Are there any nontrivial examples of C1C^1 hypersurfaces with bounded (integrable) generalized mean curvature?

The definition of generalized mean curvature on C1 hypersurfaces is given as follows:

Let M be a closed orientable C1 hypersurface in Rn+1 and μ be the n-dimensional Hausdorff measure. We say the Rn+1 valued vector function H is a generalized mean curvature on M, if for any smooth vector fields XRn+1, the following identity holds:
MdivMXdμ=MXHdμ,where dimMX is the tangential gradient, that is, if e1,,en form an orthomormal tangent bundle on M, then divMX=ni=1eiXei.

My question is, is there a concrete example that M is only C1 and the generalized mean curvature can be constructed?

This question has constantly intrigued me for a long time. The first time when I was interested in this question was when I was reading the paper by Micheal and Simon In the paper, the authors introduced the generalized mean curvature possibly defined on very rough manifolds, and then they proved the generalized Micheal-Simon Sobolev inequality. Later on I constantly occured papers with the setting that the initial surface is just rectifiable and has bounded generalized mean curvature. I just don’t want things to be pused into the most general setting.

As far as I know, even given a C1,α manifold, the generalized mean curvature can only be defined as a measure, and cannot be written as a vector valued function, see Can anyone give me some concrete nontrivial examples, and why people study flows with such weak assumption? For example, Huisken and Ilmanen deals with inverse mean curvature flow on C1 manifolds with bounded mean curvature, see I’ve no idea what is the motivation…

I’ll really appreciate if anyone can share any ideas. Thanks!


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