Timeline for How does the inverse mean curvature flow start with minimal surface?
Current License: CC BY-SA 3.0
21 events
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| Jan 17, 2017 at 2:52 | comment | added | student | If I have more questions, I'll definitely contact you by email. It's so kind of you to be such patient and helpful! | |
| Jan 17, 2017 at 2:51 | comment | added | student | Thank you so much! After spending a day I eventually understood a little bit. So the key is defining a variational solution of inverse mean curvature flow, and then prove the monotonicity of Hawking mass is preserved even at jumps. Of course there are a lot of difficulties to welcome, but now I can get the picture. Previously it's just too hard for me to picture the jump case, but since the solution is variational sense, the jump really works. This is a genius trick. The paper is hard to read, but I can understand little by little now. | |
| Jan 14, 2017 at 14:58 | comment | added | Otis Chodosh | btw, you are free to contact me by email (found on my website, linked from my profile) if you would like answers longer than 500 characters | |
| Jan 14, 2017 at 14:57 | comment | added | Otis Chodosh | .... This can be seen from property 1.4. note that huisken ilmanen only consider flows from smooth initial data, since they can use their approximation theory (Lemma 5.6) to get information about C^1,1 outer minimizing hulls, which is what they actually need for part of the penrose inequality | |
| Jan 14, 2017 at 14:57 | comment | added | Otis Chodosh | As for your second question, "why is an outward minimizing surface the level set of a function" ? Note that (assuming a subsolution at infinity) given any precompact open set E_0 with smooth boundary, HI produce a weak solution to IMCF in their sense (Theorem 3.1). Note that this produces a function u so that int{u<=0} is the outer minimizing hull of E_0. ... | |
| Jan 14, 2017 at 14:51 | comment | added | Otis Chodosh | ... but it might be best to stick with their first paper so as to not get confused. | |
| Jan 14, 2017 at 14:51 | comment | added | Otis Chodosh | Huisken and Ilmanen's paper is indeed a hard read, but I find that I learn something each time I open it. Lemma 7.1 in HI proves that in an AF manifold, the solutions become spherical in the blown-down sense. Some further information is also available in Lemma 7.4. If you want more information about the asymptotics of the flow, HI have written a followup paper (that only considers Euclidean space) "higher regularity of the inverse mean curvature flow" pubman.mpdl.mpg.de/pubman/item/escidoc:52807/component/… ... | |
| Jan 13, 2017 at 23:13 | comment | added | student | Thank you very much for the reference. I looked through Huisken's paper (2001) a little bit, but I couldn't find the asymptotic convergence result of an outward minimizing surface under mean curvature flow. Yes, the solution exists for all time if initial surface is outward minimizing, but why it must converge to a sphere? Is it just an implication from some result? Could you please specify a page? Also why an outward minimizing surface must be a level surface of some function? I'm getting more and more confused. Thanks for your time! | |
| Nov 12, 2016 at 17:53 | comment | added | Otis Chodosh | An open set $\Omega$ (with finite perimeter) is outer minimizing if for any open set $\Omega'$ containing $\Omega$, we have $|\partial^*\Omega|\leq |\partial^*\Omega'|$. For your second question, in the asymptotically flat case, this is true (see ams.org/mathscinet-getitem?mr=1916951), but not in general (see ams.org/mathscinet-getitem?mr=2629514). | |
| Nov 10, 2016 at 20:23 | comment | added | student | Could you tell me what do you mean by "outer minimizing" in your answer? Does it mean the set is a supersolution with obstacle inside? I learned in your first reference that star-shaped hypersurfaces evolve into a sphere by inverse mean curvature flow, so is it true that the "outer minimizing" set also evolve to a sphere by inverse mean curvature flow? Thanks! | |
| Mar 29, 2014 at 6:54 | vote | accept | Jer | ||
| Mar 28, 2014 at 16:13 | comment | added | Otis Chodosh | You're welcome! | |
| Mar 28, 2014 at 11:56 | comment | added | Jer | Ok, I will compute the inverse mean curvature flow relevant to the Riemannian Schwarzschild metric and read the papers you recommended to me. Thank you very much! | |
| Mar 26, 2014 at 4:08 | history | edited | Otis Chodosh | CC BY-SA 3.0 |
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| Mar 26, 2014 at 2:30 | comment | added | Jer | Oh,I am sorry...that's right. I should use "we assumed" | |
| Mar 25, 2014 at 3:43 | comment | added | Otis Chodosh | @Jer, I'm sorry but I am having difficulty understanding your question. Could you please clarify? In particular, by "we admit" do you mean "we assumed"? | |
| Mar 25, 2014 at 2:09 | comment | added | Jer | I am wondering that in classic case, we admitted this flow could be start from a minimal surface and using this initial condition to make other conclusions. But till Huisken and Ilmanen, people then actually make it clear with weak inverse mean curvature flow. is it right? | |
| Mar 24, 2014 at 14:05 | comment | added | Otis Chodosh | Sorry, I don't understand your question, can you rephrase it? | |
| Mar 24, 2014 at 8:40 | comment | added | Jer | Does that mean that in the classic condition we have admitted the existence of the inverse mean curvature flow starting from a minimal surface? | |
| Mar 23, 2014 at 17:44 | history | edited | Otis Chodosh | CC BY-SA 3.0 |
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| Mar 23, 2014 at 17:39 | history | answered | Otis Chodosh | CC BY-SA 3.0 |