Timeline for Negative curvature in the middle of $R^{3}$
Current License: CC BY-SA 3.0
9 events
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| Jun 25, 2013 at 3:02 | review | First posts | |||
| Jun 25, 2013 at 5:13 | |||||
| Jun 15, 2013 at 4:18 | comment | added | delenda | @Igor Yes! Section 2 does exactly what I asked for. Also, through the References, I found jstor.org.sci-hub.org/stable/2118620 (Section 3) with a different solution. Both are quite complicated, though! I wish there was a simpler example. I'll work my way through these papers, anyway. Thanks. | |
| Jun 15, 2013 at 3:19 | comment | added | Igor Belegradek | I think the paper mathnet.or.kr/mathnet/thesis_file/BKMS-49-3-581-588.pdf in [Bull. Korean Math. Soc. 49 (2012), No. 3, pp. 581–588] does exactly what you want. | |
| Jun 15, 2013 at 3:09 | history | edited | delenda | CC BY-SA 3.0 |
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| Jun 15, 2013 at 3:01 | comment | added | delenda | @Igor By "standard Euclidean metric outside" I mean it IS the standard metric in each point outside N, not just up to isometry. Your construction is of little use here, because it won't generally produce standard flat space outside of a compact set, not even up to isometry, unless $\int f dA=0$, which implies $f$ has positive AND negative values (or is constantly zero). This can be seen by Gauss-Bonnet, as follows. Suppose the space to be standard $R^{2}$ outside N. Take a polygon enclosing N: its angular excess is 0, so $\int k dA=0$ inside the polygon, where $k$ is the curvature. | |
| Jun 15, 2013 at 1:48 | comment | added | Igor Belegradek | What do you mean by the "standard Euclidean metric outside"? Do you mean that after removing your $N$ there is an isometry to the standard $\mathbb R^3$ minus a compact set? BTW, in dimension 2 any smooh nonpositive function on $\mathbb R^2$ can be realized as the scalar (or equaivalently sectional) curvature of a complete Riemannian metric. One simply solves the Jacobi equation $f_{xx}+Kf=0$ where $K$ is the sectional curvature, and then the metric is $dx^2+f^2dy^2$. See page 217 of [Kazdan-Warner, "Curvature Functions for Open 2-Manifolds", Annals of Math. 99, No. 2, (1974), pp. 203-219]. | |
| Jun 15, 2013 at 0:55 | history | edited | delenda | CC BY-SA 3.0 |
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| Jun 15, 2013 at 0:22 | history | edited | delenda | CC BY-SA 3.0 |
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| Jun 14, 2013 at 23:54 | history | asked | delenda | CC BY-SA 3.0 |