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.

     

Minimal webs in Riemannian manifolds

For a given combinatorial graph G a geometrization (G, g) of the graph is obtained by considering each edge of the graph as a 1-dimensional manifold with an associated metric g. In this paper we are concerned with minimal isometric immersions of geometrized graphs (G, g) into Riemannian manifolds (N ⁿ , h). Such immersions we call minimal webs. They admit a natural ‘geometric’ extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which are of instrumental importance for the applications. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence theorems together with the comparison techniques for distance functions in Riemannian geometry and obtain bounds for the first Dirichlet eigenvalues, the exit times and the capacities as well as isoperimetric type inequalities for so-called extrinsic R-webs of minimal webs in ambient Riemannian manifolds with bounded curvature.      

Finite metric spaces of strictly negative type

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.     

The classical version of Stokes' theorem revisited

Using only fairly simple and elementary considerations–essentially from first year undergraduate mathematics–we show how the classical Stokes' theorem for any given surface and vector field in ?³ follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the given surface. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version of Stokes' theorem for differential forms on manifolds. However, the main point in the present article is first, that this latter fact usually does not get within reach for students in first year calculus courses and second, that calculus textbooks in general only just hint at the correspondence alluded to above. Our proof that Stokes' theorem follows from Gauss' divergence theorem goes via a well-known and often used exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of this article uses only integration in 1, 2 and 3 variables together with a ‘fattening’ technique for surfaces and the inverse function theorem.     

Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity ofextrinsic domains of minimal submanifolds P^m in ambient Riemannianmanifolds Nⁿ with a pole p. The upper bounds are given in termsof the torsional rigidities of corresponding Schwarz symmetrizationsof the domains in warped product model spaces. Our main resultsare obtained via previously established isoperimetric inequalities,which are here extended to hold for this more general settingbased on warped product comparison spaces. We also characterizethe geometry of those situations in which the upper bounds forthe torsional rigidity are actually attained and give conditionsunder which the geometric average of the stochastic mean exittime for Brownian motion at infinity is finite. 2000 MathematicsSubject Classification 53C42, 58J65, 35J25, 60J65.     

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.     

Comparison of Exit Moment Spectra for Extrinsic Metric Balls

We prove explicit upper and lower bounds for the L ¹-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds P ^m in ambient Riemannian spaces N ⁿ . We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds N ⁿ themselves.