Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mn is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R0 > nKmax, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if Mn(n  3) is a compact manifold whose (n . 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric(n.2) min > n(n .2)R0, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on\cite {Th}. We give new results about the stability condition, and propose a Jordan-Hlder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embedding into the reference spacetime, the Killing fields of the reference spacetime are transplanted back to the surface in the physical spacetime to complete the definitions of quasi-local conserved quantities. We also compute how the corresponding total conserved quantities evolve under the Einstein equation with a cosmological constant.

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.

We study certain natural differential forms [] and their G equivariant extensions on the space of connections. These forms are defined using the family local index theorem. When the base manifold is symplectic, they define a family of symplectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations).