In this paper, we derive the CR Reilly'’s formula and its applications to studying of the fi…rst eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the …first Dirichlet eigenvalue estimate in a compact pseudo-hermitian manifold with boundary and the fi…rst eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian manifold of vanishing torsion.

We show an example of a non-archimedean version of the existence part of the Calabi-Yau theorem in complex geometry. Precisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.

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 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.

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide the solution to a problem by Zhang.