On the global, smooth isometric embedding of a torus into $\mathbb{R}^3$

I posted this question recently on math stackexchange (https://math.stackexchange.com/questions/2291382/c2-isometric-embedding-of-the-flat-torus-into-mathbbr3) but have not received any satisfactory answer yet. I am framing a refined version of the problem in the following.

Question 1 Is there any smooth, or at least $C^2$, global isometric embedding of the flat torus into $\mathbb{R}^3$?

We know that the answer is no. The standard reason given for the non-existence is that the given flat metric $a_0$ on the torus $\mathbb{T}$ violates the necessary requirement for any global, sufficiently smooth, isometrically embedded compact 2-manifold inside $\mathbb{R}^3$ that, there must be one elliptic point, i.e., a point where the Gaussian curvature of $a_0$ is positive. On the flat torus, there is no such point. Here, we can make a note that the flat torus, however, satisfies the Gauss-Bonnet theorem; in other words, the flat metric $a_0$ is indeed a valid metric on the torus.

This has led me to the following question:

Question 2 (On the global isometric embedding of a torus into $\mathbb{R}^3$) Let a sufficiently smooth Riemannian metric $a$ be given on the torus of genus 1, $\mathbb{T}$. $(\mathbb{T}, a)$ is, thus, a sufficiently smooth 2-dimensional Riemannian manifold. Can we pose the statement the subset of $\mathbb{T}$ where the Gaussian curvature $K$ of $a$ is positive has to be non-empty as a sufficient condition on the given metric $a$ for the existence of a global isometric embedding of $(\mathbb{T}, a)$ into $\mathbb{R}^3$?

I am following this paper: Han and Lin, On the isometric embedding of torus in $\mathbb{R}^3$, Methods and Applications of Analysis 15, pp. 197-204, 2008.

Here, the sufficient conditions for the existence of a global smooth isometric embedding of $(\mathbb{T}, a)$ are given. But these conditions, as can be clearly seen, are given for the existence of the standard embedded torus in $\mathbb{R}^3$ (the standard illustration of the embedded torus in the literature), which is too strict. The original question is much weaker; it seeks the existence of some (sufficiently smooth) isometrically embedded torus inside $\mathbb{R}^3$, not necessarily the standard torus. So there must be a different set of sufficient conditions than the ones given in the above reference. This is the brief motivation that has led me to Question 2. Of course, the local sufficient requirements imposed by the Gauss-Codazzi equations must hold. But, interestingly, we need a valid second fundamental form $b$ to check the Gauss-Codazzi equations. Taking $b$ as identically zero is a choice. But is it a valid choice? This would then imply that a local, at least $C^2$, isometric embedding is possible. But there seems to be a problem somewhere in this logic.

Thanks in advance for any help or reference!

• In the article that you are citing another necessary condition is given: the integral of the positive part of the Gauss curvature must be at least $4\pi$. So the answer to Question 2 is "no". And this condition seems to be not sufficient either, the problem is hard and maybe intractable. – Ivan Izmestiev May 22 '17 at 7:33
• @IvanIzmestiev: I see the point. They have assumed that the Gaussian curvature is already positive somewhere. But, still, the condition "the integral of the positive part of the Gauss curvature must be at least 4$\pi$" is too strict, isn't it? – Ayan May 22 '17 at 7:39
• It is necessary for the existence of an isometric embedding but probably not sufficient. – Ivan Izmestiev May 22 '17 at 8:28