Timeline for answer to Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space? by Willie Wong
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Aug 18, 2013 at 4:20 | vote | accept | Ritwik | ||
| Aug 16, 2013 at 15:36 | history | edited | Willie Wong | CC BY-SA 3.0 |
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| Aug 16, 2013 at 15:33 | comment | added | Willie Wong | @Ritwik The devil is in the details! What Banach space are you going to use? Even if you mod out symmetries, because of the exponential weight $e^{2y}$, the class of $C^k$ functions relative to the hyperbolic metrics is quite different from the class relative to the Euclidean metric. (Similarly for Lebesgue type spaces.) Because of the exponential weight you will probably need a function space that can accommodate fast growth at infinity. Now, I've never tried your argument, since I don't even know where to start with the choice of functions spaces. So I cannot tell you why it is hard. | |
| Aug 16, 2013 at 15:07 | comment | added | Ritwik | I see. So in this case, proving openness is the difficult part. Proving the set is closed is obvious, as you have pointed out. Is there a reason why proving openness is hard here? Typically one uses the implicit function theorem on some appropriate Banach space to get openness. Is there some obvious reason that approach can't work? | |
| Aug 16, 2013 at 14:48 | history | answered | Willie Wong | CC BY-SA 3.0 |