Vol. 299, No. 2, 2019

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Rigidity theorems of hypersurfaces with free boundary in a wedge in a space form

Juncheol Pyo

Vol. 299 (2019), No. 2, 489–510
DOI: 10.2140/pjm.2019.299.489
Abstract

This paper presents some rigidity results about compact hypersurfaces with free boundary in a wedge in a space form. First, we prove that every compact immersed stable constant mean curvature hypersurface with free boundary in a wedge is part of an intrinsic sphere centered at a point of the edge of the wedge. Second, we show that the same rigidity result holds for a compact embedded constant higher-order mean curvature hypersurface with free boundary in a wedge. Finally, we extend this result to a compact immersed hypersurface with free boundary in a wedge that has the additional property that the ratio of two higher-order mean curvatures is constant.

The same conclusions hold for a compact hypersurface with free boundary that lies in a half-space in a space form.

Dedicated to Professor Jaigyoung Choe in honor of his 65th birthday.

Keywords
intrinsic spheres, surfaces with free boundary, higher-order mean curvature
Mathematical Subject Classification 2010
Primary: 53C24, 49Q10
Milestones
Received: 25 December 2016
Revised: 19 March 2018
Accepted: 24 March 2018
Published: 21 May 2019
Authors
Juncheol Pyo
Department of Mathematics
Pusan National University
Busan
South Korea
Korea Institute for Advanced Study
Seoul
South Korea