Open Access
July, 2001 Hypersurfaces with mean Curvature given by an Ambient Sobolev Function
Reiner Schätzle
J. Differential Geom. 58(3): 371-420 (July, 2001). DOI: 10.4310/jdg/1090348353


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.


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Reiner Schätzle. "Hypersurfaces with mean Curvature given by an Ambient Sobolev Function." J. Differential Geom. 58 (3) 371 - 420, July, 2001.


Published: July, 2001
First available in Project Euclid: 20 July 2004

zbMATH: 1055.49032
MathSciNet: MR1906780
Digital Object Identifier: 10.4310/jdg/1090348353

Rights: Copyright © 2001 Lehigh University

Vol.58 • No. 3 • July, 2001
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