Related papers: Parabolic frequency monotonicity on compact manifo…

The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing…

We consider a class of fully non-linear parabolic equations on compact Hermitian manifolds involving symmetric functions of partial Laplacians. Under fairly general assumptions, we show the long time existence and convergence of solutions.…

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are…

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…

We introduce a new notion of Almgren's frequency which is adapted to solutions of a sub-Laplacian (harmonic functions) on a Carnot group of arbitrary step $\bG$. With this notion we investigate some new functionals associated with the…

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-\kappa$ for $\kappa\geq 0$. In the elliptic case, Wang and…

This is a continuation of the research in [16]. Let $(\overline{M},g_{-1})$ be a closed geodesic $r_0$-ball in the hyperbolic space $(\mathbb{H}^n,g_{-1})$. Let $m\neq1$ be a positive constant. In this paper, we show that for $n\geq3$,…

We investigate the effect of frequency on the principal eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with…

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\ppt f=\Delta f-f \ln f +Rf.$$…

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar…

In this article we establish a local parabolic almost monotonicity formula for two phase free boundary problems on Riemannian manifolds, which is an extension of a work of Edquist-Petrosyan.

The main goal of this paper is to show how some monotonicity methods related with the subdifferential of suitable convex functions and its extensions as m-accretive operators in Banach spaces lead to new and unexpected results showing, for…

We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation…

This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow…

We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear…

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation…

In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to…

We prove forward and backward parabolic boundary Harnack principles for nonnegative solutions of the heat equation in the complements of thin parabolic Lipschitz sets given as subgraphs $E=\{(x,t): x_{n-1}\leq f(x'',t),x_n=0\}\subset…

We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap…

Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. Some results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds…