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The p-th moment matrix is defined for a real random vector, generalizing the classical covariance matrix. Sharp inequalities relating the p-th moment and Renyi entropy are established, generalizing the classical inequality relating the second moment and the Shannon entropy. The extremal distributions for these inequalities are completely characterized.

In this paper we study the p-adic analytic geometry of the basic unitary group Rapoport–Zink spaces MK with signature (1, n − 1). Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandteMathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/ Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special fiber Mred defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain DK in MK for the action of G(Qp)×Jb(Qp), the product of the associated p-adic reductive groups, and prove that MK admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720, 2013).

We prove that for every non-negative integer g, there exists a bound on the number of ends of a complete, embedded minimal surface M in R3 of genus g and finite topology. This bound on the finite number of ends when M has at least two ends implies that M has finite stability index which is bounded by a constant that only depends on its genus.

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