In this paper, we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding toolthe hyperbolic geometric flow. This kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds. It possesses many interesting properties from both mathematics and physics. Several applications of this method have been found.

We study harmonic maps from an admissible flat simplicial complex to a non-positively curved Riemannian manifold. Our main regularity theorem is that these maps are C1, at the interfaces of the top-dimensional simplices in addition to satisfying a balancing condition. If we assume that the domain is a 2-complex,then these maps are C1. As an application, we show that the regularity, the balancing condition and a Bochner formula lead to rigidity and vanishing theorems for harmonic maps. Furthermore, we give an explicit relationship between our techniques and those obtained via combinatorial methods.

The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k . We also discuss some applications of our estimates.

Let $\sum$ be a minimal submanifold of $\mathbb{R}^{n+m}$ that can be represented as the graph of a smooth map $f : \mathbb{R}^n \to \mathbb{R}^m$. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which $\sum$ must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a “universal lifting theorem” for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.