Minimal hypersurfaces in the ball with free boundary

2017/03/28 by Wheeler, Glen, Wheeler, Valentina-Mira
#49Q05 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1703.09367

Abstract

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces F:Mⁿ→ℝⁿ⁺¹ with free boundary on the standard unit sphere. First we show that if F is graphical with respect to any Killing field, then F(Mⁿ) is a flat disk. This result is independent of the topology or number or boundaries. Second, if Mⁿ = \mathbbDⁿ is a disk, we show the supremum of the curvature squared on the interior is bounded below by n times the infimum of the curvature squared on the boundary. These may be combined the give an impression of the curvature of non-flat minimal hyperdisks with free boundary.

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