Journal of Differential Geometry
- J. Differential Geom.
- Volume 58, Number 3 (2001), 371-420.
Hypersurfaces with mean Curvature given by an Ambient Sobolev Function
We consider n-hypersurfaces Σj with interior Ej whose mean curvature are given by the trace of an ambient Sobolev function uj ∊ W1,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 uj → u, Ej → E. 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.
J. Differential Geom., Volume 58, Number 3 (2001), 371-420.
First available in Project Euclid: 20 July 2004
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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