In this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. This latter result can be viewed as an analog of the well-known volume comparison theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d ≥ 4$ (see Theorem 2.13). These examples result from applying $CAT(0)/CAT(−1)$ filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps.

The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory (see Section 5).

Given a negatively curved geodesic metric space $M$, we study the almost sure asymptotic penetration behavior of (locally) geodesic lines of $M$ into small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objets. As a consequence in the tree setting, we obtain Diophantine approximation results of elements of non-archimedian local fields by quadratic irrational ones.

We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in Alexandrov spaces with curvature $≥ κ$. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison. Imposing an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel’man, we develop a harmonic map theory from two dimensional domains into these spaces. In particular, we show that the solution to the Dirichlet problem from a disk is Hölder continuous in the interior and continuous up to the boundary. Using this theory, we solve the Plateau Problem in this setting generalizing classical results in Euclidean space (due to J. Douglas and T. Rado) and in Riemannian manifolds (due to C.B. Morrey).