Steen Markvorsen

Finite metric spaces of strictly negative type

Abstract We prove that, if a finite metric space is of strictly negative type, then its transfinite diameter is uniquely realized by the infinite extender (load vector). Finite metric spaces that have this property include all spaces on two, three, or four points, all trees, and all finite subspaces of Euclidean spaces. We prove that, if the distance matrix is both hypermetric and regular, then it is of strictly negative type. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points. In connection with an open problem raised by Kelly, we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular, and hence of strictly negative type.

Hyperbolic spaces are of strictly negative type

We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.

Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds

Global Riemannian Geometry: Curvature and Topology

P^m

Curvature, triameter, and beyond

in ambient Riemannian manifolds

Robotic Hot-Blade Cutting

N^n

The relative volume growth of minimal submanifolds

with a pole

From PA(X) to RPAM(X)

p

Hot Blade Cuttings for the Building Industries

. The upper bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained via previously established isoperimetric inequalities, which are here extended to hold for this more general setting based on warped product comparison spaces. We also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for Brownian motion at infinity is finite.

New extremal problems for the Riemannian recognition program via Alexandrov geometry

Distance Geometric Analysis on Manifolds (Steen Markvorsen).- The Dirac Operator in Geometry and Physics (Maung Min-Oo).