Timeline for What does Colding-Minicozzi theory say about convergence with multiplicity?
Current License: CC BY-SA 4.0
9 events
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| Oct 25, 2021 at 16:47 | comment | added | Leo Moos | That's interesting, particularly the no-tilt part---thanks again! | |
| Oct 25, 2021 at 12:46 | comment | added | RBega2 | I just remembered this paper arxiv.org/abs/1503.02190 of Brian White that more or less directly deals with the uniformly bounded genus version of your question. | |
| Oct 25, 2021 at 12:08 | comment | added | Leo Moos | I wasn't aware of the paper, thanks for the reference---and indeed thanks for all your comments; they've been very helpful. | |
| Oct 24, 2021 at 12:57 | comment | added | RBega2 | Yes assuming you are restricted to embedded surfaces a uniform area bound together with simple connectedness has a sheeting property. This is actually a result of Schoen and Simon (supposedly proved in degruyter.com/document/doi/10.1515/9781400881437-007/html) though I'll admit to never reading that paper. | |
| Oct 24, 2021 at 9:52 | comment | added | Leo Moos | I've taken a look at the notes that you linked to, thanks! I still need some time to translate the arguments to the minimal setting, because I expected that the argument would be a bit simpler in this setting, e.g. by resorting to a blow-up argument. Could you confirm one thing: 'helicoid-like' portions can't actually appear when the mass is bounded, correct? | |
| Oct 22, 2021 at 22:42 | comment | added | RBega2 | Actually, reading your question more carefully, I realize you are assuming the convergence is with finite multiplicity. In this case, one should be able to use a result of Ilmanen (which is related to Colding Minicozzi's ideas) to get (under uniform genus bounds and some assumptions on boundaries) a uniform total curvature bound -- Page 32 of people.math.ethz.ch/~ilmanen/papers/notes.ps. This is then a classical removable singularities picture (as described later in the notes). | |
| Oct 22, 2021 at 9:30 | comment | added | Leo Moos | Shoot, I always forget about this. What is the right way of formalising a bound on the topology: something like $\mathrm{genus} \, M_j + \mathrm{rank} \, \pi_1 (M_j) \leq C$? By the way, I'm surprised by what you explain about the sequence needing to be "global". I assumed that---with the topology bound---a sheeting theorem would be available even in this "local" setting. | |
| Oct 21, 2021 at 21:24 | comment | added | RBega2 | Without any topological assumption their work gives you nothing (as necks could, in principle, concentrate everywhere). Even with a genus bound, their work is only going to give information if the sequence is "global" in the sense of their papers (i.e. rather than having a fixed ball the boundaries lie in balls of radius tending to infinity). You should also keep in mind that one can have necks without genus (e.g. in the catenoid or Riemann examples). | |
| Oct 21, 2021 at 20:38 | history | asked | Leo Moos | CC BY-SA 4.0 |