Minimal hypersurfaces in the ball with free boundary

Abstract

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces F : M n → R n+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 (M n ) is a flat disk. This result is independent of the topology or number or boundaries. Second, if M n = D n 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.2000 Mathematics Subject Classification. 49Q05 and 53A10.

Pinching results for minimal submanifolds and a characterization of spherical caps

“…1 since the result is independent of the number of boundary components of Σ. The proof of Proposition 5.1 is inspired by previous work of G. Wheeler and V.-M Wheeler[28] which proved that a smooth free boundary minimal graph in B n is an equatorial disk.…”

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

“…See [4,7,9,19,28,35,40], and so on, for more rigidity results for free boundary minimal submanifolds.…”

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

“…Recently, Fraser-Schoen [17] generalized Nitsche's result to free boundary minimal disks in geodesic balls of arbitrary dimension in space forms. In [46], a higher dimensional analogue for free boundary minimal hypersurfaces in the n-dimensional Euclidean unit ball B n was obtained under some graphical conditions.…”