Complete space-like hypersurfaces with constant scalar curvature

Let M ⁿ be a complete space-like hypersurface with constant normalized scalar curvature R in the de Sitter space S ^{n + 1} ₁ and denote . We prove that if the norm square of the second fundamental form of M ⁿ satisfies , then either and M ⁿ is a totally umbilical hypersurface; or , and, up to rigid motion, M ⁿ is a hyperbolic cylinder . Received: 8 February 2001 / Revised version: 27 April 2001     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Starshaped hypersurfaces¶and the mean curvature flow

Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds that are close to them. This gives a classification of the possible singularities for these surfaces in the case n= 2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity occurs. Received: 6 June 1997 / Revised version: 24 October 1997     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

Existence of constant mean curvature graphs in hyperbolic space

We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible. Received July 18, 1997 / Accepted April 24, 1998     

Ricci curvature and the topology of open manifolds

In this paper, we prove that an open Riemannian n-manifold with Ricci curvature and for some is diffeomorphic to a Euclidean n-space if the volume growth of geodesic balls around p is not too far from that of the balls in . We also prove that a complete n-manifold M with is diffeomorphic to if , where is the volume of unit ball in . Received 5 May, 1997     

Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

In this paper we prove that a submanifold with parallel mean curvature of a space of constant curvature, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, using properties of Clifford systems, we give a short and alternative proof of a result of Cartan asserting that a compact isoparametric hypersurface of the sphere with three distinct principal curvatures is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes. Received: 22 April 1998     

Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space

Let M be an n-dimensional complete non-compact submanifold in a hyperbolic space with the norm of its mean curvature vector bounded by a constant . We prove in this paper that . In particular when M is minimal we have and this is sharp because equality holds when M is totally geodesic. Received September 14, 1999; in final form November 12, 1999 / Published online December 8, 2000     

Topological obstructions to nonnegative curvature

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics. Received February 11, 2000 / Published online February 5, 2001     

On eigenvalue pinching in positive Ricci curvature

We shall show that for manifolds with Ric≥n−1 the radius is close to π iff the (n+1)st eigenvalue is close to n. This extends results of Cheng and Croke which show that the diameter is close to π iff the first eigenvalue is close to n. We shall also give a new proof of an important theorem of Colding to the effect that if the radius is close to π, then the volume is close to that of the sphere and the manifold is Gromov-Hausdorff close to the sphere. From work of Cheeger and Colding these conditions imply that the manifold is diffeomorphic to a sphere. Oblatum 29-V-1998 & 4-II-1999 / Published online: 21 May 1999     

Invariant metrics of positive Ricci curvature on principal bundles

Let be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let be a principal bundle over with compact connected structure group . If the fundamental group of is finite, we show that admits a invariant metric with positive Ricci curvature so that is a Riemannian submersion. Received 14 January 1997     

Embedded constant mean curvature surfaces with special symmetry

We give sharp, necessary conditions on complete embedded CMC surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in the moduli space of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere. Received: 23 January 1998 / Revised version: 23 October 1998     

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001     

Minimal surfaces in R3 with one end¶and bounded curvature

We study complete minimal surfaces M immersed in R ³, with finite topology and one end. We give conditions which oblige M to be conformally a compact Riemann surface punctured in one point, and we show that M can be parametrized by meromorphic data on this compact Riemann surface. The goal is to prove that when M is also embedded, then the end of M is asymptotic to an end of a helicoid (or M is a plane). Received: 13 January 1997 / Revised version: 15 September 1997     

On the area of constant mean curvature discs and annuli with circular boundaries

It is still an open question whether a constant mean curvature (CMC) disc which is bounded by a circle is necessarily a spherical cap or a flat disc. The authors together with López [1] recently showed that the only stable CMC discs which are bounded by a circle are spherical caps. In this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3-dimensional space forms. Received November 8, 1999; in final form January 18, 2000 / Published online March 12, 2001     

Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8. Received: 12 July 1999