We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.

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.

Let $F: \sum^n \times [0, T) \to \mathbb{R}^{n+m} be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $\gamma : (\sum^n, g_t) \to G(n,m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on $G(n,m)$ produces a subsolution of the nonlinear heat equation on $(\sum, g_t)$. We also show the condition that the image of the Gauss map lies in a totally geodesic submanifold of $G(n,m)$ is preserved by the mean curvature flow. Since the space of Lagrangian subspaces is totally geodesic in $G(n, n)$, this gives an alternative proof that any Lagrangian submanifold remains Lagrangian along the mean curvature flow.

We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].

No abstract uploaded!