Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation \begin{eqnarray*} \frac{\partial f}{\partial t} = -Δf + γf\log f +aSf \end{eqnarray*} where $a$ and $γ$ are constants and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover lead to new Harnack inequalities for a variety of geometric flows.
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 Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.
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 Harnack inequalities extend the previous differential Harnack inequalities for linear heat equations with potentials under the Ricci flow.
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.
We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton.
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 obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for heat kernels on manifolds with $Ricci(M)\ge -k$, which generalizes a result of L. Ni \cite{NL1,NL4}. As applications, we obtain new Harnack inequali…
The main goal of this paper is to generalize some Li-Yau type gradient estimates to Finsler geometry in order to derive Harnack type inequalities. Moreover, we obtain, under some curvature assumption, a general gradient estimate for positive solutions of the heat equation when the manifold evolving along the Finsler Ricci flow.
In this article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation provides a unified framework for some known results, in particular including corresponding results of Ni, Perelman, and Tran as special cases. Moreover it leads to new results in the setting of Ricci-Harmonic flow and mean curvature flow in Lorentzian manifolds with nonnegative sectional curvature.
In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, $\mathbf{S}-$curvature and Hessian) and suitable Ricci curvature bounds throughout the flow. One of the key tools we use is the Bochner identity for Finsler structures proved by Ohta and Sturm.
We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$ Δ- R(x,t)$", where $R$ is the scalar curvature of the Ricci flow, which is well generalised to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by product, we obtain some Li-Yau type…
Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential: \begin{eqnarray*} \frac{\partial f}{\partial t} = Δf + γ(t) f\log f +aSf, \end{eqnarray*} where $γ(t)$ is a continuous function on $t$, $a$ is a constant and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our Harnack estimates include many known results as sp…
In the current paper,under the transverse Ricci flow on a totally geodesic Riemannian foliation, we prove two types of differential Harnack inequalities (Li-Yau gradient estimate) for the positive solutions of the heat equation associated with the time dependent horizontal Laplacian operators. We also get a time dependent version of the generalized curvature dimension inequality. As a consequence of aforementioned results, we also get parabolic Harnack inequalities and heat kernel upper bounds.
We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $ω_t=Δω+aω\ln ω$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $ω_t=Δω-ω\lnω+\varepsilon Rω$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove a new differential Harnack inequality for the equation $ω_t=Δω-ω\lnω$ under the Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.
We derive a matrix version of Li \& Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did in~\cite{hamilton7} for the standard heat equation. We then apply these estimates to obtain some Harnack--type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.
In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's differential Harnack inequality.
In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows and the $K$-super Perelman Ricci flows.
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 $$\ppt f=Δf-f \ln f +Rf.$$ We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.
In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a rigidity theorem on complete Riemannian manifolds satisfying the $CD(K, m)$ condition, and extend the $W$-entropy formula to time dependent Witten Laplacian on compact Riemannian manifolds with $(K, m)$-super Perelman Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$ are…
We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an "integrated" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional "Lie" groups.
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the bundle of 2-forms Whitney sum 1-forms is nonnegative. The idea is to consider the 2-form which minimizes the associated quadratic form to obtain a symmetric 2-tensor. A long but straightforward computation implies this 2-tensor is a subsolution to heat-type equation. A standard a…
We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the the eigenvalue estimate $λ_1 \geq \log(2)/\operatorname{diam}^2$.
In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition, where $m\in [n, \infty)$ and $K\geq 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition and on compact manifolds equipped with $(-K, m)$-super Ricci flows.
In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$ to the conjugate heat equation $\frac{d}{dt}u = -Δu + Ru$ on compact Kähler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized Kähler Ricci flow, blowing up at $T < \infty$), which satisfy Perelman's differential Harnack inequality on $[0,T)$. We show $\mathcal{A}$ is nonempty. If $|\ric(g(t))| \le \frac{C}{T-t}$, which is alaways true if we have type I singularity, we prove the solution $u(x,t)$ satisfies the elliptic type Harnack inequlity, with the constants that are uniform in time. If the flo…
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-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the Kähler-Ricci flow and the Ricci flow.
