Timeline for Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Feb 22, 2016 at 16:33 | comment | added | Willie Wong | @Rbega points out an important point! Indeed that an $\epsilon$-regularity type result holds generally (see Theorem 4.2 in De Lellis's survey). The issue is more that the approach via $\epsilon$-regularity doesn't hold, due to that assumptions required to prove the necessary theorem does not hold sufficiently generally. | |
| Feb 22, 2016 at 15:53 | comment | added | Rbega | One comment. $\epsilon$-regularity of either the volume type (i.e., Allard's theorem) or total curvature type (i.e., a generalized Choi-Schoen theorem as in the question) both hold quite generally. The difficulties one encounters in developing the regularity theory in higher co-dimension lie elsewhere. The essential difficulty is to show that there are enough places where the hypotheses of the $\epsilon$-regularity results hold. | |
| Feb 22, 2016 at 14:33 | comment | added | Deane Yang | I vaguely recall that the SSY paper is pretty readable, and the proof not particularly hard by PDE standards. The proof is by Moser iteration, which uses only integration by parts, the Sobolev inequality, and the Holder inequality. As I've mentioned elsewhere, the key critical new idea (due, I believe, to Sacks-Uhlenbeck) is an improvement of Kato's inequality (which is really just Cauchy-Schwarz). Here, SSY's proof is enough but overly messy. It's easier to just work out your own proof by solving a finite-dimensional constrained minimization problem. | |
| Feb 22, 2016 at 14:22 | comment | added | Willie Wong | (Side comment: I am no expert on this, so don't ask me about technical details.) | |
| Feb 22, 2016 at 14:15 | history | answered | Willie Wong | CC BY-SA 3.0 |