Topic
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.
Papers published on a yearly basis
Papers
More filters
••
TL;DR: In this article, a measure contraction property of metric measure spaces is introduced, which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds.
Abstract: We introduce a measure contraction property of metric measure spaces which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds. It is actually equivalent to the lower bound of the Ricci curvature in the Riemannian case. We will generalize the Bonnet?Myers theorem, and prove that this property is preserved under the measured Gromov?Hausdorff convergence.
284 citations
••
TL;DR: In this article, the authors studied nonlinear Neumann problems on riemannian manifolds with dimension n ⩾ 2, where the boundary B is an (n − 1)-dimensional submanifold and M = M⧹B is the interior of M.
280 citations
••
279 citations
••
TL;DR: In this article, a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension is developed, and the existence and uniqueness of a weak (level-set) solution is easily established using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations.
Abstract: We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d — k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution, is easily established using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [7] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of De Giorgi. An introduction to the theory of barriers and his connection to the level set solutions is also provided.
279 citations
••
278 citations