Torsional rigidity of submanifolds with controlled geometry

Abstract

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P m with controlled radial mean curvature in ambient Riemannian manifolds N n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz-symmetrization of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in, e.g., Markvorsen and Palmer (Proc Lond Math Soc 93:253-272, 2006; Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, p. 39, preprint, 2007). As in that paper we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion. Mathematics Subject Classification (2000)Primary 53C42 · 58J65 · 35J25 · 60J65 S.

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

“…Balance conditions. In the paper [HMP1] we imposed a balance condition on the general model spaces M m W , that we will need in the sequel:…”

“…Comparison Constellations. We now present the precise settings where our main results take place, introducing the notion of comparison constellations as it was previously defined in [HMP1]. For that purpose we shall bound the previously introduced notions of radial curvature and tangency by the corresponding quantities attained in some special model spaces, called isoperimetric comparison spaces to be defined in the next subsection.…”

“…In this paper, we use a fundamental bootstrapping argument of S. Sato [Sa] to obtain a more explicit expression than (1.5) for the first eigenvalue of the geodesic balls in the rotationally symmetric model spaces in terms of their moment spectra. Using the comparison techniques and results developed in [HMP1] we then obtain upper and lower estimates for the first Dirichlet eigenvalue of the extrinsic balls of a submanifold in a more general setting and, as a corollary we recover McKean's result and obtain also upper and lower bounds for the fundamental tone of a Cartan-Hadamard manifold and the fundamental tone of a submanifold with controlled mean curvature along the lines of Cheung-Leung in [CL] and Bessa-Montenegro in [BM].…”

Exit time moments and eigenvalue estimates

“…where λ 1 denotes the principal Dirichlet eigenvalue and |Ω| denotes volume. They then study the domain functional F (Ω) = λ 1 (Ω)T 1 (Ω) |Ω| (4) and note that the obvious upper bound is not sharp. This work is continued in [1] where the authors improve the classical Pólya bound with the estimate…”

Comparison of Exit Moment Spectra for Extrinsic Metric Balls

“…[Pa2,MP1,MP4,HMP,MP5]. In these works we use R-balls and R-spheres in tailor made rotationally symmetric (warped product) model spaces M m w as comparison objects.…”