We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the c
omparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.
We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the fi
rst Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung--Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.
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 cur
vatures (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.
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 motion in the extrinsic balls. 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.
