Timeline for Smoothability of compact Alexandrov surfaces with curvature bounded from below
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 28, 2012 at 14:43 | comment | added | Thomas Richard | This paper of D. Ramos arxiv.org/abs/1109.5554 gives a way to smooth out the cone points without messing with gluing. It constructs a smooth solution to Ricci flow for surfaces with conical singularities. | |
| Jan 29, 2012 at 3:16 | history | edited | Vitali Kapovitch | CC BY-SA 3.0 |
deleted 67 characters in body
|
| Jan 29, 2012 at 3:14 | comment | added | Vitali Kapovitch | @Anton No I don't have a short proof. I admit it was wrong to call what I describe an "idea of the proof" as the hard part of the proof is missing but I haven't looked at Alexandrov's book in a long time and I could only remember the general structure of the argument when I wrote my answer. | |
| Jan 29, 2012 at 2:07 | comment | added | Anton Petrunin | BTW, Once you are trying to make the "idea of the proof" work, you get into tedious details. $$ $$ That means: If you have a complete clean and short proof then I am very interested. | |
| Jan 28, 2012 at 23:05 | comment | added | Vitali Kapovitch | @Igor: Alexandrov doesn't mention smoothing but that is a minor point. He proves that $X$ can be approximated by polyhedral surfaces with the same lower curvature bound. By construction the approximating surfaces are smooth and have constant curvature outside of the cone points. Cone points are isolated and it's trivial to smooth such metrics with the same lower curvature bounds the way you smooth a flat cone by a surface of revolution. The fact that he considers lower curvature bound 0 is a minor point too as all arguments are local. He discusses the general case in chapter 12. | |
| Jan 28, 2012 at 19:43 | comment | added | Igor Belegradek | @Vitali, I see nothing about smoothing in Lemmas 1-3. Also he only considers the case of positive curvature, but this must be a minor point. Smoothing at vertices is not a minor point. It may be fixed by the realization theorem in Chapter XII, section 2 which says every point in a space of curvature $\ge k$ has a neigborhood isometric to a convex surface is the 3-dimensional model space of constant curvature $k$. Of course, any convex surfaces can be approximated by smooth ones, giving the smoothing, and hopefully these smoothings can be glued. | |
| Jan 28, 2012 at 18:44 | comment | added | Vitali Kapovitch | @Igor sorry,it's in Chapter 7. In case the chapter number differs in your edition the chapter is called "Existence of a closed convex surface with a given metric". | |
| Jan 28, 2012 at 18:41 | history | edited | Vitali Kapovitch | CC BY-SA 3.0 |
added 11 characters in body
|
| Jan 28, 2012 at 17:54 | comment | added | Igor Belegradek | @Vitali, on your edit: which chapter is this in? I have the new edition of Alexandrov's "Intrinsic Geometry of Convex Surfaces" published in his collected works and almost every chapter has section 6. | |
| Jan 28, 2012 at 15:53 | history | edited | Vitali Kapovitch | CC BY-SA 3.0 |
corrected references
|
| Jan 28, 2012 at 14:15 | comment | added | Igor Belegradek | After looking at Alexandrov-Zalgaller's argument, I must admit I would not be comfortable quoting it as an answer to the OP's question. Given that the question is of considerable interest it is unfortunate that there seem to be no proof in the literature. | |
| Jan 28, 2012 at 13:32 | comment | added | Thomas Richard | Thanks, I didn't realized that Alexandrov already proved the result about the topology of 2-dimensional Alexandrov surfaces. Maybe my mistake come from "Geometry IV" of Reshetnyak, which defines surfaces of bounded integral curvature assuming at the beginning that the underlying space is a topological surface. By the way, Alexandrov-Zalgaller is on my desk these days, I'll have a look at it. Thanks again. | |
| Jan 28, 2012 at 13:27 | vote | accept | Thomas Richard | ||
| Jan 28, 2012 at 0:22 | comment | added | Vitali Kapovitch | @Igor sorry, I don't have the book and I haven't looked at in in quite a while but my recollection is that I read it there. | |
| Jan 28, 2012 at 0:06 | comment | added | Igor Belegradek | Where exactly is this argument in Alexandrov-Zalgaller's book? | |
| Jan 27, 2012 at 23:31 | history | answered | Vitali Kapovitch | CC BY-SA 3.0 |