Timeline for Isometric embedding of the modular surface
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2020 at 12:26 | comment | added | Robert Bryant | @BS.: Actually, I just realized (silly me) that an isometric immersion of $Y(3)$ is ruled out by Hilbert's Theorem. The point is that the simply-connected cover of $Y(3)$ is the entire Poincaré disk, and Hilbert's Theorem (1901) is that there is no isometric immersion of the entire Poincaré disk into $\mathbb{E}^3$. (Hilbert's Theorem didn't apply to $X(1)$ because of the two conical points. In particular, the composition $\mathbb{H}\to X(1)\to \mathbb{E}^3$ is not an immersion.) | |
| Jul 9, 2020 at 21:20 | comment | added | BS. | That would be a great example ! | |
| Jul 9, 2020 at 18:31 | comment | added | BS. | What about $Y(3)$ ? Conformally, it is a regular tetrahedron without its vertices, quotient of the upper half plane by the torsion-free group $\Gamma(3)$, kernel of the surjective morphism $PSL_2(\mathbb{Z})\to PSL_2(\mathbb{Z}/3)\simeq A_4$. | |
| Jul 5, 2020 at 0:43 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed some typos and run-on sentences. Hopefully improved the readability of the proof.
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| Jul 4, 2020 at 9:42 | vote | accept | Alex Gavrilov | ||
| Jul 3, 2020 at 14:22 | comment | added | Robert Bryant | @AlexGavrilov: Are you asking about "For an isometric immersion of a compact surface with negative Gauss curvature into $\mathbb{E}^3$, the image lies in the convex hull of the image of the boundary"? This is a common exercise: If $f:M\to\mathbb{E}^3$ is an isometric immersion and $h:\mathbb{E}^3\to\mathbb{R}$ is a linear function such that $h\circ f$ is not positive on $\partial M$, then $h\circ f$ is not positive on $M$ since, otherwise, if $h\circ f$ has a positive maximum at $p\in M$, then $f(M)$ will lie on one side of the image tangent plane at $f(p)$, which is impossible since $K<0$. | |
| Jul 3, 2020 at 12:25 | comment | added | Alex Gavrilov | About the convex hull, could you give a reference? | |
| Jul 2, 2020 at 21:45 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed some typos
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| Jul 2, 2020 at 21:26 | history | answered | Robert Bryant | CC BY-SA 4.0 |