Timeline for Explicit metrics
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 6, 2011 at 19:46 | answer | added | Robert Bryant | timeline score: 15 | |
| Jun 6, 2011 at 18:40 | comment | added | Robert Bryant | I'll just point out that Bill's proposed metric of nonpositive curvature for open sets in the plane does not, in general yield a $C^2$-metric. For example, for the unit disk in polar coordinates, the metric Bill proposes would be written in the form $g = (dr^2 + r^2\ d\theta^2)/(1-r)^2$, and a short computation shows that $K = -1/r$ for this metric, so, not surprisingly, it's not $C^2$ at the origin. (In general, for a domain in the plane, the distance to the boundary is continuous, but not even $C^1$.) | |
| Sep 25, 2010 at 6:23 | answer | added | j.c. | timeline score: 9 | |
| Sep 24, 2010 at 23:49 | comment | added | Dmitri Panov | Yes, I think this colour scheme works better. Thanks also for the example $S^3$\ Whitehead continuum! | |
| Sep 24, 2010 at 22:48 | comment | added | Bill Thurston | I changed the color scheme. Do you think this will help reduce the interpretation as a domain with holes? Is it more clear to someone who hasn't seen it before? | |
| Sep 24, 2010 at 22:43 | comment | added | Bill Thurston | @Dmitri: Thanks for telling me your initial interpretation of the figure. I overlooked that possibility because I already knew what i wanted it to mena. I think I will change the color scheme. It's trickier in higher dimensions to make it negatively curved, because of the sectional curvatures for planes parallel to the boundary. The Poincarê disk is negatively curved, so sometimes it can be done. It can't always be done, because there are contractible open subsets of $\mathbb R^3$ that are not homeomorphic to $\mathbb R^3$, e.g. $S^3 \setminus $Whitehead contrinuum | |
| Sep 24, 2010 at 22:15 | comment | added | Dmitri Panov | Dear Bill, I like very much the picture and the idea of this non-positively curved metric (though, funny enough in the beginning I thought, that these circles are wholes in the paddle), and I have a question. Does this type of construction works for domains in higher-dimensional spaces? For some domains in R^n (maybe convex)? Or. for example, suppose we want to prove that a complement to some hyperplane arrangement in $C^n$ is $K(\pi,1)$, is there a chance that by some similar kind of method we could find a complete, non-positively curved metric on its complement? (this would do the job). | |
| Sep 24, 2010 at 19:32 | comment | added | Piero D'Ancona | @Bill: oops. Of course. | |
| Sep 24, 2010 at 14:09 | answer | added | Zachary Treisman | timeline score: 6 | |
| Sep 24, 2010 at 8:43 | comment | added | Bill Thurston | @Piero: $S^2$ does not have a negatively curved metric. Any metric on a surface has total Gaussian curvature equal to $2\pi$ times its Euler characteristic, which for $S^2$ is $4 \pi$. The more general question is to find geometric definitions of metrics where the curvature has a consistent sign, either positive, zero, or negative, but for spheres, there are some easy solutions (any convex shape), and for toruses, it's too hard to do qualitatively in space, since it's an equation rather than inequality. So I just posed it for surfaces of negative Euler characteristic, like the double torus. | |
| Sep 24, 2010 at 8:34 | comment | added | Bill Thurston | @Deane, @Ryan: It sounds like you're on a path ... where it leads, I don't know. Here's an idea that occured to me but haven't pursue: what if you interpret a surface in space as the stereographic image of a surface in $S^3$? The metric on $S^3$ can be determined geometrically once you specify the image of the equatorial 2-sphere: from that, you can get all great spheres. Is there any good geometric way to ensure that the lifting to $S^3$ is negatively curved? Variant: base a conformal change somehow on the relationship to an ellipsoid, or perhaps some other comparison surface. | |
| Sep 24, 2010 at 5:44 | comment | added | Ryan Budney | Dean, okay I agree with you now. Sometimes it takes me a while to process whether an idea is cheezy or meaningful. This one feels a bit like a re-hash of an idea some author called "corrugations". :) | |
| Sep 24, 2010 at 0:48 | comment | added | Deane Yang | Ryan, sounds good and not cheesy to me. | |
| Sep 24, 2010 at 0:13 | comment | added | Ryan Budney | A slightly cheezy modification of Deane Yang's idea would be to remove a large disc from a hyperbolic surface, so large that the complement is a thin regular neighbourhood of a graph in the surface. Embed that regular neighbourhood in Euclidean space much like how one constructs zero Gauss curvature embeddings of cylinders and Moebius bands in Euclidean space. | |
| Sep 23, 2010 at 23:23 | comment | added | Deane Yang | Here's a very naive and noncomformal speculative approach to building a metric of negative curvature on the double torus: Remove a neighborhood of the positively curved part, which is just a positively curved cylinder. Replace by a negatively curved cylinder (which is basically just the positively curved cylinder turned inside out). What I don't know is whether you can adjust a neighborhood of the boundaries to make the two pieces fit together properly. If so, you get an immersed surface of negative curvature. | |
| Sep 23, 2010 at 21:38 | comment | added | Bill Thurston | @Ryan: Yes, that's a fun problem. There are many possible subgroups of $O(3)$, and many possible constructions. An interesting extra geometric condition: when are they equivariant minimal surfaces? Sometimes they're not equivariantly compressible, which I believe means they can be made into minimal surfaces. | |
| Sep 23, 2010 at 20:46 | comment | added | Ryan Budney | Although it doesn't answer your question, I've been looking at a related thing recently. Given a finite group $G$ acting on a surface $\Sigma$, try to find an embedding of $\Sigma$ in $S^3$ such that the action of $G$ on $\Sigma$ extends to an action of $G$ on $S^3$. So there's all kinds of restrictions on when this is possible, but looking at examples can be quite pleasant. | |
| Sep 23, 2010 at 20:46 | comment | added | Bill Thurston | I wouldn't assume that nobody has figured it out. It's such an obvious question, someone could easily have done it, and I could easily not be aware. But if so, I'd like to become aware of it. I haven't made a concerted effort to find the metric---I've thought about it casually from time to time. I wouldn't have posted it if I thought it was so thorny as to be unlikely that someone would know about it or solve in this forum. It's something that should be fun to think about. | |
| Sep 23, 2010 at 20:42 | comment | added | j.c. | Never mind, I read the example more carefully and I understand what you mean - one can easily imagine (though explicit constructions will require some cleverness) a scale function related to the distance in the Euclidean metric from a certain set on the embedded surface which changes the pullback metric on the surface to one with negative curvature everywhere. | |
| Sep 23, 2010 at 20:40 | comment | added | Deane Yang | This is a great question, and it is indeed somewhat surprising that nobody has figured anything like this out yet. I find it rather difficult to envision a negatively curved metric on the double torus, given what it looks like when embedded in $R^3$. We know that it is possible to change the "visual" metric into a negatively curved metric via a conformal factor, but it is highly unclear how to do this without using heavy analytic machinery. I imagine that Bill and other visually oriented mathematicians have already tried to do this, so I doubt it's an easy question to answer. | |
| Sep 23, 2010 at 20:33 | comment | added | j.c. | I'm not sure I understand what you mean by a metric that is "related to an embedding"; for instance, regarding your final example, what kinds of modifications to the metric of an embedded double torus are part of the game? Perhaps that is part of the point of the question? | |
| Sep 23, 2010 at 20:23 | history | asked | Bill Thurston | CC BY-SA 2.5 |