Timeline for A basic question about compact $C^1$ surfaces with boundary
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2019 at 15:58 | vote | accept | MathLearner | ||
| Oct 25, 2019 at 15:50 | answer | added | Ben McKay | timeline score: 2 | |
| Oct 25, 2019 at 15:47 | comment | added | MathLearner | @BenMcKay Yes, I just added this assumption too. | |
| Oct 25, 2019 at 15:47 | history | edited | MathLearner | CC BY-SA 4.0 |
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| Oct 25, 2019 at 15:46 | comment | added | Ben McKay | Is the surface $S$ compact? | |
| Oct 25, 2019 at 15:46 | history | edited | MathLearner | CC BY-SA 4.0 |
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| Oct 25, 2019 at 15:44 | comment | added | MathLearner | @BenMcKay I have done that. We assume the surface is simply-connected. | |
| Oct 25, 2019 at 15:03 | comment | added | Ben McKay | @MathLearner: if you don't assume that there is a $C^1$ diffeomorphism between the manifold and a closed ball, what do you assume? Every $C^1$ manifold is locally $C^1$ diffeomorphic to a ball, so the annulus is a counterexample to existence of a global $C^1$ diffeomorphism. Can you perhaps rewrite the question to make your hypotheses precise? | |
| Oct 25, 2019 at 13:22 | comment | added | MathLearner | @BenMcKay We have only assumed assumed that locally there is a C^1 map between the manifold and closed ball. My problem is how to make this map a global map. Getting a bound after seems straightforward. Thanks. | |
| Oct 25, 2019 at 8:20 | review | Close votes | |||
| Nov 7, 2019 at 3:05 | |||||
| Oct 25, 2019 at 8:03 | comment | added | Ben McKay | You have assumed the existence of a $C^1$ diffeomorphism between a closed ball and your submanifold. Hence they are homeomorphic, so both are compact. Every $C^1$ map between manifolds has $C^0$ bound locally on its differential. By compactness, this local bound gives a globally bounded differential. | |
| Oct 25, 2019 at 4:42 | comment | added | MathLearner | @RyanBudney Thank you Ryan, but I still can't figure out how to construct such map? Could you please elaborate a little more on this? | |
| Oct 25, 2019 at 3:20 | comment | added | Ryan Budney | Partitions of unity come for free unless you are using some non-standard definition of a surface. | |
| Oct 24, 2019 at 23:38 | comment | added | MathLearner | @RyanBudney Compactness and probably partition of unity? | |
| Oct 24, 2019 at 23:10 | comment | added | Ryan Budney | So you are asking that there is a diffeomorphism between the surface and a ball, together with what looks like some control on the norm of the derivative? If that's what you want, yes, it follows from compactness. | |
| Oct 24, 2019 at 22:36 | history | edited | MathLearner | CC BY-SA 4.0 |
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| Oct 24, 2019 at 22:25 | comment | added | MathLearner | @RyanBudney I added the assumption that $S$ is simply connected co-dimension 1 surface. | |
| Oct 24, 2019 at 22:24 | history | edited | MathLearner | CC BY-SA 4.0 |
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| Oct 24, 2019 at 22:23 | comment | added | MathLearner | @BenMcKay Yes, you are right. Assume that $S$ is diffeomorphic to a closed ball. | |
| Oct 24, 2019 at 20:06 | comment | added | Ryan Budney | Your question is rather vague, and appears to be using terminology different than what people use in differential topology/geometry. Generally co-dimension one submanifolds are not discs or balls, if that's what your question is concerned with. | |
| Oct 24, 2019 at 19:59 | comment | added | Ben McKay | When you ask about constructing such a map, what data do you provide? For example, do you know $S$ as a Riemannian manifold, with an explicit metric, and then I have to find $F$? | |
| Oct 24, 2019 at 19:28 | comment | added | Ben McKay | What is $S$ is an annulus, not $C^1$ diffeomorphic to a closed ball? | |
| Oct 24, 2019 at 19:26 | comment | added | Ben McKay | Is $S$ a hypersurface in Euclidean space? | |
| Oct 24, 2019 at 19:23 | history | asked | MathLearner | CC BY-SA 4.0 |