Complete Hypersurfaces Research Articles

A note on complete hypersurfaces of non-positive Ricci curvature

In this note we point out that a recent result of Leung concerning hypersurfaces of a Euclidean space has a simple generalisation to hypersurfaces of complete simply-connected Riemannian manifolds of non-positive constant sectional curvature.

Complete Hypersurfaces with RS = 0 In E n+1

Complete Hypersurfaces with RS = 0 In E n+1

Complete hypersurface of non-positive Ricci curvature

We conjecture that a complete hypersurface of non-positive Ricci curvature in the Euclidean space must be unbounded. We prove this under the additional assumption that all sectional curvatures of the hypersurface are bounded away from negative infinity.

Complete hypersurfaces with 𝑅𝑆=0 in 𝐸ⁿ⁺¹

A locally symmetric Riemannian manifold satisfies R R = 0 RR = 0 and in particular R S = 0 RS = 0 . The purpose of this paper is to show that the conditions R R = 0 RR = 0 and R S = 0 RS = 0 are equivalent for complete hypersurfaces in E n + 1 {E^{n + 1}} and to give by R S = 0 RS = 0 some characterizations of locally symmetric hypersurfaces in E n + 1 {E^{n + 1}} .

Complete hypersurfaces in the space form with three principal curvatures

Complete hypersurfaces in the space form with three principal curvatures

A characterization of totally geodesic hypersurfaces of $S\sp{n+1}$\ and ${\bf C}P\sp{n+1}$

Let $M$ be a complete hypersurface in ${S^{n + 1}}$ (or ${\mathbf {C}}{P^{n + 1}}$). Assume that through each point $x$ of $M$ a (local) $\mu (x)$-dimensional totally geodesic submanifold ${S_x}$ of ${S^{n + 1}}$ (or ${\mathbf {C}}{P^{n + 1}}$) exists in $M$. A sufficient condition for $M$ itself to be totally geodesic is given in terms of $\mu (x)$.

A Characterization of Totally Geodesic Hypersurfaces of S n+1 and CP n+1

Let M be a complete hypersurface in S+' (or CP'+ ). Assume that through each point x of M a (local) ,(x)-dimensional totally geodesic submanifold Sx of 1 (or CP'+ 1) exists in M. A sufficient condition for M itself to be totally geodesic is given in terms of ,u(x). Introduction. In this paper, we give a sufficient condition for a hypersurface of Sn+' (or CPn+I in the complex case) to be totally geodesic. Obviously, if a hypersurface M of Sn+' (or CPn+l) is locally totally geodesic everywhere in M, then M is globally totally geodesic. We ask, therefore, what we can say about M if M contains a lower dimensional totally geodesic submanifold (local) through each point of M. The specific statement of this assumption is given as condition (*) (or (* *) for the complex case) in the following sections. Our main results are stated as Theorems A and A'. These theorems basically tell us that we can often conclude that M is totally geodesic, even if the dimensions of the local totally geodesic submanifolds are considerably lower than that of M itself. In particular, the dimensions of local totally geodesic submanifolds in the complex case may be as low as a half of that of M + 1. Our condition (*) (or (**)) may also be regarded as an extremely simplified version of the so-called axiom of spheres for hypersurfaces of Sn+I1 (or CPn+ 1); see Cartan [5] or Leung-Nomizu [6] for this notion. Hypersurfaces of Sn+ . Let us denote by Sn+I the standard (n + 1)-dimensional sphere of constant curvature c. Let M be a complete Riemannian manifold of dimension n. Furthermore, assume that there is an isometric immersionf of M into S' . Call the pair (M, f) a hypersurface of 1. Our condition (*) is stated as follows. (*) Through each point x of M exists a k(x)-dimensional submanifold S,x (2k(x) > n) of M which is mapped under f isometrically into a k(x)-dimensional totally geodesic sphere of 1. Here k(x) is a positive integer-valued function of M. Let (M, f) be as above. Denote by V and V the Riemannian connections of Sn 1 and M, respectively. Then, for any tangent vector fields X and Y of M, we have Vf*(X)f*(Y) = f*(Vx Y) + a(X, Y), Received by the editors February 18, 1980. 1980 Mathematics Subject Classification. Primary 53C40. ? 1981 American Mathematical Society 0002-9939/81/0000-01 72/$02.00

