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.
KeywordsSubmanifolds Extrinsic balls Radial convexity Radial tangency Mean exit time Isoperimetric inequalities Volume bounds Parabolicity
Mathematics Subject Classification (2000)53C42 58J65 35J25 60J65
Unable to display preview. Download preview PDF.
- 10.Karp, L., Pinsky, M.: Mean exit time from an extrinsic ball. In: From Local Times to Global Geometry, Control and Physics, Coventry, 1984/1985. Pitman Res. Notes Math. Ser., vol. 150, pp. 179–186. Longman, Harlow (1986) Google Scholar
- 14.Markvorsen, S.: Distance geometric analysis on manifolds. In: Global Riemannian Geometry: Curvature and Topology. Adv. Courses Math. CRM Barcelona, pp. 1–54. Birkhäuser, Basel (2003) Google Scholar
- 24.Pigola, S., Rigoli, M., Setti, A.G.: Maximum Principles on Riemannian Manifolds and Applications. Memoirs of the American Mathematical Society, vol. 174(822), American Mathematical Society, Providence (2005) Google Scholar