共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper, we study the Ricci flow on a closed manifold equipped with a warped product metric \((N\times F,g_{N}+f^2 g_{F})\) in which \((F,g_{F})\) is Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of Perelman’s differential Harnack inequality (The entropy formula for the Ricci flow and its geometric applications, 2002). 相似文献
2.
In the thesis, we study the differential Harnack estimate for the heat equation of the Hodge Laplacian deformation of (p, p)-forms on both fixed and evolving (by Kähler-Ricci flow) Kähler manifolds, which generalize the known differential Harnack estimates for (1, 1)-forms. On a Kähler manifold, we define a new curvature cone Cp and prove that the cone is invariant under Kähler-Ricci flow and that the cone ensures the preservation of the nonnegativity of the solutions to Hodge Laplacian heat equation. After identifying the curvature conditions, we prove the sharp differential Harnack estimates for the positive solution to the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow after obtaining some interpolating matrix differential Harnack type estimates for curvature operators between Hamilton’s and Cao’s matrix Harnack estimates. Similarly, we define another new curvature cone, which is invariant under Ricci flow, and prove another interpolating matrix differential Harnack estimates for curvature operators on Riemannian manifolds. 相似文献
3.
Chow Bennett 《偏微分方程(英文版)》1998,11(2):137-140
We establish a one-parameter family of Harnack inequalities connecting Li and Yau's differential Harnack inequality for the heat equation to Hamilton's Harnack inequality for the Ricci flow on a 2-dimensional manifold with positive scalar curvature. 相似文献
4.
We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolved under the Ricci flow. These upper bounds seem to be the first general ones that match the well-known lower bounds which have been around for some time. As an application, we discover a local, time reversed Harnack type inequality for bounded positive solutions of the heat equation. 相似文献
5.
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 the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest. 相似文献
6.
Mihai Bailesteanu 《Journal of Functional Analysis》2010,258(10):3517-2919
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. 相似文献
7.
Shilong Kuang 《Journal of Functional Analysis》2008,255(4):1008-1023
We establish a point-wise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li-Yau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A classical Harnack inequality follows. 相似文献
8.
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li–Yau differential Harnack inequality
for the heat equation to the constrained trace Chow–Hamilton Harnack inequality for the Ricci flow on a 2-dimensional closed
manifold with positive scalar curvature, and thereby generalize Chow’s interpolated Harnack inequality (J. Partial Diff. Eqs.
11 (1998), 137–140). 相似文献
9.
研究了在Yamabe流下演化的一个完备非紧黎曼流形,对流形上热方程的正解给出了两种局部的梯度估计.作为应用,可以得到这个热方程的Harnack不等式. 相似文献
10.
Lin-Feng Wang 《Archiv der Mathematik》2011,96(5):473-481
We give a monotonicity entropy formula for the linear heat equation on complete manifolds with Ricci curvature bounded from
below. As its applications, we get a differential Harnack inequality and a lower bound estimate about the heat kernel. 相似文献
11.
Seick Kim Soojung Kim Ki-Ahm Lee 《Calculus of Variations and Partial Differential Equations》2014,49(1-2):669-706
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré’s result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature. 相似文献
12.
Jiayong WU 《数学年刊B辑(英文版)》2020,41(2):267-284
This paper deals with constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation ωt = ?ω + aωln ω on closed manifolds. A new interpolated Harnack inequality for ωt = ?ω-ωln ω +εRω on closed surfaces under ε-Ricci flow is also derived. Finally, the author proves a new differential Harnack inequality for ωt= ?ω-ωln ω under 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. 相似文献
13.
We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs. 相似文献
14.
In this paper we consider an n-dimensional manifold M n evolving under the Ricci flow and establish gradient estimates for positive solutions of porous medium equations on M n . As applications, we derive Harnack type inequalities. In particular, our results generalize gradient estimates for positive solutions of the heat equations in Liu (Pacific J Math 243:165–180 [18]). 相似文献
15.
We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic
is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show
that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us
to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow. 相似文献
16.
We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing
on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the
volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume
non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum
value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the
manifold is isometric to the Euclidean space. 相似文献
17.
Jia-Yong Wu 《Journal of Mathematical Analysis and Applications》2012,396(1):363-370
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 the 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 [J.-Y. Wu, Y. Zheng, Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface, Arch. Math., 94 (2010) 591–600]. 相似文献
18.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(9):1037-1042
We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality. 相似文献
19.
In this paper, we study the global behavior of the Ricci flow equation for two classes of homogeneous manifolds with two isotropy summands. Using methods of the qualitative theory of differential equations, we present the global phase portrait of such systems and derive some geometrical consequences on the structure of such manifolds under the action of the Ricci flow. 相似文献
20.
Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications 下载免费PDF全文
Wen WANG 《数学学报(英文版)》2017,33(5):620-634
In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results. 相似文献