Timeline for Harmonic spinors on closed hyperbolic manifolds
Current License: CC BY-SA 3.0
31 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 7, 2019 at 17:17 | vote | accept | Danny Ruberman | ||
| May 7, 2019 at 17:17 | |||||
| May 7, 2019 at 12:33 | comment | added | Danny Ruberman | @GorapadaBera Recently, McKee Krumpak and I found examples of closed hyperbolic 3-manifolds with harmonic spinors. A paper is in preparation. I don't know examples where it is shown that the kernel of the Dirac operator is trivial. | |
| May 7, 2019 at 3:04 | comment | added | Gorapada Bera | Hello,Professor Danny , since this question was active here 1 year ago I am asking here for three dimension agin. Are there any updates for closed hyperbolic three manifolds having trivial or non trivial harmonic spinor? | |
| Dec 5, 2018 at 2:29 | vote | accept | Danny Ruberman | ||
| Dec 5, 2018 at 2:29 | |||||
| Mar 21, 2018 at 2:49 | answer | added | Danny Ruberman | timeline score: 12 | |
| Mar 21, 2018 at 2:33 | comment | added | Danny Ruberman | I now have examples (with Ratcliffe and Tschantz) in dimension 4; I will put some details in an answer below. | |
| Mar 27, 2017 at 0:40 | comment | added | Danny Ruberman | Thanks; I stand corrected. (Hitchin's paper, of course, attributes this to Atiyah, so I should have known better.) Your second question sounds hard; it seems that you're asking about whether certain elements in the spin cobordism group can be represented by hyperbolic manifolds. | |
| Mar 26, 2017 at 15:32 | comment | added | Ben Wieland | The result you attribute to Hitchin is due to Atiyah (or Riemann). . . He systematically studies how the parity the changes as the spin structure changes. Do you know what happens with 9- or 10-dimensional hyperbolic manifolds? Do they have uniformly even parity? | |
| Jan 7, 2017 at 3:49 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 15, 2016 at 6:58 | comment | added | mme | I guess one has examples of harmonic spinors when the index theorem does not demand it in high dimension as a result of Crowley-Schick-Steimle. Of course, still nothing in dimensions 3 or 4, or examples that are actually hyperbolic. | |
| Dec 8, 2016 at 3:24 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 8, 2016 at 2:40 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 9, 2016 at 1:51 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 9, 2016 at 1:20 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 10, 2016 at 1:16 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 11, 2016 at 0:37 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 11, 2016 at 0:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 11, 2016 at 23:57 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 17, 2016 at 19:09 | comment | added | Danny Ruberman | Misha: Thanks. We have a visitor next year who does automorphic forms, and I am hoping to take this up with him. I was referring to a more topological approach of mine, which I tried to get a student interested in. | |
| Apr 16, 2016 at 16:44 | comment | added | Misha | Danny: My suggestion is to outsource this question not to your student but to somebody in automorphic forms. There should be plenty of these guys in the greater Boston area. | |
| Apr 12, 2016 at 20:29 | comment | added | Danny Ruberman | I have a feeling that you can construct examples, and even an idea for a construction. I tried to get a student to follow up on this plan, but, well, kids these days... I'd be happy to share my idea off-line if you're interested. I also think that the suggestion from @Misha is a good one. In any event the question is still open. | |
| Apr 12, 2016 at 7:20 | comment | added | Ryan Budney | Hi Danny, I just noticed this question of yours now... a year late. Do you have a guess as to which way the answer might go, or is this pretty much a wide-open question? | |
| Nov 24, 2015 at 18:58 | comment | added | Misha | I know nothing about harmonic spinors, but my suggestion is to talk to people who study automorphic forms. There is a neat trick (using theta-series and congruence subgroups) for constructing compact hyperbolic manifolds supporting nonvanishing automorphic forms of certain kind, which may work in your setting. I learned this trick from Gordan Savin years ago, see Proposition on page 204 of his paper "Cusp forms", Israel Math Journal, 1992. You have to get lucky for this trick to work with harmonic spinors since it requires absolute convergence, but it's worth asking. | |
| May 19, 2015 at 16:54 | comment | added | Danny Ruberman | I spent a couple of minutes looking for a reference, and found that it comes from Chern's paper, On Curvature and Characteristic Classes of a Riemann Manifold. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg October 1955, Volume 20, Issue 1-2, pp 117-126. Of course, this predates the Dirac operator and the index theorem, but the characteristic class computation is there. | |
| May 19, 2015 at 10:37 | comment | added | Matthias Ludewig | Ah, that is interesting. I didn't know that. | |
| May 18, 2015 at 14:54 | comment | added | Danny Ruberman | @MatthiasLudewig As far as I recall Pontrjagin classes of a hyperbolic manifold are trivial in real cohomology, using Chern-Weil theory. (I don't have a handy reference, but compare Igor Belgradek's comments in mathoverflow.net/questions/107716/…). This implies that the A-hat class is zero, so the index of the Dirac operator is 0. That's kind of the point of the question, to find examples where the kernel is non-zero in a setting where the A-S index theorem doesn't help. | |
| May 17, 2015 at 21:27 | comment | added | Matthias Ludewig | On any even-dimensional spin manifold with non-vanishing $\hat{A}$-genus, there exist harmonic spinors, due to Atiyah-Singer, whatever metric you choose. Now you just need to find a negatively curved spin manifold with non-vanishing $\hat{A}$-genus... | |
| Apr 17, 2015 at 17:46 | comment | added | Igor Rivin | By the way, in an earlier paper of Baer, he shows that every 3-manifold has a metric which has harmonic spinors. Of course, it is not clear that the hyperbolic metric is it... | |
| Apr 17, 2015 at 14:13 | answer | added | Igor Rivin | timeline score: 1 | |
| Apr 17, 2015 at 5:48 | history | edited | Ricardo Andrade |
edited tags
|
|
| Apr 17, 2015 at 1:09 | history | asked | Danny Ruberman | CC BY-SA 3.0 |