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Minimal hypersurfaces in the ball with free boundary

Journal Article


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=Dn 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 on the boundary. These may be combined the give an impression of the curvature of non-flat minimal hyperdisks with free boundary.

Publication Date


  • 2019

Citation


  • Wheeler, G. & Wheeler, V. (2019). Minimal hypersurfaces in the ball with free boundary. Differential Geometry and Its Applications, 62 120-127.

Scopus Eid


  • 2-s2.0-85056231181

Number Of Pages


  • 7

Start Page


  • 120

End Page


  • 127

Volume


  • 62

Place Of Publication


  • Netherlands

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=Dn 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 on the boundary. These may be combined the give an impression of the curvature of non-flat minimal hyperdisks with free boundary.

Publication Date


  • 2019

Citation


  • Wheeler, G. & Wheeler, V. (2019). Minimal hypersurfaces in the ball with free boundary. Differential Geometry and Its Applications, 62 120-127.

Scopus Eid


  • 2-s2.0-85056231181

Number Of Pages


  • 7

Start Page


  • 120

End Page


  • 127

Volume


  • 62

Place Of Publication


  • Netherlands