Timeline for A possible generalization of the exponential map
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2018 at 5:07 | comment | added | JSCB | A typo in the comment above, I meant to write $f(t,C)$ is a minimal submanifold under the metric $g_t$. | |
| Jun 23, 2018 at 4:03 | comment | added | JSCB | Hence by the inverse function theorem, $f(t,x)$ is a diffeomorphism for small $t$, so we get a foliation. | |
| Jun 23, 2018 at 4:01 | comment | added | JSCB | I don't quite see why we need the maximum principle here. Fixing a $\mathbf{v}$, let $f(t,x)$ as a function from $[0,1]\times B$ to $B$ such that $f(0, \cdot)$ is the identity and for each $t$, (1) $f(t, \cdot)=f(0, \cdot)$ on $\partial B$, and (2) for each slice $C\subset B$ (having its first coordinate being constant), $f(C)$ is a minimal submanifold. Since $\det \partial f(t,x)/\partial x\ne 0$ at $x=0$, and “everything is continuous” (I hope I am correct about this), we have $\det \partial f(t,x)/\partial x\ne 0$ for small $t$. | |
| Jun 22, 2018 at 20:16 | comment | added | Rbega | @ᴊᴀsᴏɴ Yes, I am only addressing the first question, the second seems much harder. I have also fixed the typo. Regarding higher co-dimension, it might work as you say, they only part I'm unsure about is whether the perturbations are still foliations (it's easier to see this in the codimension one case as you can use the maximum principle). | |
| Jun 22, 2018 at 20:12 | history | edited | Rbega | CC BY-SA 4.0 |
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| Jun 22, 2018 at 17:35 | comment | added | JSCB | I think for higher codimension n-k, we can just use n-k dimensional foliation. And we can consider the degree of the map $\phi_t$ between the Grassmanian manifold Gr(n,n-k). | |
| Jun 22, 2018 at 17:09 | comment | added | JSCB | Just to be clear, you have addressed my first question, right? Also on paragraph 4 line 3, do you mean $\partial \mathcal{F}_{\mathbf{v}}(t)=\mathcal{G}_{\mathbf{v}}$? | |
| Jun 22, 2018 at 12:06 | history | edited | Rbega | CC BY-SA 4.0 |
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| Jun 22, 2018 at 11:49 | history | answered | Rbega | CC BY-SA 4.0 |