Complete self-shrinkers confined into some regions of the space
- 201 Downloads
We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H.D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented.
KeywordsBounded self-shrinkers Hyperplane intersection Weighted manifolds Drifted Laplacian
Mathematics Subject Classification (1991)53C21
The authors would like to thank Pacelli Bessa, Debora Impera and Giona Veronelli for their interest in this work and for several suggestions that have improved the presentation of the paper. Further suggestions concerning the presentation are due to the anonymous referee.
- 1.Anderson, M.: The compactification of a minimal submanifold by its Gauss map. Preprint. http://www.math.sunysb.edu/anderson/compactif.pdf
- 7.Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115. Academic Press Inc., Orlando (1984)Google Scholar
- 10.Ding, Q., Xin, Y.L.: Volume growth, eigenvalue and compactness for self-shrinkers. Preprint. http://arxiv.org/abs/1101.1411v1.pdf
- 14.Lichnerowicz, A.: Variétés riemanniennes à tenseur C non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)Google Scholar
- 17.Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 174(822) (2005)Google Scholar
- 19.Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10, 757–799 (2011)Google Scholar
- 21.Rimoldi, M.: A classification theorem for self-shrinkers. Proc. Am. Math. Soc. (to appear)Google Scholar
- 22.Sjögren, P.: Ornstein–Uhlenbeck theory in finite dimension. Lecture Notes, University of Gothenburg. http://www.math.chalmers.se/donnerda/OU.pdf