The definition of generalized mean curvature on $C^1$ hypersurfaces is given as follows:

Let $M$ be a closed orientable $C^1$ hypersurface in $\mathbb{R}^{n+1}$ and $\mu$ be the $n$-dimensional Hausdorff measure. We say the $\mathbb{R}^{n+1}$ valued vector function $\textbf{H}$ is a generalized mean curvature on $M$, if for any smooth vector fields $X \in \mathbb{R}^{n+1}$, the following identity holds: $$\int_{M} div_MX d\mu= \int_{M} X \cdot \textbf{H} d\mu,$$where $\dim_MX$ is the tangential gradient, that is, if $e_1, \cdots, e_n$ form an orthomormal tangent bundle on $M$, then $div_M X= \sum_{i=1}^n\nabla_{e_i} X \cdot e_i$.

My question is, is there a concrete example that $M$ is only $C^1$ 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 http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160260305/abstract. 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 $C^{1,\alpha}$ manifold, the generalized mean curvature can only be defined as a measure, and cannot be written as a vector valued function, see https://eudml.org/doc/252366. 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 $C^1$ manifolds with bounded mean curvature, see https://projecteuclid.org/euclid.jdg/1226090483. I've no idea what is the motivation...

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

  • 1
    $\begingroup$ Huisken and Ilmanen use the flow to prove the Penrose Inequality (projecteuclid.org/euclid.jdg/1090349447). In that paper you will see how previously Jang-Wald showed that if no singularity formed, the flow could be used to prove the result. The reason they needed the weak formulation is that in general the flow will develop singularities and they were able to make the argument work even in this case thus completing the proof. $\endgroup$
    – Paul Bryan
    Jun 4 '17 at 4:45
  • $\begingroup$ @Paul Bryan, thanks a lot Paul. This comment is quite useful. After some reading, I just realized that the hypersurface with weak mean curvature is sort of an intermediate state during evolution of curvature flows. Things surprisingly work very well. I also found a very well written paper by F. Schulze of the same flavor: projecteuclid.org/euclid.jdg/1211512640. However I do have a curious question: Is it possible to star flows from weakly regular hypersurfaces instead of viewing weakly regular hypersurfaces as intermediate state? $\endgroup$
    – student
    Jun 4 '17 at 20:16
  • $\begingroup$ Absolutely one can start flows from weakly regular initial data. The way to do it is by approximating by something smooth - e.g. a smooth hypersurface within say $C^{1,\alpha}$ distance $\epsilon$ of the weakly regular initial data by convolution. Then flow the smooth data and obtain a-priori estimates depending only on the initial weakly regular hypersuface that are uniform in $\epsilon$. Then one sends $\epsilon \to 0$ and interchanges limits (the a priori estimates allow for this) to obtain a smooth solution with only weakly regular initial data. Don't have time to chase up a reference now. $\endgroup$
    – Paul Bryan
    Jun 5 '17 at 5:22
  • $\begingroup$ @Paul Bryan, this is exactly Huisken and Ilmanen did in the paper "higher regularity of the inverse mean curvature flow" projecteuclid.org/euclid.jdg/1226090483. They first proved the uniform lower bound of mean curvature, which is a weak sharp harnack inequality, then they are able to run inverse mean curture flow on $C^1$ hypersurface with bounded mean curvature. The idea is by Allard's regularity theorem, bounded mean curvature implies $C^{1,\alpha}$, then they did exactly what you said. However, I do have a curious question about curvature flows starting at weaker assumptions... $\endgroup$
    – student
    Jun 5 '17 at 5:47
  • $\begingroup$ @Paul Bryan, just a more coment, there is weak mean curvature flow on varifold, the so called Brakke's mean curvature flow. A more recent paper is by Kasai and Tonegawa, arxiv.org/abs/1111.0824. They corrected the mistake of Krabbe's notes and proved a new regularity theorem. I believe the existence they proved is very nice but too weak to find application such as proving some geometric inequalities. Their regularity theorem is interior, so I don't think the regularity theory can be applied to very rough initial hypersurface. Anyway, these are all I know about weak mean curvature. $\endgroup$
    – student
    Jun 5 '17 at 5:52

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