Timeline for Extensions of minimal hypersurfaces
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2020 at 17:31 | comment | added | Leo Moos | @RBega2 In any case you're right the Nadirashvili example is troublesome, I'm happy to concede. By rescaling and Allard regularity you can have $M$ arbitrarily close to the plane which does not admit 'global' extensions. However I'll leave the question up, in the hope that somebody might comment on Question 2 in particular. | |
| Dec 19, 2020 at 17:24 | comment | added | Leo Moos | @RBega2 Thank you for the comment; I had forgotten about Nadirashvili's surfaces. I was a bit vague in my statement, and avoided the term 'proper immersion' because I was worried that the small singular set might make this fail on a technicality. I wanted to avoid using too many terms from GMT, but to be precise 'globally defined' would mean something like the following: the surface $\tilde{M}$ has locally finite area densities, and defines a stationary varifold in $\mathbf{R}^{n+1}$. I guess the local finiteness would work like a GMT-analog of properness. | |
| Dec 19, 2020 at 17:07 | comment | added | RBega2 | What does "globally defined" mean? A proper immersion? If so then one can use Nadirashvilli's examples to see that one can't even ensure this. In any event I doubt there is any checkable condition. One sometimes sees real analyticity up to the boundary to extend a little bit, but that is far from what you are asking. | |
| Dec 19, 2020 at 16:59 | history | asked | Leo Moos | CC BY-SA 4.0 |