛ ũ ɛ σ 2 ( W ɛ )]dv g ɛ, [ φ ɛ φ ɛ σ 1 ( W ɛ ) 2 φ ɛ u ɛ σ 1,1 (W ɛ, W ɛ )]dv g ɛ +R ɛ 1, [φ ɛ φɛ σ 1 (W ɛ ) 2φ ɛ ũ ɛ σ 1,1 ( W ɛ, W ɛ )]dv g ɛ +R ɛ 2, where the error terms R ɛ i, i = 1, 2 involves only the differences of derivatives of gɛ and g ɛ up to second order. Therefore, R ɛ i 0 as ɛ 0 for i = 1, 2. (10) follows for C 2 isometric star-shaped hypersurfaces by letting ɛ 0. With integral formulae (10), we follow a similar argument to Herglotz in [6] (see also [7]) and using the Garding s inequality as in [1] to prove that two isometric star-shaped compact hypersurfaces in N n+1 () share the same second fundamental form. We now proceed to prove the main result below. The term congruency will be used to describe two hypersurfaces in N n+1 () that differ by an isometry of the ambient space. Theorem 1. Two C 2 isometric compact star-shaped hypersurfaces in N n+1 () with the normalized scalar curvature R strictly larger than are congruent if = 1, 0. Two C 2 isometric compact star-shaped hypersurfaces in S n+1 with normalized scalar curvature strictly larger than +1 are congruent if the hypersurface are contained in some (may be different) hemispheres. Proof. We may assume that and are two star-shaped hypersurfaces with respect to a fixed point p N n+1 (). We may use polar coordinates in (1) and assume p = 0. In this setting, and are in the same N+ n+1 (). Subtracting the first equation in (10) from the third and the second from the fourth, φ uσ 2 (W ) = φ ũσ 1,1 (W, W ), φ ũσ 2 ( W ) = φ uσ 1,1 ( W, W )

5 A RIGIDITY THEORE FOR HYPERSURFACES 5 As σ 2 (W ) = σ 2 ( W ) and σ 1,1 (W, W ) = σ 1,1 ( W, W ), it follows that (12) ( φ u + φ ũ)(σ 2 (W ) σ 1,1 (W, W )) = 0. Since the support functions u and ũ are strictly positive and the hypersurfaces are in the same N+ n+1 (), φ (x) φ (x) > 0, x. That is, φ u+φ ũ is nowhere vanishing on. We conclude from (12) and Lemma 1 that on, σ 2 (W ) σ 1,1 (W, W ) 0. Again by Lemma 1, W = W on. Thus, the first and second fundamental forms of and are the same. This implies that and are congruent. Remark 1. After the paper appeared, our attention was brought to the global rigidity results established in [9] for hypersurfaces R n+1 and their generalizations to space form in [3]. In particular, Theorem 6.14 in [3] states that the rigidity result holds for compact hypersurface (without curvature condition) in space form N n+1 () if the set B of totally geodesic points does not disconnect with n 3 and 0 (resp. n 4 and > 0). This result directly implies Theorem 1 if we assume R > (except the case > 0 and n = 3). Our proof is different from those in [9, 3]. We believe that the integral formulae established in this paper should be effective for dealing with the case R in general, without any condition on the totally geodesic points B (and without dimensional restriction in the case > 0). References [1] S. S. Chern, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, J. ath. ech, 8, (1959), [2] S. Cohn-Vossen, Unstarre geschlossene Flächen, ath. Ann., 102 (1929), [3]. Dajczer, Submanifolds and isometric immersions. Based on the notes prepared by auricio Antonucci, Gilvan Oliveira, Paulo Lima-Filho and Rui Tojeiro. athematics Lecture Series, 13. Publish or Perish, Inc., Houston, TX, [4] L. Garding, An inequality for hyperbolic polynomials, J. ath. ech, 8, (1959), [5] P. Guan and J. Li, A mean curvature type flow in space forms, to appear in International athematics Research Notices. [6] G. Herglotz, Ueber die Starrheit von Eiflchen. Abh. ath. Sem. Univ. Hamburg, 15, (1943) [7] H. Hopf, Differential Geometry in the Large:Seminar Lectures New York University 1946 and Stanford University 1956, Lecture Notes in athematics, 1000, Springer Verlag, (1983). [8] C. Hsiung and J. Liu, A generalization of the rigidity theorem of Cohn-Vossen. J. London ath. Soc. (2) 15 (1977), no. 3, [9] R. Sacksteder, The rigidity of hypersurfaces, J. of ath. ech. 11 (1962), [10]. Spivak, A comprehensive introduction to differential geometry, Volume 5. Publsh or Perish, Inc. (1979). Department of athematics and Statistics, cgill University, ontreal, Quebec. H3A 26, Canada. address: guan@math.mcgill.ca, xi.shen@mail.mcgill.ca