Suppose M is a m-dimensional submanifold without umbilic points in the (m+ p)-dimensional unit sphere S m+ p. Four basic invariants of M m under the Mbius transformation group of S m+ p are a symmetric positive definite 2-form g called the Mbius metric, a section B of the normal bundle called the Mbius second fundamental form, a 1-form called the Mbius form, and a symmetric (0, 2) tensor A called the Blaschke tensor. In the Mbius geometry of submanifolds, the most important examples of Mbius minimal submanifolds (also called Willmore submanifolds) are Willmore tori and Veronese submanifolds. In this paper, several fundamental inequalities of the Mbius geometry of submanifolds are established and the Mbius characterizations of Willmore tori and Veronese submanifolds are presented by using Mbius invariants.

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.

In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the KhlerEinstein metric, we show that the logarithmic cotangent bundle over the DeligneMumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the KhlerEinstein metric by using the KhlerRicci flow and a priori estimates of the complex Monge-Ampere equation.

In this paper, we generalize the Cao-Yau'’s gradient estimate for the sum of squares of vector …elds up to higher step under ssumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel. 1.

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n−2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X^{(k)} with k ≤ n−2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.