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16
SOME GRADIENT ESTIMATES FOR THE HEAT EQUATION ON DOMAINS AND FOR AN EQUATION BY PERELMAN
, 2006
"... Abstract. In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local LiYau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bo ..."
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Cited by 28 (2 self)
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Abstract. In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local LiYau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound. Contents
Sharp logarithmic Sobolev inequalities on gradient solitons and applications
, 2008
"... We show that gradient solitons, expanding, shrinking or steady, for the Ricci flow have potentials leading to suitable reference probability measures on the manifold. Under suitable conditions these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by ..."
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Cited by 27 (0 self)
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We show that gradient solitons, expanding, shrinking or steady, for the Ricci flow have potentials leading to suitable reference probability measures on the manifold. Under suitable conditions these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. 1
A new matrix LiYauHamilton estimate for KählerRicci flow
 J. Differential Geom. FLOW ON COMPLETE MANIFOLDS
"... In this paper we prove a new matrix LiYauHamilton (LYH) estimate for KählerRicci flow on manifolds with nonnegative bisectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated in [Ch] by Chow. This new inequality is shown to be connected with ..."
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Cited by 18 (4 self)
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In this paper we prove a new matrix LiYauHamilton (LYH) estimate for KählerRicci flow on manifolds with nonnegative bisectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated in [Ch] by Chow. This new inequality is shown to be connected with Perelman’s entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its corresponding estimate for linear heat equation. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian [LT], a manifold version of Stoll’s theorem [St] on the characterization of ‘algebraic divisors’, and a localized monotonicity formula for analytic subvarieties, which sharpens the Bishop volume comparison theorem. Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman [P], we prove a sharp lower bound heat kernel estimate for the timedependent heat equation, which is, in a certain sense, dual to Perelman’s monotonicity of the reduced volume. As an application of this new monotonicity formula, we show that the blowdown limit of a certain type longtime solution is a gradient expanding soliton. We also illustrate the connection between the new LYH estimate and the Hessian comparison theorem of [FIN] on the forward reduced distance. Local monotonicity formulae on entropy and forward reduced volume are also derived. 1.
Pseudolocality for Ricci flow and applications
"... Abstract. In [26], Perelman established a differential LiYauHamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result ..."
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Cited by 15 (2 self)
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Abstract. In [26], Perelman established a differential LiYauHamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the KählerRicci flow flow on complete nonnegatively curved Kähler manifolds.
GRADIENT ESTIMATES FOR THE HEAT EQUATION UNDER THE RICCIHARMONIC MAP FLOW
, 2013
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COMPLETE MANIFOLDS WITH NONNEGATIVE CURVATURE OPERATOR
"... In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifo ..."
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Cited by 12 (4 self)
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In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension ≥ 3) whose curvature operator is bounded and satisfies the pinching condition R ≥ δ tr(R) I> 0, for some δ> 0, must 2n(n−1) be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces. 1.
Positive complex sectional curvature, Ricci flow and the differential shpere theorem
, 2007
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Differential Harnack Estimates for Parabolic Equations
"... Abstract 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 ∂ t f = Δ f − f ln f +R f. We also comment on an earlier result of the first author on ..."
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Cited by 7 (0 self)
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Abstract 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 ∂ t f = Δ f − f ln f +R f. We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow. 1