The Frankel property for self-shrinkers from the viewpoint of elliptic PDEs

Author:

Impera Debora1,Pigola Stefano2,Rimoldi Michele1

Affiliation:

1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange” , Politecnico di Torino , Corso Duca degli Abruzzi 24, I-10129 Torino , Italy

2. Dipartimento di Matematica e Applicazioni , Università degli Studi di Milano-Bicocca , Via Cozzi 55, 20126 Milano , Italy

Abstract

Abstract We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Reference26 articles.

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3. H. I. Choi and A. N. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983), no. 3, 559–562.

4. T. H. Colding and W. P. Minicozzi, II, Generic mean curvature flow I: Generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833.

5. Q. Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443–456.

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