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67
Collapsing riemannian manifolds while keeping their curvature bounded
 I, J. Differential Geometry
, 1986
"... This is the second of two papers concerned with the situation in which the injectivity radius at certain points of a riemannian manifold is "small" compared to the curvature. In Part I [3], we introduced the concept of an Fstructure of positive ..."
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Cited by 119 (6 self)
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This is the second of two papers concerned with the situation in which the injectivity radius at certain points of a riemannian manifold is "small" compared to the curvature. In Part I [3], we introduced the concept of an Fstructure of positive
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 69 (4 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
Gaussian Upper Bounds For The Heat Kernel On Arbitrary Manifolds
, 1997
"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the co ..."
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Cited by 61 (2 self)
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In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 55 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Uniqueness of the Ricci flow on complete noncompact manifolds
, 2005
"... The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result t ..."
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Cited by 52 (7 self)
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The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.
The entropy formula for linear heat equation
 J. Geom. Anal
, 2004
"... ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp ..."
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Cited by 36 (11 self)
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ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’ − � M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to � ∂ ∂t − � � u(x, t) = 0 with �
metrics on the moduli space of Riemann surface
, 2004
"... In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional ..."
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Cited by 35 (4 self)
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In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the Kähler– Einstein metric, we show that the logarithmic cotangent bundle over the Deligne–Mumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the Kähler–Einstein metric by using the Kähler–Ricci flow and aprioriestimates of the complex MongeAmpere equation. 1.
Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 34 (4 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Curvature and injectivity radius estimates for Einstein 4manifolds
 J. Amer. Math. Soc
"... It is of fundamental interest to study the geometric and analytic properties of compact Einstein manifolds and their moduli. In dimension 2 these problems are well understood. A 2dimensional Einstein manifold, (M2, g), has constant curvature, which after normalization, can be taken to be −1, 0 or 1 ..."
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Cited by 30 (2 self)
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It is of fundamental interest to study the geometric and analytic properties of compact Einstein manifolds and their moduli. In dimension 2 these problems are well understood. A 2dimensional Einstein manifold, (M2, g), has constant curvature, which after normalization, can be taken to be −1, 0 or 1. Thus, (M2, g) is the quotient of a space form and the metric, g, is completely determined by the conformal structure. For fixed M2, the moduli space of all such g admits a natural compactification, the DeligneMumford compactification, which has played a crucial role in geometry and topology in the last two decades, e.g. in establishing GromovWitten theory in symplectic and algebraic geometry. In dimension 3, it remains true that Einstein manifolds have constant sectional curvature and hence are quotients of space forms. An essential portion of Thurston’s geometrization program can be viewed as the problem of determining which 3manifolds admit Einstein metrics. The moduli space of Einstein metrics on a 3dimensional manifold is also well understood. As a consequence of Mostow rigidity, the situation is actually simpler than in twodimensions.