Volume entropy based on integral Ricci curvature and volume ratio

For a compact Riemannian manifold M, we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.     

Rigidity of some Weyl manifolds with nonpositive sectional curvature

We provide a list of all locally metric Weyl connections with nonpositive sectional curvatures on two types of manifolds, -dimensional tori and with the standard conformal structures. For we prove that it carries no other Weyl connections with nonpositive sectional curvatures, locally metric or not. In the case of we prove the same in the more narrow class of integrable connections.

     

The manifolds with nonnegative Ricci curvature and collapsing volume

Let be a complete noncompact -manifold with collapsing volume and . The paper proves that is of finite topological type under some restrictions on volume growth.

     

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

On compact submanifolds of nonpositive external curvature in Riemannian spaces

In this paper we consider compact multidimensional surfaces of nonpositive external curvature in a Riemannian space. If the curvature of the underlying space is ≥ 1 and the curvature of the surface is ≤ 1, then in small codimension the surface is a totally geodesic submanifold that is locally isometric to the sphere. Under stricter restrictions on the curvature of the underlying space, the submanifold is globally isometric to the unit sphere. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 3–10, July, 1996.     

Bounds on the fundamental groups with integral curvature bound

We extend the polynomial growth of the fundamental group, the first Betti number estimate and finiteness of fundamental groups of compact Riemannian manifolds from pointwise Ricci lower bound to integral Ricci lower bound, using a volume comparison for star-shaped domains.      

On the average curvature of a convex curve in a surface of nonpositive Gaussian curvature

In this paper, the upper bound of the average curvature of a convex curve in a simply connected surface of nonpositive Gaussian curvature is obtained.

     

Open manifolds with nonnegative Ricci curvature and large volume growth

In this paper, we study complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We prove among other things that such a manifold is diffeomorphic to a Euclidean n-space if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in . Received: August 17, 1998.     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

On the fundamental group of manifolds with almost nonnegative Ricci curvature

Gromov conjectured that the fundamental group of a manifold with almost nonnegative Ricci curvature is almost nilpotent. This conjecture is proved under the additional assumption on the conjugate radius. We show that there exists a nilpotent subgroup of finite index depending on a lower bound of the conjugate radius.

     

Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits

The 3-dimensional Heisenberg group H together with its standard sub-Riemannian metric g₀ is viewed as the limit of a family of Riemannian manifolds, (H,g_u), u>0. For each u>0, we consider some invariant surfaces with constant mean curvature in (H,g_u). These surfaces of (H,g_u) have very nice limits as u0. We then define the mean curvature of a hypersurface in (H,g₀) to be the limit of its mean curvature in (H,g_u). We show that in a more general case, this definition is appropriate. Mathematics Subject Classification (2000) Principal 53C17; Secondary 22E25     

Crossed flux homomorphisms and vanishing theorems for Flux groups

We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits. Received: March 2005 Revision: August 2005 Accepted: September 2005     

Volume growth and holonomy in nonnegative curvature

The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between the codimension of the soul and the maximal dimension of an orbit of the action of the normal holonomy group of the soul. Additionally, an example of a simply-connected soul with a non-compact normal holonomy group is constructed.

     

Heat content and horizontal mean curvature on the Heisenberg group

We identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg–Le Gall and van den Berg–Gilkey to the sub-Riemannian context, and identifies the first few coe?cients in the sub-Riemannian heat content in terms of the horizontal perimeter and the total horizontal mean curvature of the boundary. The proof is probabilistic, and relies on a characterization of the heat content in terms of Brownian motion.     

The volume preserving mean curvature flow near spheres

By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.

     

A note on the fundamental group of Finsler manifolds with nonnegative flag curvature

In this note we prove that the fundamental group of any forward complete Finsler manifold with nonnegative flag curvature is finitely generated provided the line integral of T-curvature is small. In particular, the fundamental group of any forward complete Berwald manifold with nonnegative flag curvature is finitely generated.     

The Length Spectrum of a Compact Constant Curvature Complex is Discrete

In this short note we prove that the length spectrum of a compact constant curvature complex is discrete. After recalling the relevant definitions and reducing to a relatively simple situation, the result follows easily from a foundational result about real semi-algebraic sets and the Morse–Sard theorem. We conclude with a conjecture which remains open, a few remarks and an easy application.Partially supported by NSF grant no. DMS-0123344 (N. Brady).Partially supported by NSF grant no. DMS-0101506 (J. McCammond)