Interpolating Between Li-Yau's and Hamilton's Harnack Inequalities on a Surface

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.     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

流形上的微分Harnack 估计

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 C_p 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.     

Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface

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).     

Harnack inequalities for graphs with non-negative Ricci curvature

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.     

Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere

We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We apply this using the Rosenau solution as the model metric to deduce sharp time-dependent curvature bounds for arbitrary solutions of the normalized Ricci flow on the two-sphere. This gives a simple and direct proof of convergence to a constant curvature metric without use of any blowup or compactness arguments, Harnack estimates, or any classification of behaviour near singularities.     

关于射影平坦Finsler空间

本文研究了射影平坦Finsler空间的几何量及其几何性质。证明了射影平坦Finsler空间的Ricci曲率可完全由射影因子简洁地刻画出来。同时还证明了，在射影平坦Finsler空间中，平均Berwald曲率S=0意味着Ricci曲率Ric是二次齐次的。此外，给出了一个射影平坦Finsler空间成为常曲率空间或局部Minkowski空间的充分条件。     

Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

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.     

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.     

Differential Harnack estimates for backward heat equations with potentials under the Ricci flow

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.     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.     

On deformation of Riemannian metrics quadratic in the Ricci tensor

We prove a theorem on the short-time existence of a flow quadratic in the Ricci tensor for Riemannian metrics on compact manifolds under certain conditions. Also, we construct formulas of the deformation of the Ricci curvature tensor for this flow.     

On Harnack estimates for positive solutions of the heat equation on a complete manifold

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.     

Remarks on scalar curvature of Yamabe solitons

In this article, we consider the scalar curvature of Yamabe solitons. In particular, we show that, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that a complete non-compact Yamabe soliton with the quadratic decay at infinity of its Ricci curvature has non-negative scalar curvature. A new proof of Kazdan?CWarner condition is also presented.     

Flow by mean curvature inside a moving ambient space

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.     

Three-manifolds with small L 2-norm of traceless-Ricci curvature pinching

In this paper, we consider a fourth-order gradient flow of the quadratic Riemannian functional ɛ of traceless Ricci curvature on closed 3 -manifolds with a fixed conformal class. We show that the L ²-curvature pinching locally conformally flat 3-manifolds can be deformed to space forms through such gradient flow. More precisely, for the suitable small initial energy functional ɛ, the gradient flow exists for all times and converges smoothly to space forms as the time goes to infinity. As a consequence, we prove the stability for any background metric whose such gradient flow converges to an Einstein metric.Mathematics Subject Classifications (2000): Primary: 53C21; Secondary: 58JOS.Communicated by: Claude LeBrun (Stony Brook)     

A new entropy formula for the linear heat equation

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.     

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

We study the injectivity radius bound for 3-d Ricci flow with bounded curvature. As applications, we show the long time existence of the Ricci flow with positive Ricci curvature and with curvature decay condition at infinity. We partially settle a question of Chow-Lu-Ni [Hamilton’s Ricci Flow, p. 302].     

Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons

We show that recent work of Ni and Wilking (in preparation) [11] yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler–Ricci shrinkers by Feldman, Ilmanen, and Knopf (2003) [7] exhibit that this result is sharp. We also prove a similar result for certain noncompact steady gradient Ricci solitons.     

The conjugate heat equation and Ancient solutions of the Ricci flow

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.