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In this paper, we establish various L 2 -estimates for the exterior differential operator on p-convex Riemannian manifolds in the sense of Harvey and Lawson. As applications, we establish a Carleman type estimate which is uniform with respect to both weight functions and domains, and we also obtain topological restrictions for a Riemannian manifold to be p-convex.

F. Labourie [arXiv:1212.5015] characterized the Hitchin components for PSL(n,R) for any n>1 by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank n swapping algebra, which is the quotient of the swapping algebra by the (n+1)×(n+1) determinant relations. The main results are the well-definedness of the rank n swapping algebra and the "cross-ratio" in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank n swapping algebra generated by these "cross-ratios" to characterize the PSL(n,R) Hitchin component for a fixed n>1. We also show the relation between the rank 2 swapping algebra and the cluster X space.

In this paper we consider proper holomorphic embeddings of finitely connected planar domains into ℂ^{$n$}that approximate given proper embeddings on smooth curves. As a side result we obtain a tool for approximating a $\mathcal{C}^{\infty}$ diffeomorphism on a polynomially convex set in ℂ^{$n$}by an automorphism of ℂ^{$n$}that satisfies some additional properties along a real embedded curve.

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature 􏰢 1 in the sense of Alexandrov.