Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

Motion by Mean Curvature from the Ginzburg-Landau Interface Model

Abstract: We consider the scalar field φ t with a reversible stochastic dynamics which is defined by the standard Dirichlet form relative to the Gibbs measure with formal energy . The potential V is even and strictly convex. We prove that under a suitable large scale limit the φ t -field becomes deterministic such that locally its normal velocity is proportional to its mean curvature, except for some anisotropy effects. As an essential input we prove that for every tilt there is a unique shift invariant, ergodic Gibbs measure for the -field.

Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Abstract: We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Holder continuous way along the geodesic. We give examples that show that the Holder exponent, along with essentially all the other consequences that follow from this estimate, are sharp. Among the applications is that the regular set is convex for any non- collapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

On complete manifolds with nonnegative Ricci curvature

Abstract: Complete open Riemannian manifolds (Mn, g) with nonnegative sectional curvature are well understood. The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CG1]. The Splitting Theorem has been extended to manifolds of nonnegative Ricci curvature [CG2]. On the other hand, the Soul Theorem does not extend even topologically, according to recent examples in [GM2]. A different method to construct manifolds which carry a metric with Ric > 0, but no metric with nonnegative sectional curvature, has been given by L. Berard Bergery [BB]. This leads to the question (cf. also [Y1]): Is there any finiteness result for complete Riemannian manifolds with Ric > 0 ? The answer is certainly affirmative in the low-dimensional special cases n = 2, where all notions of curvature coincide, and n = 3, where nonnegative Ricci curvature has been studied by means of stable minimal surfaces [MSY, SY]. On the other hand, J. P. Sha and D. G. Yang [ShY] have constructed complete manifolds with strictly positive Ricci curvature in higher dimensions. For example they can choose the underlying space to be R4 x S3 with infinitely many copies of S3 x CP 2 attached to it by surgery; cf. also [ShY 1]. It is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions. The purpose of this paper is to establish the following main result.