Showing papers on "Curvature of Riemannian manifolds published in 2018"
TL;DR: In this paper, functional inequalities for diffusion semigroups on Riemannian manifolds are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, and log-Sobolev inequalities.
Abstract: In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, $$L^p$$
-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities.
18 citations
Book•
28 Jun 2018
TL;DR: In this article, the authors introduce linear algebra, differential forms and tensors, Riemannian geometry, contact geometry, symplectic geometry, and Symplectic Geometry.
Abstract: Basic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.
16 citations
TL;DR: In this article, a classification of connected complete, locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing L 2 -forms is given.
13 citations
TL;DR: In this paper, the authors give an optimal transport characterization of sectional curvature lower and upper bounds for smooth n-dimensional Riemannian manifolds, which roughly consists on a convexity condition of the pRenyi entropy along L 2 -Wasserstein geodesics, where the role of reference measure is played by the p-dimensional Hausdorff measure.
13 citations
TL;DR: In this article, the spectral gap of the Ornstein-Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature was investigated.
10 citations
01 Aug 2018
TL;DR: In this paper, a Lie group G equipped with a left invariant Riemannian metric g is considered and the Levi-Civita connection, sectional curvature and Ricci tensor formulas are presented.
Abstract: In the present article we consider a Lie group G equipped with a left invariant Riemannian metric g. Then, by using complete and vertical lifts of left invariant vector fields we induce a left invariant Riemannian metric \(\widetilde{g}\) on the tangent Lie group TG. The Levi-Civita connection and sectional curvature of \((TG,\widetilde{g})\) are given, in terms of Levi-Civita connection and sectional curvature of (G, g). Then, we present Levi-Civita connection, sectional curvature and Ricci tensor formulas of \((TG,\widetilde{g})\) in terms of structure constants of the Lie algebra of G. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given.
10 citations
Posted Content•
TL;DR: It is proved that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function to be geodesic curvature of the boundary of a flat metric of some flat metric on a compact connected surface with boundary.
Abstract: Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on $M$ (resp. metric on $M$ with geodesic boundary). In order to provide analogous results for this problem with $n\geq 3,$ we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on $\partial M$ (resp. on $M$) is a mean curvature of a scalar flat metric on $M$ (resp. scalar curvature of a metric on $M$ and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.
8 citations
TL;DR: In this paper, the authors prove R-bisectoriality and boundedness of the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on k-forms.
Abstract: We prove R-bisectoriality and boundedness of the $$H^\infty $$
-functional calculus in $$L^p$$
for all $$1
4 citations
TL;DR: For complete non-compact Riemannian manifolds (M n, g ) with harmonic curvatures, this paper proved that g is Einstein under an inequality involving L n 2 -norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.
4 citations
Journal Article•
TL;DR: For weakly isotropic Riemannian manifolds, the existence of subsonic travelling wave solutions to nonlinear Schrodinger and Klein-Gordon equations was shown in this paper.
Abstract: We study travelling wave solutions to nonlinear Schrodinger and Klein-Gordon equations on complete Riemannian manifolds, which have a bounded Killing field $X$. For a natural class of power-type nonlinearities, we use standard variational techniques to demonstrate the existence of travelling waves on complete weakly homogeneous manifolds. If the manifolds in question are weakly isotropic, we prove that they have genuine subsonic travelling waves, at least for a non-empty set of parameters. Finally we establish that a slight perturbation of the Killing field $X$ will result in a controlled perturbation of the travelling wave solutions (in appropriate $L^p$-norms).
3 citations
TL;DR: In this article, the principal eigenvalue of a drift Laplacian on a compact Hermitian manifold M is derived, where M depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M and a norm, expressed in curvature, that determines how much M fails to be Kahler.
Abstract: We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kahler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.
TL;DR: In this paper, the number of cusps of complete Riemannian manifolds with finite volume was shown to be bounded by the volume of the manifold if some geometric conditions hold true.
Abstract: In this paper, we count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.