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Consider a complete $C^\infty$ Riemannian metric on $\mathbb R^2$ of positive sectional curvature.

  1. Is the metric embeddable as the boundary of a convex subset of $\mathbb R^3$?

  2. Is the embedding unique?

  3. Are there generalizations of 1-2 to complete noncompact surfaces of nonnegative sectional curvature?

  4. What are good references for these matters?

UPDATE:

$\bullet\ $ after doing some reading on the subject I found that the assertion 1 is true in the sense that the surface is isometric, as a metric space, to the boundary of a convex body in $\mathbb R^3$ (as proved by Alexandrov back in 1942). The matter of uniqueness is well-understood.

$\bullet\ $ However, one should not expect the boundary to be smooth, e.g. there are examples of $C^\infty$ metrics of nonnegative curvature on $S^2$ which cannot be isometrically $C^3$-embedded into $\mathbb R^3$.

$\bullet\ $ If the curvature is positive, then smoothness can be achieved as proved by Pogorelov and Nirenberg (independently in the 1950s).

$\bullet\ $ Local smooth isometric embedding for nonnegatively curved surfaces was established by Lin in 1985.

$\bullet\ $ A more recent reference for these matters is the book by Burago and Zalgaller, Geometry III, Encyclopedia of Mathematical Sciences.

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Why is this titled "On Alexandrov embedding theorem"? What is the Alexandrov embedding theorem? (I am probably supposed to know this. In my defense, it is possible that I once knew it years ago, but have forgotten.) –  Deane Yang Apr 21 '10 at 22:34
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Alexandrov embedding theorem says that any positively curved metric on $S^2$ is isometric to the boundary of a convex body (in fact, very little regularity is assumed about the metric: it should be inner and have positive curvature in comparison sense). I know there is a version of this theorem for positively curved metrics on the plane, possibly with some condition on total curvature. Anyway, I think those who are able to answer know the relevant background; I do not. –  Igor Belegradek Apr 21 '10 at 22:52
    
I googled it, and found this as the top hit: mathoverflow.net/questions/22122/… :) –  Nate Eldredge Apr 21 '10 at 22:54
    
@Igor: Thanks! I knew only the theorem for smooth metrics, which is due to Pogorelov and Nirenberg. I imagine that Alexandrov proved it under weaker regularity assumptions. But did he do this before, simultaneously, or after Pogorelov and Nirenberg? –  Deane Yang Apr 22 '10 at 2:18
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@Igor: Thanks for the update on your question. –  Deane Yang Apr 29 '10 at 2:20
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2 Answers

up vote 7 down vote accepted

Is the metric embeddable as the boundary of a convex subset of 3?

YES, it is a limit case of standard Alexandrov's theorem. Moreover one can choose any embedding of cone at infinity and construct the embedding. This is a theorem of Olovyanishnikov --- one of three students of Alexandrov who died in the war.

Is the embedding unique?

NO, but I suspect it is unique once you fixed the convex embedding of the cone at infinity. It might follow from the proof of Pogorelov's theorem but I was not able to check his proof.

Are there generalizations of 1-2 to complete noncompact surfaces of nonnegative sectional curvature?

I'm not sure what you mean --- if it has strictly positive curvature at one point then it is automatically $\mathbb R^2$. If it is $\mathbb R^2$ then it is all the same.

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Just to confirm, is it true that any complete metric of nonnegative sectional curvature on $\mathbb R^2$ is isometric to the boundary of a convex subset of $\mathbb R^3$? I have never seen this stated in the literature. –  Igor Belegradek Apr 22 '10 at 3:16
    
@Igor, it is in the paper linked to theorem of Olovyanishnikov. –  Anton Petrunin Apr 22 '10 at 3:34
    
@Anton, I cannot find the relevant statement in Olovyanishnikov's paper but on the same site I found an earlier paper of Alexandrov: "Existence of a convex polyhedron and of a convex surface with a given metric. (Russian. English summary) Rec. Math. [Mat. Sbornik] N.S. 11(53), (1942). 15--65.", which in Theorem 3 proves what I asked for for nonnegatively curved metrics on the plane with positive curvature at one point. –  Igor Belegradek Apr 22 '10 at 4:33
    
@Igor, see §3 "Основная теорема" –  Anton Petrunin Apr 22 '10 at 14:54
    
Thanks, Anton! By the way, this site for Mat.Sbornik is amazing. I never expected to find PDF's of articles from 1940s. And they even have English summary; unbelievable. –  Igor Belegradek Apr 22 '10 at 17:40
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I am not sure about some details, but there is a standard reference to most results of this type: "Extrinsic geometry of convex surfaces" by A.V. Pogorelov which is about 700 pp. in either English or Russian. As I later discovered, this book is essentially a union of 3-4 previous books that Pogorelov wrote on different topics. He even copied entire chapters from older books, including a chapter dealing with extensions of the Alexandrov embedding theorem to various functionals of curvatures and some global parameters. Anyway, this book is hard to read, but it a great source of material and references.

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I was going to get Pogorelov's book but I was hoping that some expert could give me answers before I read the book. –  Igor Belegradek Apr 21 '10 at 23:06
    
I'm pretty sure that very few people besides Pogorelov understand the contents of this book, but I second Igor's recommendation. I haven't looked at the book in a long time and never really understood it, but there is a lot in there that I think deserves to be better understood and known. –  Deane Yang Apr 22 '10 at 2:15
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