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.

We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we obtain a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

We characterize conformally flat spaces as the only compact self-dual manifolds which are U(1)-equivariantly and conformally decomposable into two complete self-dual Einstein manifolds with common conformal infinity. A geometric characterization of such conformally flat spaces is also given.

Following the work of Hamilton, Ilmanen and the first author [4], in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability (also called ν-stability in this paper) restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more ν-stable and ν-unstable examples than previously known and also the first ν-stable examples, other than the standard spheres, whose second variations are negative definite.

One of the main purposes of this paper is to understand the geometry of the moduli and the Teichmller spaces of Riemann surfaces. The most interesting results we have in this paper are the detailed understanding of two new complete Khler metrics with nice properties and the KhlerEinstein metric on the Teichmller and the moduli spaces of Riemann surfaces. The two new metrics, the Ricci metric and the perturbed Ricci metric, are naturally defined as the negative Ricci curvature of the WeilPetersson metric and a combination of it with the WeilPetersson metric. We prove that these new metrics and the KhlerEinstein metric on the Teichmller and moduli spaces all have Poincar type boundary behavior, and further, in [7], we prove that they all have bounded geometry. Note that the KhlerEinstein metric is the key link between the differential geometric and algebraic geometric aspects of these spaces