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105
On the structure of spaces with Ricci curvature bounded below. I
 J. DIFFERENTIAL GEOM
, 1997
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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 43 (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.
Fredholm operators and Einstein metrics on conformally compact manifolds
"... Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, t ..."
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Cited by 43 (2 self)
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Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for selfadjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.
The space of embedded minimal surfaces of fixed genus in a 3manifold V; Fixed genus
, 2005
"... This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for nonsimply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small nec ..."
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Cited by 41 (10 self)
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This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for nonsimply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely–oriented double spiral staircases; see Figure 1. The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves; see Figure 2. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub–disks removed (a topological disk with two sub–disks removed is called a pair of pants). Both of these structures occur as different extremes in the twoparameter family of minimal surfaces known as the Riemann examples. The results of [CM3]–[CM6] have already been used by many authors; see, e.g., the surveys [MeP], [P], [Ro] and the introduction in [CM6] for some of these applications. There is much current research on minimal surfaces with infinite topology. Some of the results of the present paper were announced previously and have already been widely used to study
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 37 (3 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.
Ricci Flow And The Uniformization On Complete Noncompact Kähler Manifolds
 J. Differential Geom
, 1997
"... this paper proved the following result in his Ph.D. thesis [43] in 1990: ..."
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Cited by 37 (0 self)
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this paper proved the following result in his Ph.D. thesis [43] in 1990:
Complete Manifolds With Positive Spectrum, II
, 2003
"... In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for ndimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of ..."
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Cited by 26 (11 self)
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In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for ndimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of a theorem of Cheng.
Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature
 J. DIFFERENTIAL GEOM
, 2003
"... In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate ..."
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Cited by 26 (18 self)
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In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are then applied to prove several structure theorems on complete noncompact Kähler manifolds with nonnegative bisectional or sectional curvature.
KählerRicci flow and the PoincaréLelong equation
 Comm. Anal. Geom
"... In [MSY], MokSiuYau studied complete Kähler manifolds with nonnegative holomorphic bisectional curvature by solving the PoincaréLelong equation ..."
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Cited by 24 (16 self)
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In [MSY], MokSiuYau studied complete Kähler manifolds with nonnegative holomorphic bisectional curvature by solving the PoincaréLelong equation