Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below

Let M be a complete open n-manifold with a base point p, at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface. We discuss the maximal diameter theorem for the compactification of M by attaching the ideal boundary. Under certain conditions we prove that p becomes a pole and that M is isometric to the n-model. Received: 24 September 2000; in final form: 21 November 2001 / Published online: 17 June 2002 Dedicated to Professor Su Bu-Chin on the occasion of his one hundredth birthday The work of the first author was partially supported by the Grant-in-Aid for Scientific Research, No. 12440021 and for Exploratory Research, No. 13874012     

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     

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     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

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     

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 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     

Remarks on η-Einstein unit tangent bundles

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is the space of constant sectional curvature 1 or 2. Authors’ addresses: Y. D. Chai, S. H. Chun, J. H. Park, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea; K. Sekigawa, Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan     

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     

Homogeneous spaces of curvature bounded below

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces. Dedicated to the memory of A. D. Alexandrov     

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     

Enclosure theorems for generalized mean curvature flows

We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002     

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 space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

This is an addendum to the recent Cambridge Tract “Harmonic maps between Riemannian polyhedra”, by J. Eells and the present author. H?lder continuity of locally energy minimizing maps from an admissible Riemannian polyhedron X to a complete geodesic space Y is established here in two cases: (1) Y is simply connected and has curvature (in the sense of A.D. Alexandrov), or (2) Y is locally compact and has curvature , say, and is contained in a convex ball in Y satisfying bi-point uniqueness and of radius (best possible). With Y a Riemannian polyhedron, and in case (2), this was established in the book mentioned above, though with H?lder continuity taken in a weaker, pointwise sense. For X a Riemannian manifold the stated results are due to N.J. Korevaar and R.M. Schoen, resp. T. Serbinowski. Received: 10 October 2001 / Accepted: 20 November 2001 / Published online: 6 August 2002     

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

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