We show that, for all nonnegative integers k, Г, m and n, theYamabe invariant of #k(RP3)#Г(RP2 】 S1)#m(S2 】 S1)#n(S2. 】S1) is equal to the Yamabe invariant of RP3, provided k + Г ∶ 1. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of RP3. More precisely, we show that such manifolds are either S3 or finite connected sums #m(S2 】 S1)#n(S2. 】S1),where S2. 】S1 is the nonorientable S2-bundle over S1. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yam- abe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.

We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.

Mean curvature ows of hypersurfaces have been extensively studied and there are various dierent approaches and many beautiful results. However, relatively little is known about mean curvature ows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature ows in general dimensions and co-dimensions. The related techniques in the so called \blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature ows of two-dimensional surfaces in four-dimensional ambient spaces.

Let T be the Teichm¨uller space of marked genus g, n punctured Riemann surfaces with its bordification T the augmented Teichm ¨uller space of marked Riemann surfaces with nodes, [Abi77,Ber74]. Provided with the WP metric, T is a complete CAT(0) metric space, [DW03,Wol03, Yam04]. An invariant of a marked hyperbolic structure is the length ℓ of the geodesic α in a free homotopy class. A basic feature of Teichm¨uller theory is the interplay of two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the behavior of geodesic-length functions. Our goal is to develop an understanding of the intrinsic local WP geometry through a study of the gradient and Hessian of geodesic-length functions. Considerations include expansions for the WP pairing of gradients, expansions for the Hessian and covariant derivative, comparability models for the WP metric, as well as the behavior of WP geodesics, including a description of the Alexandrov tangent cone at the augmentation. Approximations and applications for geodesics close to the augmentation are developed. An application for fixed points of group actions is described. Bounding configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a discrete group of isometries and bounding sums of the inverse square exponential-distance.

We prove exponential estimates for plurisubharmonic functions with respect to Monge-Amp`ere measures with H¨older continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h. observables, and the large deviations theorem for bounded d.s.h. observables and H¨older continuous observables.