On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $μ$ and for $K$ a constant, the curvature condition $\Ric-\nn Z\ge K$ together with the convexity of the boundary is equivalent to the heat kernel entropy inequality $$\int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} μ(\d z)\le \ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M,$$ where $\rr$ is the Riemannian dist…
We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve [1] Arnaudon Thalmaier, C., in order to generalize Harnack inequality with power -for inhomogeneous heat equation - introduced by F.Y Wang. In the case of Ricci ow, we will derive on-diagonal bound of the Heat kernel along Ricci ow (and also for the usual Heat kernel on complete M…
Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical soliton of the coupled flow within the canonical Ricci soliton. We show that this provides a link between Hamilton's differential Harnack estimate for the mean curvature flow and Hamilton's differential Harnack estimate for the Ricci flow. Moreover the second fundamental form of our…
The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold $M$ evolving under the Ricci flow, coupled with the harmonic map flow between $M$ and a second manifold $N$. We prove Li-Yau type Harnack inequalities and we consider the cases when $M$ is a complete manifold without boundary and when $M$ is compact, without boundary.
The paper considers the Ricci flow, coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analog of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev imbedding theorem in $\mathbb{R}^n$.
We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $Ω\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of $Ω$ to be made of various pieces of different codimension. We assume that ${\mathcal L}$ has a generalized first eigenfunction of which we know two sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator ${\mathcal L}$, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnac…
We derive a Harnack inequality for positive solutions of the $f$-heat equation and Gaussian upper and lower bounds for the $f$-heat kernel on complete smooth metric measure spaces $(M, g, e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser theory. As applications, we prove an $L^1_f$-Liouville theorem for $f$-subharmonic functions and an $L^1_f$-uniqueness theorem for $f$-heat equations when $f$ has at most linear growth. We also obtain eigenvalues estimates and $f$-Green's function estimates for the $f$-Laplace ope…
Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,μ)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. When $N=\infty$, we also establish a sharp upper bound of the heat kernel by using the dimension free Harnack inequality. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the $L^p$ boundedness of…
In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gradient estimates and Harnack inequalities for positive solutions to the heat equation under Ricci flow are derived. These results in the paper can be regard as generalizing the gradient estimates of Li-Yau, J. Y. Li, Hamilton and Li-Xu to the Ricci flow. Our results also improve t…
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 optimal Li-Yau type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Finally, as an application, we obtain sharp Harnack inequalities on hyperbolic spaces.
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 heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.
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 $κ$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $κ$-solutions in dimension 3, in which case that the curvature operator is nonnegative (follows from Hamilton-Ivey curvature pinching estimate). The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under…
The paper considers a manifold $M$ evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on $M$. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where $M$ is a complete manifold without boundary and the case where $M$ is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on $M$.
In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation $ \left(\frac{2t}{N} \cdot \frac{\partial^{2}}{\partial t^{2}} + \frac{tR(t, x)}{N} \frac{\partial}{\partial t} - Δ_{\tilde{g}^{(N)}(t)} \right) u = R(t, x) $ outside its wavefront set onto $ \mathbb{R}_{t} \times M_{x, g(t)} $ solves the backward heat equation $ \partial_{t} u + Δ_{x, g(t)} u = -R(t, x) $ wi…
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 Hamilton gradient estimates.
Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in mathbb{R}$ and $N\in [1,\infty)$. Suppose that $(X,d)$ is connected, complete and separable, and $\supp μ=X$. We prove that the Li-Yau inequality for the heat flow holds true on $(X,d,μ)$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on $(X,d,μ)$ for general $K\in \mathbb{R}$. Large time behaviors of heat kernels are also studied.
We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions.
We show some computations related to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the possible generalization of Huisken's monotonicity formula and its connection with the validity of some Li--Yau--Hamilton differential Harnack--type inequalities in a moving Riemannian manifold.
Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $$ u_t=Δ_f u+au\log u+bu +Au^p+Bu^{-q} $$ and \[ u_t=Δ_f u+Ae^{pu}+Be^{-pu}+D \] are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.
(a). Using time analyticity result, we address a basic question for a nonhomogeneous backward heat equation (exact control problem) in the setting of smooth domains and compact manifolds, namely: when is essentially time independent control possible? i.e. The control function is 0 on one time interval and stationary on the other. For general $L^2$ initial values, the answer is: if and only if the full space domain is used for the control function. Also an explicit formula for the control function is found in the form of an infinite series involving the heat kernel, which converges rapidly. (b)…
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 method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.
