We study analytic properties of harmonic maps from Riemannian polyhedra into CAT(κ) spaces for κ∈{0,1}. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT(κ) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an Lp Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT(κ) spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for CAT(1) targets.

We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

Let M be a compact Hamiltonian T−space, with finite fixed point set MT . An equivariant class is determined by its restriction to MT , and to each fixed point p ∈ MT and generic component of the moment map, there corresponds a canonical class τp. For a special class of Hamiltonian T−spaces, the value τp,q of τp at a fixed point q can be determined through an iterated interpolation procedure, and we obtained a formula for τp,q as a sum over ascending chains from p to q. In general the number of such chains is huge, and the main result of this paper is a procedure to reduce the number of relevant chains, through a systematic degeneration of the interpolation direction. The resulting formula for τp,q resembles, via the localization formula, an integral over a space of chains, and we prove that, for complex Grassmannians, τp,q can indeed be expressed as the integral of an equivariant form over a smooth Schubert variety.

We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

A hyperbolic 3-manifold M which fibers over the circle admits a flow called the suspension flow. We show that such a flow can be isotoped to be uniformly quasigeodesic in the hyperbolic metric on M; i.e., the flow lines lifted to hyperbolic space are K-bilipschitz embeddings of R (K > 1 fixed).