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89
Einstein metrics with prescribed conformal infinity on 4manifolds
, 2008
"... This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that gen ..."
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Cited by 34 (12 self)
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This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.
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 28 (9 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
Entropy and reduced distance for Ricci expanders
 J. Geom. Anal
"... ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij /∂t =−2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expan ..."
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Cited by 25 (6 self)
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ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij /∂t =−2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnacklike quantity for the conjugate heat equation, and leads to functionals µ+ and ν+. The forward reduced volume θ+ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t →∞to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known apriori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/t n/2 (Hamilton) and ¯λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like t n/2 (maximal volume growth) then W+, θ+ and ¯λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture. Small, large and distant parts of a Ricci flow are known to be modeled by various kinds of Ricci solitons: Steady, shrinking, and expanding. Perelman has discovered monotone quantities
Moduli spaces of critical Riemannian metrics in dimension four
"... Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can ..."
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Cited by 23 (0 self)
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Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding metrics with orbifold singularities. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound. 1.
Sobolev Spaces, Laplacian, And Heat Kernel On Alexandrov Spaces
, 1998
"... We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approx ..."
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Cited by 17 (6 self)
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We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DCstructure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel.
Weak curvature conditions and functional inequalities
 J. of Funct. Anal
, 2007
"... Abstract. We give sufficient conditions for a measured length space (X, d, ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d, ν), defined in terms of transport of measures. We show that DM, together with a doubling conditi ..."
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Cited by 16 (2 self)
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Abstract. We give sufficient conditions for a measured length space (X, d, ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d, ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scaleinvariant local Poincaré inequality. We show that if (X, d, ν) has nonnegative NRicci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant 2 N. The condition DM is preserved by measured GromovHausdorff limits. We then prove a Sobolev inequality for measured length spaces with NRicci curvature bounded below by K> 0. Finally we derive a sharp global Poincaré inequality. There has been recent work on giving a good notion for a compact measured length space (X, d, ν) to have a “lower Ricci curvature bound”. In our previous work [10] we gave a notion of (X, d, ν) having nonnegative NRicci curvature, where N ∈ [1, ∞) is an effective dimension. The definition was in terms of the optimal transport of measures on X. A notion was also given of (X, d, ν) having ∞Ricci curvature bounded below by K ∈ R; a closely related definition in this case was given independently by Sturm [13]. In a recent
Some geometric properties of the BakryÉmeryRicci tensor
 Comment. Math. Helv
"... Abstract. The BakryÉmery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the BakryÉmery tensor. We show that the BakryÉmery tensor is n ..."
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Cited by 16 (2 self)
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Abstract. The BakryÉmery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the BakryÉmery tensor. We show that the BakryÉmery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the BakryÉmery tensor and measured GromovHausdorff limits. 1.