Journal of Differential Geometry

Hypersurfaces with mean Curvature given by an Ambient Sobolev Function

Reiner Schätzle

Abstract

We consider n-hypersurfaces Σj with interior Ej whose mean curvature are given by the trace of an ambient Sobolev function ujW1,p(ℝn+1)

(0.1) \bar HΣj = ujνEj on Σj,

where νEj denotes the inner normal of Σj. We investigate (0.1) when Σj → Σ weakly as varifolds and prove that Σ is an integral n-varifold with bounded first variation which still satisfies (0.1) for uju, EjE. p has to satisfy

p > 1/2 (n + 1)

and p ≥ 4/3 if n = 1. The difficulty is that in the limit several layers can meet at Σ which creates cancellations of the mean curvature.

Article information

Source
J. Differential Geom., Volume 58, Number 3 (2001), 371-420.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090348353

Digital Object Identifier
doi:10.4310/jdg/1090348353

Mathematical Reviews number (MathSciNet)
MR1906780

Zentralblatt MATH identifier
1055.49032

Citation

Schätzle, Reiner. Hypersurfaces with mean Curvature given by an Ambient Sobolev Function. J. Differential Geom. 58 (2001), no. 3, 371--420. doi:10.4310/jdg/1090348353. https://projecteuclid.org/euclid.jdg/1090348353


Export citation