arXiv research

In this paper, we obtain a Li-Yau type gradient estimate with time dependent parameter for positive solutions of the heat equation, so that the Li-Yau type gradient estimate of Li-Xu are special cases of the estimate. We also obtain improvements of Davies' Li-Yau type gradient estimate. The argument is different with t…

Derives Li & Yau estimates for heat equations on manifolds.

Directly proves Li-Yau estimates on manifolds with negative Ricci curvature.

In this paper, motivated by finding sharp Li-Yau type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold. Then, an opt…

In this paper, motivated by the works of Bakry et. al in finding sharp Li-Yau type gradient estimate for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li-Yau type gradient estimate and show that the validity of suc…

In this paper, we obtain Li-Yau type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu \cite{LX} and Qian \cite{Qi}. As an application of the estimate, we also obtained improvements of Davies' Li-Yau type gradient estimate.

Proves estimates for Kähler-Ricci flow solutions.

We derive an interpolation version of constrained matrix Li-Yau-Hamilton estimate on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get a corresponding estimate for forward conjugate heat equation on…

New gradient estimates for heat equation on Riemannian manifolds.

We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for these …

Paper proves inequality for Green function on Kähler manifolds.

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

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

Derives gradient bounds for f-heat equations on manifolds with Bakry-Emery Ricci curvature.

We prove the Bochner-Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

We proved a matrix Li-Yau-Hamilton type gradient estimates for the positive solutin of the heat equation on complete Kaehler manifolds with nonnegative bisectional curvature. As a consequence we obtain a comparison theorem for the distance function under this curvature assumption.

In this paper, we study Li-Yau gradient estimates for the solutions $u$ to the heat equation $\partial_tu=Δu$ on graphs under the curvature condition $CD(n,-K)$ introduced by Bauer et al. in \cite{BHLLMY}. As applications, we derive Harnack inequalities and heat kernel estimates on graphs. Also we present a type of Ham…

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality $CDE'(n,K)$, which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, hea…

Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.

The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold $M$ with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution $\nabla(t)$ of the Yan…

This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant.

Estimates graph curvature and diameter using Laplacian eigenvalues.

In this paper we derive Cheng-Yau, Li-Yau, Hamilton estimates for Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry-Emery Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemann…

Survey on heat equation estimates on manifolds.

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

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(…

In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a Kähler manifold with a fixed Kähler metric. Then we consider the estimate on Kähler manifolds with Kähler metrics evolving under the rescaled Kähler-Ricci flow. Both of the estima…

We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that arise as curvature tensors of space-time connections satisfying the Ricci flow with respect to the natural space-time degenerate metric. As a special case, we employ scal…

Probability versions of Li-Yau inequalities for manifolds with boundary.

Improved heat equation estimates without gradient curvature assumption.

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

In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalit…

Eigenfunction gradients on curved spaces imply rigid structure.

Estimates for eigenvalues on Riemannian manifolds using classical inequalities.

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR he…

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar meth…

In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smoot…

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and …

Paper improves volume gap between minimal submanifolds and unit spheres.

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

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

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, i…

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature assumption: \begin{equation} D^2 \sup_{x\in {\bf N}} \left( \oint_{B(x,D)} |Ric^-|^…

The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.