Timeline for The differentiability of the distance function on asymptotically flat manifolds
Current License: CC BY-SA 4.0
3 events
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| Mar 21, 2022 at 12:58 | comment | added | Willie Wong | The mean curvature in general doesn't provide a bound on the injectivity radius of the normal exponential map. At least you would need that the whole second fundamental form is positive definite. | |
| Mar 19, 2022 at 17:23 | comment | added | Laithy | Under your sufficient conditions, one could also prove, using a continuity argument, that $trA$ is bounded from below, where $A$ is the second fundamental form on $S_r$. $trA$ obeys a first order Riccatti type ODE in which $R_{00}$ provides the forcing term, and if that's small, then $trA = \frac{2}{r} + o(\frac{1}{r})$. I am not sure however if that implies that $r$ is differentiable everywhere. | |
| Mar 19, 2022 at 16:47 | history | answered | Willie Wong | CC BY-SA 4.0 |