Klaus Ecker
We generalize an entropy calculation of Perelman to the case of domains evolving inside a Ricciflow solution. In the case of Euclidean space as ambient manifold an interesting relation with Harnack inequalities emerges.
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Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds
该研究针对黎曼流形上过滤方程的光滑正解，通过Bochner判别式结合Nash–Moser迭代推导微分Harnack估计，恢复了两类方程的经典估计并拓展了过滤律类型。
We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0<p\leq 1.$ If $f$ is the mean curvature, we obtain stronger Harnack inequalities.
We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.
In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality. Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity.
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 obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space.
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimensional closed manifold. This result can be regarded as a nonlinear version of the previous work of Y. Zheng and the author (Arch. Math. 94 (2010), 591-600).
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our proof, solely based on the ABP estimate, does not involve any Sobolev inequalities nor gradient estimate. We also propose a question regarding the characterization of Ricci lower bound by the Harnack …
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
We consider a subset $S$ of the complex Lie algebra $\so(n,\C)$ and the cone $C(S)$ of curvature operators which are nonnegative on $S$. We show that $C(S)$ defines a Ricci flow invariant curvature condition if $S$ is invariant under $\Ad_{\SO(n,\C)}$. The analogue for Kähler curvature operators holds as well. Although the proof is very simple and short it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact Kähler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thu…
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 introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.
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.
We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton's differential Harnack inequality for mean curvature flow and, more generally, Andrews' differential Harnack inequalities for certain "$α$-inverse-concave" flows.
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 offer an algorithmic approach for determining Harnack quantities for the curve shortening flow and we show how, following this procedure, one can obtain Hamilton's Harnack inequality for this flow $κ_t + \frac{1}{2t}κ\geq \frac{κ_s^2}κ$, where $κ$ is the curvature of the curve being deformed by the flow.
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 short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for `positive' solutions of some parabolic PDE, and can be verified quickly by grinding out a computation and applying a maximum principle. In this note we emphasise the geometry behind the Harnack inequalities, which typically turn out to be assertions of the convexity of some natural object…
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…
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.
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.
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in Hörmander's form. In this case a Harnac…
By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the consequent $L^2$-gradient inequality, are proved to be equivalent to the pointwise curvature lower bound condition together with the convexity or absence of the boundary. Some applications of the log-Harnack inequality are also introduced.
We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The primary method is construction of blow-ups of degenerating metrics, followed by a geometric/topological triviality result from a previous work.
Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow
该研究针对一般几何流演化流形上的抛物方程，建立正解的时空梯度估计与有界正解的椭圆型梯度估计，推导对应Harnack不等式，并给出特定抛物方程的梯度估计应用。
In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds $(M, g_0)$ of dimension $n\ge 3$. For $n/2 <k<n$, we prove a sharp Harnack inequality for admissible metrics when $(M,g_0)$ is not conformally equivalent to the unit sphere $S^n$ and that the set of all such metrics is compact. When $(M,g_0)$ is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering a recent result of Gursky and Viaclovski on the solvabilit…
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…
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…
Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows
本文利用 Schrödinger 型共轭半群导数的概率表示，在演化 Riemann 流形上建立无维数 Harnack 不等式，并推出热核上界与典范对数 Sobolev 不等式。
In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.
Recall that if $(M^n,g)$ satisfies $\mathrm{Ric}\geq 0$, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative $f:M\to \mathbb{R}^+$, with $f_t$ its heat flow, that $\frac{Δf_t}{f_t}-\frac{|\nabla f_t|^2}{f_t^2} +\frac{n}{2t}\geq 0.$ Our main result will be to generalize this to path space $P_xM$ of the manifold. A key point is that instead of considering infinite dimensional gradients and Laplacians on $P_xM$ we will consider a family of finite dimensional gradients and Laplace operators. Namely, for each $H^1_0$-function $φ:\mathbb{R}^+\to \mathbb{R}$ we will define t…
Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in …
In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as consequences, we give lower bound estimates for the volume of these submanifolds in terms of the warping function. We conclude the paper with an isoperimetric inequality for minimal surfaces.
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