Glen Wheeler
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 Sep 2 awarded Revival Aug 31 answered Calculating the area of a special hexagon Aug 26 comment The integral of the mean curvature vector over a closed immersed surface Thanks for the reply. I'll have to think a bit more before I can digest this fully. Aug 26 accepted The integral of the mean curvature vector over a closed immersed surface Aug 26 comment The integral of the mean curvature vector over a closed immersed surface Hi Willie, thanks for the answer. Could you elaborate a little more on why there is no canonical vector space in which the mean curvature of a submanifold lives? I would have naively thought the Euclidean analogue works under mild conditions on the ambient space. Aug 25 comment The integral of the mean curvature vector over a closed immersed surface @Theo Thanks for the summary. Exact components in which basis, the standard Euclidean basis? I didn't think of trying that. The generalisation you mention really answers the question I have posed. Aug 25 revised The integral of the mean curvature vector over a closed immersed surface clarification Aug 25 comment The integral of the mean curvature vector over a closed immersed surface @Willie I meant by this that $\Sigma$ has a Riemannian structure induced by $f$. I shall make it more explicit. Aug 25 comment The integral of the mean curvature vector over a closed immersed surface @anon Since I can't see the proof, it may also be that they are assuming the surface is embedded and has genus 0. Sometimes this went without saying. Aug 25 comment The integral of the mean curvature vector over a closed immersed surface @anon My university doesn't have a subscription to that journal, and it would be nice to see what kind of conditions they impose on $f$ and $\Sigma$ to obtain their result. Aug 25 asked The integral of the mean curvature vector over a closed immersed surface Aug 15 awarded Excavator Aug 11 comment derivative must be bounded on interior for a differentiable function on a closed interval @Didier I've been thinking of $f\in C^1$. Aug 11 comment derivative must be bounded on interior for a differentiable function on a closed interval @Olivier I see that this is my miscommunication. All this time I have been 'explaining' my statement in my first comment. I completely ignored the OP's definition. Aug 11 comment derivative must be bounded on interior for a differentiable function on a closed interval @Olivier Let $f$ be your function. $|f'(\epsilon)| > c\epsilon^{-1}$ for some finite $c$. Aug 11 comment derivative must be bounded on interior for a differentiable function on a closed interval @Olivier This just bounds the function itself at zero. The derivative is still unbounded in any open interval containing zero, and since by hypothesis $f$ is differentiable, zero cannot be in $[a,b]$. Aug 11 comment derivative must be bounded on interior for a differentiable function on a closed interval @Olivier Not really, the derivative of that function is bounded on any closed interval not containing zero, and zero is not allowed in the interval because the function is not differentiable at zero. I think you may be confused. Aug 11 comment derivative must be bounded on interior for a differentiable function on a closed interval @Olivier Sometimes $f$ being differentiable is shorthand for $f\in C^1$, and if $f\in C^1$ over a closed interval, then $f'\in C^0$ and is bounded. Jul 20 comment Nonlinear Fubini-Tonelli? @Shai In this question let's leave it at $f(y)$, since that's where all the discussion has headed. I had $f(x,y)$ in mind, which one could probably guess from my comments, but my mistake for not writing it correctly in the first place :). Jul 20 comment Nonlinear Fubini-Tonelli? This is certainly related. One can use several standard techniques to estimate the integral (from above and below) and then the cases of equality gives expressions equivalent to that above in special cases. But Fubini-Tonelli is fundamentally an equality. Still, thanks for the answer. +1