For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

We introduce a new geometric flow, called the chord shortening flow, which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord not orthogonal to the boundary would shrink to a point in finite time under the flow.

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact 4-manifolds with the same homology as S1×S3. This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of H3(X;Z) with the 4-dimensional Casson invariant λSW(X) defined by Mrowka-Ruberman-Saveliev. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point p in a spacetime N , we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p . For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.

The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a two-manifold M, based on a geodesic metric. We present a formal definition of the skeleton S(\Omega) for a shape V in M and show several properties that make S(\Omega) distinct from its Euclidean counterpart in R^2. We further prove that for a shape sequence {\Omega_i} that converge to a shape V in M, the mapping \Omega\rightarrow S(\Omega_i) is lower semi-continuous. A direct application of this result is that we can use a set P of sample points to approximate the boundary of a 2D shape V in M, and the Voronoi diagram of P inside \Omega\subset M gives a good approximation to the skeleton S(\Omega). Examples of skeleton computation in topography and brain morphometry are illustrated.