Timeline for geodesic balls in the conformal change
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 30, 2022 at 13:42 | review | Close votes | |||
| Dec 5, 2022 at 3:02 | |||||
| Nov 3, 2018 at 4:26 | comment | added | mathpde | Thanks, I just want $\|\phi\|_{C^2}$ to be bounded with that lower and upper bound. Could you please show me a little more calculation about that minimizing geodesic for $\tilde{g}$ of length $\alpha$ is a curve of length at most $C\alpha$ with respect to $g$? or just use the definition of minimizing geodesic $min \int^b_a | r'(t)|\,dt$? where $r(t)$ is the curve between $p$ and $x$. Thanks! | |
| Nov 3, 2018 at 3:58 | comment | added | Willie Wong | If you just postulate upper and lower bounds for $\phi$, then the minimizing geodesic for $\tilde{g}$ of length $\alpha$ is a curve of length at most $C \alpha$ with respect to $g$. So this guarantees the inclusion of balls. // The relation to exponential map is much trickier: play around with $\mathbb{S}^2$; you can rather explicitly write down the conformal changes in that case. | |
| Nov 3, 2018 at 3:50 | comment | added | Willie Wong | As stated, I doubt it. Why is $\|\phi\|_{C^2}$ the thing that is bounded? You can take $\phi$ very small and highly oscillatory so that the first and second derivatives are huge. Just by measuring distances this would give a counterexample. Do you just want $\|\phi\|_{C^2}$ to be bounded and that $\phi$ itself to have some upper and lower bound? | |
| Nov 3, 2018 at 2:56 | history | edited | mathpde | CC BY-SA 4.0 |
added 13 characters in body
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| Nov 2, 2018 at 22:08 | history | asked | mathpde | CC BY-SA 4.0 |