Abstract
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.
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A. Hurtado was supported by Spanish Micinn-DGI grant No. MTM2007-62344 and J.A. grants FQM-325 and P09-FQM-5088.
S. Markvorsen was supported by the Danish Natural Science Research Council and Spanish Micinn-DGI grant No. MTM2007-62344.
V. Palmer was supported by Spanish Micinn-DGI grant No. MTM2007-62344 and Caixa Castelló Foundation.
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Hurtado, A., Markvorsen, S. & Palmer, V. Comparison of Exit Moment Spectra for Extrinsic Metric Balls. Potential Anal 36, 137–153 (2012). https://doi.org/10.1007/s11118-011-9223-3
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DOI: https://doi.org/10.1007/s11118-011-9223-3
Keywords
- Riemannian submanifolds
- Extrinsic balls
- Torsional rigidity
- L 1-moment spectra
- Exit time
- Isoperimetric inequalities