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.

Axial isometries of manifolds of non-positive curvature

Abstract: Let M be a complete C ~~ Riemannian manifold of non-positive sectional curvature. We say that a geodesic 9: IR~ M bounds a fiat strip of width c > 0 (a fiat half plane) if there is a totally geodesic, isometric immersion i: [0, c) x IR~M(i: [0, oo) x IR~M) such that i(0, t) = 9(0. A 9eodesic without fiat strip (without fiat half plane) is a geodesic, which does not bound a flat strip (a flat half plane). We will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of M. In fact, many of the properties of a manifold of strictly negative curvature (resp. of a visibility manifold) still remain true if one assumes only the existence of a closed geodesic without flat half plane. We will discuss the existence of free (non-Abelian) subgroups of gl(M), the existence of infinitely many closed geodesics, the density of closed geodesics, and a transitivity property of the geodesic flow. It is, therefore, interesting to give conditions which ensure the existence of a closed geodesic without flat half plane. We will prove that M has a closed geodesic without flat half plane if vol(M)< oo and if M contains a geodesic without flat half plane. Note that a geodesic is not boundary of a flat strip (and a fortiori not boundary of a flat half plane) if it passes through a point p e M such that the sectional curvature of all tangent planes at p is negative. In the proofs of our results we investigate the action of rtl(M ) as group of isometries on the universal covering space H of M. In the proofs of many of our results we do not use the fact that this action is properly discontinuous and free. We, therefore, formulate these results for arbitrary groups D of isometries of H. The paper is organized as follows: In Sect. 1 we fix some definitions and notations and quote some standard results of non-positive curvature. Section 2 is the central section of this paper. We investigate the properties of those isometries of H which correspond to closed geodesics in M. We also prove

A generalization of Liu-Yau's quasi-local mass

Abstract: In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam \cite{st}. In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski space and a future-directed time-like quasi-local energy-momentum is obtained.

Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space

Abstract: Spacelike hypersurfaces with prescribed mean curvature have played a major role in the study of Lorentzian manifolds Maximal mean curvature zero hypersurfaces were used in the rst proof of the positive mass theorem Constant mean curvature hypersurfaces provide convenient time gauges for the Einstein equations For a survey of results we refer to In and it was shown that entire solutions of the maximal surface equation