We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤ π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤ π, possibly with boundary consisting of totally geodesic hyperbolic turnovers.To that end we first generalize the local rigidity result contained in [Wei] to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the techniques developed in [BLP] to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidity, cf. [Mos], [Pra], in the hyperbolic case,resp. [deR], cf. also [Rot], in the spherical case. This strategy has already been implemented successfully by [Koj] in the compact hyperbolic case if the singular locus is a link using HodgsonKerckhoff local rigidity, cf. [HK].

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization L_{ε} of a perturbed p-Laplace operator. By deriving an L_{ε}-type Bochner's formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type Theorem for p-harmonic morphisms.

We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree $1$, concave and of class $C^{m,\al}$, $m\ge4$.

A Schubert class  in the Grassmannian G(k, n) is rigid if the only proper subvarieties representing that class are Schubert varieties . We prove that a Schubert class  is rigid if and only if it is defined by a partition  satisfying a simple numerical criterion. If  fails this criterion, then there is a corresponding hyperplane class in another Grassmannian G(k0, n0) such that the deformations of the hyperplane in G(k0, n0) yield non-trivial deformations of the Schubert variety . We also prove that, if a partition  contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian G(k0, n0), then there does not exist a smooth subvariety of G(k, n) representing .