This text is a presentation of the general context and results of [Oll07] and the preprint [Oll-a], with comments on related work. The goal is to present a notion of Ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive Ricci curvature, such as spectral gap estimates, concentration of measure or log-Sobolev inequalities. T...

Abstract We study an eigenvalue problem for the biharmonic operator with Neumann boundary conditions on domains of Riemannian manifolds. discuss weak formulation and classical conditions, we describe a few properties eigenvalues. Moreover, establish upper bounds compatible Weyl’s law under given lower bound Ricci curvature.

This text is a presentation of the general context and results of [Oll07] and [Oll09], with comments on related work. The goal is to present a notion of Ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive Ricci curvature, such as spectral gap estimates, concentration of measure or log-Sobolev inequalities. The necessary ...

Dragomir Tsonev (UFAM, Manaus) Title: ON THE SPECTRA OF GEOMETRIC OPERATORS EVOLVING WITH GEOMETRIC FLOWS Abstract: In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain geometric operators under specified geometric flows. Given a compact Riemannian manifold (M, g(t)) and a smooth function η ∈ C(M) we consider the family of operators∆−...

in this paper, the matsumoto metric with special ricci tensor has been investigated. it is proved that, if  is ofpositive (negative) sectional curvature and f is of  -parallel ricci curvature with constant killing 1-form ,then (m,f) is a riemannian einstein space. in fact, we generalize the riemannian result established by akbar-zadeh.

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics...

In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the...

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics...

Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are analogs Almgren’s function. When the manifold is Euclidean space drift operator Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s for ordinary heat equation. self-similarly by Ricci flow, we solutions For Gaussian soliton, gives directly mo...

Abstract. In this note, we construct families of functionals of the type of F-functional and W-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman’s no non-trivial expanding breat...