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.
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TL;DR: In this article, it was shown that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the C2-norm and the total Leray-Schauder degree of all solutions is equal to -1.
Abstract: This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the C2-norm and the total Leray-Schauder degree of all solutions is equal to -1. Then we deduce from this compactness result the existence of at least one solution to our problem.
Mathematics Subject Classification (2000): 35J60, 53C21, 58G30
69 citations
TL;DR: In this article, the authors define the higher eta-invariant of a Dirac-type operator on a nonsimply connected closed manifold and discuss its variational properties and how it would fit into a higher index theorem for compact manifolds with boundary.
Abstract: AMtraet. We define the higher eta-invariant of a Dirac-type operator on a nonsimply-connected closed manifold. We discuss its variational properties and how it would fit into a higher index theorem for compact manifolds with boundary. We give applications to questions of positive scalar curvature for manifolds with boundary, and to a Novikov conjecture for manifolds with boundary.
69 citations
01 Jan 2013
TL;DR: In this article, the Calabi-Yau theorem and Donaldson's problem of existence of geodesics in Mabuchi's space of Kaehler metrics are discussed.
Abstract: We will discuss two main cases where the complex Monge–Ampere equation (CMA) is used in Kaehler geometry: the Calabi–Yau theorem which boils down to solving nondegenerate CMA on a compact manifold without boundary and Donaldson’s problem of existence of geodesics in Mabuchi’s space of Kaehler metrics which is equivalent to solving homogeneous CMA on a manifold with boundary. At first, we will introduce basic notions of Kaehler geometry, then derive the equations corresponding to geometric problems, discuss the continuity method which reduces solving such an equation to a priori estimates, and present some of those estimates. We shall also briefly discuss such geometric problems as Kaehler–Einstein metrics and more general metrics of constant scalar curvature.
69 citations
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TL;DR: In this paper, the authors consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemanian manifold with frequencies from a short interval (asymptotically fixed frequency).
Abstract: We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of $M$ tends to a limit current.
69 citations
TL;DR: In this article, the authors show how to define Riemannian metrics and connections on a non-commutative torus in such a way that an analogue of Levi-Civita's theorem holds.
Abstract: Received July 26, 2013, in final form November 19, 2013; Published online November 21, 2013http://dx.doi.org/10.3842/SIGMA.2013.071Abstract. We show how to define Riemannian metrics and connections on a noncommu-tative torus in such a way that an analogue of Levi-Civita’s theorem on the existence anduniqueness of a Riemannian connection holds. The major novelty is that we need to use twodifferent notions of noncommutative vector field. Levi-Civita’s theorem makes it possibleto define Riemannian curvature using the usual formulas.Key words: noncommutative torus; noncommutative vector field; Riemannian metric; Levi-Civita connection; Riemannian curvature; Gauss{Bonnet theorem2010 Mathematics Subject Classi cation: 46L87; 58B34; 46L08; 46L08
69 citations