In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.

Statistical procedures designed for analysing multivariate data sets often emphasize different sample statistics. While some procedures emphasize the estimates of both the mean vector and the covariance matrix , others may emphasize only one of these two sample quantities. In effect, while an unusual observation in a data set has a deleterious impact on the results from an analysis that depends heavily on the covariance matrix, its effect when dependence is on the mean vector may be minimal. The aim of this paper is to develop diagnostic measures for identifying influential observations of different kinds. Three diagnostic measures, based on the local influence approach, are constructed to identify observations that exercise undue influence on the estimate of , of , and of both together. Real data sets are analysed and results are presented to illustrate the effectiveness of the proposed measures.

In this paper we prove that given a point p ∈ Mn, where Mn is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop lp(Mn) at this point is bounded above by 2nd, where d is the diameter of Mn. Moreover, we show that on a closed simply connected Riemannian manifold Mn with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of Mn, or the length of a shortest closed geodesic on Mn is bounded from above by 4d.

Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) al- most normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: a minimal triangulation of a closed irreducible or a boundedhyperbolic 3-manifold contains no non-trivial k-normal sphere; every triangulation of a closed manifold with at least 2 tetra-hedra contains some non-trivial normal surface; every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces Lp,q.

We study three-dimensional Riemannian manifolds with nonnegative scalar curvature. We find new topological obstruction for such manifolds. Our method turns out to be useful in studying the positive mass conjecture in general relativity.