Related papers: General Relativity and the Ricci Flow

Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton…

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We…

We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and…

We consider the K\"ahler-Ricci flow $(X, \omega(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the K\"ahler manifold $X$ to some analytic variety $Y$…

In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type…

We prove the equivalence between the several notions of generalized Ricci curvature found in the literature. As an application, we characterize when the total generalized Ricci tensor is symmetric.

We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only…

The renormalization group equations of two-dimensional sigma models describe geometric deformations of their target space when the world-sheet length changes scale from the ultra-violet to the infra-red. These equations, which are also…

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $\kappa$-solutions of the Ricci…

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is,…

In this paper, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying a mean curvature flow in a Ricci flow background. One of…

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing…

In order to reconcile the wave-function collapse in quantum mechanics with the finiteness of signals' propagation in general relativity, we delve into a stochastic version of the Ricci flow and study its non-relativistic limit in presence…

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and…

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with…

In this paper, we establish Li-Yau-type and Hamilton-type estimates for positive solutions to the heat equation associated with the generalized Ricci flow, under a less stringent curvature condition. Compared with [25] and [35], these…

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and…

Purpose: This essay is a retelling of general relativity in a language in which space-time geometry is expressed as a fluid. This trivial and useful reformulation gives 1) a non-perturbative covariant description of cosmological…

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero…

We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we…