Timeline for Diameter of immersed surfaces with bounded from above mean curvature
Current License: CC BY-SA 3.0
18 events
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| Dec 4, 2015 at 12:53 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 3, 2015 at 21:15 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 3, 2015 at 18:10 | comment | added | Yasha | Let us continue this discussion in chat. | |
| Dec 3, 2015 at 18:06 | vote | accept | Yasha | ||
| Dec 3, 2015 at 17:40 | comment | added | Yasha | That's write I am not claiming that, so everything depends on the geometry of $(X,g)$. I would also be interested in the version of the question with $f$, (also depending on geometry) but it is not essential and probably too complicated. | |
| Dec 3, 2015 at 16:53 | comment | added | Robert Bryant | I see that you have removed the function $f$ and replaced it by $\epsilon$ and $\delta$. That changes the question completely. You now aren't claiming (are you?) that $\delta$ can be chosen in terms of $\epsilon$ without regard to the geometry of $X$. | |
| Dec 3, 2015 at 16:40 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 3, 2015 at 16:27 | comment | added | Yasha | In your example there is a lower bound on the volume for embeddings in the class of $\Sigma \times \{1\} \subset X$, so how can it be a counterexample? | |
| Dec 3, 2015 at 16:27 | answer | added | Rbega | timeline score: 2 | |
| Dec 3, 2015 at 16:24 | comment | added | Robert Bryant | Sadly, no. See my comment directly above. | |
| Dec 3, 2015 at 16:21 | comment | added | Yasha | Edited again, does this help? | |
| Dec 3, 2015 at 16:19 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 3, 2015 at 16:10 | comment | added | Robert Bryant | And what about $(X,g)$? If you take $X = \Sigma\times S^1$ and let $g$ be a product metric (for some metric on $\Sigma$), then $\Sigma\times\{1\}\subset X$ is totally geodesic (and embedded to boot), so the mean curvature is identically $0$, and yet there obviously is no relationship between the diameter of $\Sigma$ (and hence it's diameter in $X$) and its volume. | |
| Dec 3, 2015 at 15:50 | answer | added | ε-δ | timeline score: 3 | |
| Dec 3, 2015 at 15:50 | comment | added | Yasha | Yes sorry I meant everything to be compact without boundary of course, I edited to say closed. | |
| Dec 3, 2015 at 15:48 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 3, 2015 at 15:37 | comment | added | Robert Bryant | You really need to sharpen your question: For example, are you only considering compact surfaces $\Sigma$ without boundary? Are you making any hypotheses on the ambient Riemannian manifold $(X,g)$? Without some hypotheses such as these, it is hopeless to prove any such estimate, since counterexamples are easily constructed. | |
| Dec 3, 2015 at 15:29 | history | asked | Yasha | CC BY-SA 3.0 |