arXiv research

New heat equation method solves intertwining problems in CR geometry.

Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.

Bounds on Hessian of heat equation coupled with Ricci flow.

This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.

In this paper, we study the Poisson equation and heat equation in a model matrix geometry $M_n$. Our main results are about the Poisson equation and global behavior of the heat equation on $M_n$. We can show that if $c_0$ is the initial positive definite matrix in $M_n$, then $c(t)$ exists for all time and is positive …

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

Estimates heat equation on shrinking Ricci solitons with uniform bounds.

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

Article proves Liouville theorem for heat equation in super Ricci flow.

Extends gradient estimates for heat equation under Finsler geometric flows.

This paper proposes an unsupervised learning method to solve heat equations on chips.

Study on biharmonic heat equation on manifolds with curvature constraints.

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.

In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.

Survey on heat equation estimates on manifolds.

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …

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 equations.

Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.

The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.

A quaternionic contact (qc) heat equation and the corresponding qc energy functional are introduced. It is shown that the qc energy functional is monotone non-increasing along the qc heat equation on a compact qc manifold provided certain positivity conditions are satisfied.

In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.

The paper improves heat equation estimates under weaker Ricci curvature conditions.

New Hessian estimates for heat equations on manifolds.

Sharp conditions found for solving heat equation on Riemannian manifolds.

Article provides Bernstein gradient estimates for heat equations with potential terms.

In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete non-compact Riemannian manifold M or Kahler-Ricci flow. We show that under a natural assumption, a new partial convexity property for smooth solutions to the heat equation is preserved.

Second part of a study on heat equations on special manifolds, focusing on parametrix construction.

Liouville theorems extended to graphs with bounded geometry.

Trivial solution proof for heat equation on certain manifolds.

Improved heat equation estimates without gradient curvature assumption.

In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear $f$-…

Optimizes heat equation estimates on noncompact manifolds.

Proves estimates for Kähler-Ricci flow solutions.

Graphs prove curvature condition with modified heat equation.

The paper estimates gradients on graphs under specific conditions and applies these estimates to heat equations.

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.

New Harnack inequality for heat equation on compact manifolds.

Classifies self-similar solutions for heat equations with positive speed.

We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. Thi…

The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.

The study sets limits on heat equation solutions' Hessians on curved spaces.

In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence classification which admits an extension by one dimension of the principal Lie alge…

Derives Li & Yau estimates for heat equations on manifolds.