## Journal of Differential Geometry

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

### Hypersurfaces with mean Curvature given by an Ambient Sobolev Function

#### Abstract

We consider *n*-hypersurfaces Σ_{j} with interior *E*_{j} whose mean curvature are given by the trace of an ambient Sobolev function *u*_{j} ∊ *W*^{1,p}(ℝ^{n+1})

(0.1) \bar H_{Σj} = *u*_{j}ν_{Ej} on Σ_{j},

where ν_{Ej} denotes the inner normal of Σ_{j}. We investigate (0.1) when Σ_{j} → Σ weakly as varifolds and prove that Σ is an integral *n*-varifold with bounded first variation which still satisfies (0.1) for *u*_{j} → *u*, *E*_{j} → *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.

#### 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