Characterization of totally umbilical hypersurfaces

This paper gives a sufficient condition for a complete hypersurface of a Riemannian manifold of constant curvature to be umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature. K. Nomizu and B. Smyth in [3] established a formula for the Laplacian of the second fundamental form of a hypersurface M immersed with constant mean curvature in a space M of constant sectional curvature c. Later, M. Okumura in [4] characterized under certain conditions a totally umbilical hypersurface of a Riemannian manifold of nonnegative constant curvature by an inequality between the length of the second fundamental tensor and the mean curvature of the hypersurface. In the present article we prove the following theorem. THEOREM A. Let M be an n-dimensional (n > 3) connected complete hypersurface immersed with constant mean curvature in an (n + 1)-dimensional Riemannian manifold M of positive constant curvature c. If the second fundamental tensor L satisfies trace L2 cn or M is totally geodesic. 1. Preliminaries. Let M be an (n + 1)-dimensional Riemannian manifold of constant curvature c. Let q: M -, M be an isometric immersion of an n-dimensional manifold M into M. In what follows we identify M with p(M) and p E M with (p(p) E rp(M) c M. The tangent space TpM is also identified with a subspace of T9,(p)M. The Riemannian metric g of M is induced from the Riemannian metric Received by the editors December 27, 1979 and, in revised form, February 26, 1980. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.

Characterization of Totally Umbilical Hypersurfaces

This paper gives a sufficient condition for a complete hypersurface of a Riemannian manifold of constant curvature to be umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

A Note on the Asymptotic Behaviour of the Sum of Principal Radii of Curvature on Noncompact Complete Hypersurfaces

It is shown that under certain hypotheses the following conjecture is correct: on a noncompact complete hypersurface in Euclidean space the two conditions below cannot hold simultaneously: (i) the sum of principal radii of curvature is bounded; (ii) the support function is uniformly continuous.

A note on the asymptotic behaviour of the sum of principal radii of curvature on noncompact complete hypersurfaces

It is shown that under certain hypotheses the following conjecture is correct: on a noncompact complete hypersurface in Euclidean space the two conditions below cannot hold simultaneously: (i) the sum of principal radii of curvature is bounded; (ii) the support function is uniformly continuous.

On complete hypersurfaces of nonnegative sectional curvatures and constant 𝑚th mean curvature

The main result is that if M = M n M = {M^n} is a complete Riemann manifold of nonnegative sectional curvature and X : M → R n + 1 X:\,M \to {R^{n + 1}} is an isometric immersion such that X ( M ) X(M) has a positive constant mth mean curvature, then X ( M ) X(M) is the product of a Euclidean space R n − d {R^{n - d}} and a d-dimensional sphere, m ⩽ d ⩽ n m \leqslant d \leqslant n .

A Class of Riemannian Manifolds Satisfying R(X ,Y)·R = 0

Let (M,g) be a Riemannian manifold and let R be its Riemannian curvature tensor. If (M, g) is a locally symmetric space, we have(*) R(X,Y)·R = 0 for all tangent vectors X,Ywhere the endomorphism R(X,Y) (i.e., the curvature transformation) operates on R as a derivation of the tensor algebra at each point of M. There is a question: Under what additional condition does this algebraic condition (*) on R imply that (M,g) is locally symmetric (i.e., ∇R = 0)? A conjecture by K. Nomizu [5] is as follows : (*) implies ∇R = 0 in the case where (M, g) is complete and irreducible, and dim M ≥ 3. He gave an affirmative answer in the case where (M,g) is a certain complete hypersurface in a Euclidean space ([5]).

Pogorelov's theorem for complete convex hypersurfaces

Pogorelov's theorem for complete convex hypersurfaces

Complete hypersurfaces in Euclidean spaces with finite strong total curvature

Complete hypersurfaces in Euclidean spaces with finite strong total curvature

Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, which was first shown by Guan and Spruck.