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Abstract. This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. 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. 1. Introduction. This paper is concerned with the existence of conformally compact Einstein metrics on a given manifold M with boundary ∂M. The main results are restricted to dimension 4, although some of the results hold in all dimensions. This existence problem was raised by Fefferman and Graham in [19] in connection with a study

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Abstract. 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 two-parameter 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

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 Harnack-like 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 a-priori 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

Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual 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.

. 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 DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel. 1. Introduction Consider a family M of n-dimensional closed Riemannian manifolds with a uniform lower bound of sectional curvature and a uniform upper bound of diameter for a fixed n 2 N . In order to investigate various properties of manifolds in M, it is very useful to study its closure M with respect to the Gromov-Hausdorff distance dGH , which is compact by the Gromov compactness theorem [15]. Since the closure M consists of Alexandrov spaces introduced in [2], the study of Alexandrov spaces is nowadays an important topic i...

Abstract. The Bakry-Émery 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 Bakry-Émery tensor. We show that the Bakry-Émery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Émery tensor and measured Gromov-Hausdorff limits. 1.

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 scale-invariant local Poincaré inequality. We show that if (X, d, ν) has nonnegative N-Ricci 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 Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci 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 N-Ricci 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

Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.