Harmonic Maps with Prescribed Singularities on Unbounded Domains

The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities \p\colon\R^3\sm\Sigma\to\H^{k+1}_\C into the $(k+1)$-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed singularities \p\colon\R^n\sm\Sigma\to\H, where $\Sigma$ is an unbounded smooth closed submanifold of $\R^n$ of codimension at least $2$, and \H is a real, complex, or quaternionic hyperbolic space. As a corollary, we prove the existence of solutions to the reduced stationary and axially symmetric Einstein/Abelian-Yang-Mills Equations.Comment: LaTeX2e (amsart) with packages: amssymb, euscript, xspace, 11 page

A counterexample to a Penrose inequality conjectured by Gibbons

We show that the Brill-Lindquist initial data provides a counterexample to a Riemannian Penrose inequality with charge conjectured by G. Gibbons. The observation illustrates a sub-additive characteristic of the area radii for the individual connected components of an outermost horizon as a lower bound of the ADM mass

A priori bounds for co-dimension one isometric embeddings

We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$. These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case $n=2$ and positive curvature, and then by P. Guan and the first author for non-negative curvature and $n=2$. Using $C^{2,\alpha}$ interior estimates of Evans and Krylov for concave fully nonlinear elliptic partial differential equations, these bounds allow us to obtain the following convergence theorem: For any $\epsilon>0$, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by $\epsilon$ which are isometrically embedable in Euclidean space $R^{n+1}$ is closed in the H\"older space $C^{4,\alpha}$, $0<\alpha<1$. These results are obtained in an effort to understand the following higher dimensional version of the Weyl embedding problem which we propose: \emph{Suppose that $g$ is a smooth metric of non-negative sectional curvature and positive scalar curvature on \S^n$which is locally isometrically embeddable in$R^{n+1}$. Does$(S^n,g)$then admit a smooth global isometric embedding into$R^{n+1}$?