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.

A Computed example of nonuniqueness of mean curvature flow in R3

Abstract: A family of surface (M{sub t}){sub t{element_of}R} in R{sup n} is said to be moving by mean curvature provided. Here H(x) is the mean curvature vector of M{sub t} at x. Is there a smooth hypersurface in some Euclidean space whose mean curvature flow admits nonuniqueness after the onset of singularities? In this note we present compelling numerical evidence for nonuniqueness starting from a certain smooth surface in R{sup 3}. In contrast to other references, we do not have a complete proof for our construction.

Mean curvature flow through singularities for surfaces of rotation

Abstract: In this paper, we study generalized “viscosity” solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Einstein metrics and complex singularities

Abstract: This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkahler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kahler metric (which is hyperkahler if and only if K X is trivial), and that if K X is strictly nef, then X also admits a complete (non-Kahler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number. Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable. All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

Nontrivial vacua in higher-derivative gravitation.

Abstract: A discussion of an extended class of higher-derivative classical theories of gravity is presented. A procedure is given for exhibiting the new propagating degrees of freedom, at the full nonlinear level, by transforming the higher-derivative action to a canonical second-order form. For general fourth-order theories, described by actions which are general functions of the scalar curvature, the Ricci tensor and the full Riemann tensor, it is shown that the higher-derivative theories may have multiple stable vacua. The vacua are shown to be, in general, nontrivial, corresponding to de Sitter or anti-de Sitter solutions of the original theory. It is also shown that around any vacuum the elementary excitations remain the massless graviton, a massive scalar field, and a massive ghostlike spin-two field. The analysis is extended to actions which are arbitrary functions of terms of the form ${\ensuremath{ abla}}^{2k}R$, and it is shown that such theories also have a nontrivial vacuum structure.