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. 



On Alexandrov embedding theorem
*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](http://www.mathnet.ru/php/journal.phtml?wshow=paper&jrnid=sm&paperid=6287&year=1946&volume=60&issue=3&fpage=429&lpage=440&option_lang=rus) --- 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.


