Related papers: Growth conditions for conformal transformations pr…

In this paper, we are concerned with optimal control problems evolved on Riemannian manifolds, where the initial and final states satisfy some inequality and equality type constraints, and the control set is a separable metric space. We…

In this paper we provide a sufficient condition for a Furstenberg measure generated by a finitely supported measure to be absolutely continuous. Using this, we give a very broad class of examples of absolutely continuous Furstenberg…

The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian…

We study how convergence of an observer whose state lives in a copy of the given system's space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance…

We study the completeness of a metric which is related to the Bergman metric of a bounded domain. We provide a criterion for its completeness in the spirit of the Kobayashi criterion for the completeness of the Bergman metric. In particular…

We show that on a compact Riemannian manifold with boundary there exists $u \in C^{\infty}(M)$ such that, $u_{|\partial M} \equiv 0$ and $u$ solves the $\sigma_k$-Ricci problem. In the case $k = n$ the metric has negative Ricci curvature.…

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and Hardy-Littlewood, based on the order of growth at infinity along the positive real axis of certain entire functions, are here imbedded in a general…

A Riemannian metric on a closed manifold is said to be geometrically formal if the wedge product of any two harmonic forms is harmonic; equivalently, the interior product of any two harmonic forms is harmonic. Given a Riemannian foliation…

In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define…

We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.

Given a Finslerian metric $F$ on a $C^4$-manifold, conformal deformations of $F$ preserving the $R$-Einstein criterion are presented. In particular, locally conformal invariance between two Finslerian $R$-Einstein metrics is characterized.

A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start…

We introduce conformal transformations in the synthetic setting of metric spaces and Lorentzian (pre-)length spaces. Our main focus lies on the Lorentzian case, where, motivated by the need to extend classical notions to spaces of low…

Einstein's theory of general relativity is written in terms of the variables obtained from a conformal--traceless decomposition of the spatial metric and extrinsic curvature. The determinant of the conformal metric is not restricted, so the…

We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of…

For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q curvature and dimension at least 5, we prove the existence of a conformal metric with constant Q curvature. Our approach is based on the study of extremal…

We give obstructions for a noncompact manifold to admit a complete Riemannian metric with (nonuniformly) positive scalar curvature. We treat both the finite volume and infinite volume cases.

We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations. This is the first result…

The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC…

Any three-dimensional Riemannian metric can be locally obtained by deforming a constant curvature metric along one direction. The general interest of this result, both in geometry and physics, and related open problems are stressed.