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In mean curvature flow there is an important tool, namely the Huisken's monotonicity formula:
For a solution of the mean curvature flow $F: M^n \times [0,T) \rightarrow \mathbb{R}^m$ we have $$ \frac{d}{dt} \int_{M} \rho \: d \mu_t = - \int_{M} \left| H + \frac{\langle x, \nu \rangle}{2(T-t)} \right|^2 \rho \: d \mu_t, $$ where $$ \rho(x,t) =\frac{1}{(4\pi(T-t))^{\frac{n}{2}}} e^{-\frac{|x|^2}{4 (T-t)}}. $$ However in the literature of mean curvature flow I can find many statements of the formula where they all assume different conditions:

  1. Can the codimension of the solution, i.e. $m-n$ be bigger than $1$?
  2. Does $M$ have to be compact? If not, does $M$ have to be complete?
  3. Does $M$ have to be orientable?
  4. ...

In every proof I have seen it seems that the codimension does not play a role and that it may be arbitrary. For the second: I know that Stoke's Theorem is used to show the monotonicity formula in many proofs and that for this we need orientability and compactness. However none of the references did explicitly require the orientability? The Stoke's theorem is used in the form: $$ \int_M \Delta \rho=0. $$ Is there any need to introduce the Hausdorff measure for a rigorous statement?

EDIT: To make my question a little bit more precise: The question would be completely solved if someone can state (and maybe reference) a statement which allows noncompact manifolds in any codimension but possibly with additional assumptions: does $M$ have to be complete or orientable, does $F$ have to be an embedding,...?

To give some specific references in the literature:
Huisken proved the statement when $M$ is compact and in the codimension $1$ case in the paper Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geometry, 31 (1990) 285-299.
Later K. Ecker wrote a book where he proved a generalization to the noncompact case but where the integral of $\rho$ has to exist over $M$ (this is clearly natural), however in the beginning of the book he states that throughout the book he works in the case where $F$ is at each time a hypersurface which seems not to be necessary.

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  • $\begingroup$ It would be better to cite a particular treatment in the literature and ask about how premises are used to derive conclusions. Despite the presence of a few question marks in the body of your Question, I'm having difficulty picking,out what needs a response. $\endgroup$ – hardmath Apr 18 '18 at 21:08
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    $\begingroup$ I've added a more specific question. I don't really have particular treatments of which I have questions but all of the references which state some more general version of the monotonicity formula refer to the article of Huisken where he only proved it in the compact codimension $1$ case. $\endgroup$ – abcdef Apr 18 '18 at 21:36
  • $\begingroup$ I'm not a specialist but the compactness seems to me a natural condition for defining the integrals to be finite. $\endgroup$ – hardmath Apr 18 '18 at 21:43
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Regarding codimension, there is a quite thorough treatment by Ambrosio & Soner (1996,1997) both from the Level-Set and measure theoretical view-point. Their work includes Brakke's treatment with varifolds.

For the compactness question, I don't think this is necessary, as long at the integrals make sense. There is some information on this in Smoczyk (2011). But you have to be careful with topological change where "fattening" (as in Evans, Soner, Souganidis, 1992) could occur. Fattening (a level-set approach feature) is essentially a drop in the codimension that is a consequence of monotonicity (maximum principle) and can lead to infinite measure even if compactness is ensured; it doesn't occur in "immersion" models, such as yours. On the other hand your model cannot in general go past topological changes whereas the level-set method does.

I'm not aware about work without completeness (again Smoczyk 2011 hat some pertinent points).

I'm not sure about nonorientable manifolds, but it is an interesting question. For example, if you allow for interpenetration of the manifold then you could have a curvature flow (e.g., knotted curves in 3-dimensional space will "unknot" as they flow by mean curvature).

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