Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory
- 109 Downloads
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 comparison 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.
KeywordsFirst Dirichlet eigenvalue Capacity Effective resistance Minimal submanifolds Fundamental tone Minimal submanifolds
Mathematics Subject Classification (2010)53A 53C
Unable to display preview. Download preview PDF.
- 1.Anderson, M.T.: The compactification of a minimal submanifold in euclidean space by the gauss map. unpublished preprint (1984)Google Scholar
- 3.Chavel, I.: Eigenvalues in Riemannian geometry, volume 115 of Pure and Applied Mathematics Including a chapter by Burton Randol With an appendix by Jozef Dodziuk. Academic Press Inc., Orlando, FL (1984)Google Scholar
- 4.Isaac Chavel: Riemannian geometry—a modern introduction, volume 108 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1993)Google Scholar
- 8.Dynkin, E.B.: Markov processes. Vols. I, II, volume 122 of Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. Academic Press Inc., Publishers, New York (1965)Google Scholar
- 11.Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole, volume 699 of Lecture Notes in Mathematics. Springer, Berlin (1979)Google Scholar
- 13.Grigor’yan, A.: Isoperimetric inequalities and capacities on Riemannian manifolds In The Maz’ ya anniversary collection, Vol. 1 (Rostock, 1998) Birkhäuser, Basel. Oper. Theory Adv. Appl. 109, 139–153 (1999)Google Scholar
- 19.Lima, B.P., Montenegro, J.F., Mari, L., Vieira, F.B.: Density and spectrum of minimal submanifolds in space forms (2014). arXiv:1407.5280
- 25.O’Neill, B.: Semi-Riemannian geometry, volume 103 of Pure and Applied Mathematics, With applications to relativity. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1983)Google Scholar
- 26.Sakai, T.: Riemannian geometry, volume 149 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, Translated from the 1992 Japanese original by the author (1996)Google Scholar