All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid? Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open:

Given $S\subseteq\mathbb{R}^3$ a smooth (i.e. $C^\infty$) compact boundaryless connected surface (therefore orientable), let $f:S\times[0,1]\to \mathbb{R}^3$ be a differentiable map with $S_t:=f(S\times\{t\})$ being a surface and $f_t:S\to S_t$ an isometry (where $f_t(p):=f(p,t)$), where $f_0\equiv \mathrm{Id}_S$. Then there are rigid motions $A_t:\mathbb{R}^3\to\mathbb{R}^3$ with $f_t(S) = A_t(S)$.

More generally, in the cases of the sphere $S^2$ (Liebmann's Theorem) and "ovaloids", i.e. surfaces with Gaussian curvature $K$ strictly positive (Cohn-Vossen's rigidity theorem), it is known that isometric embeddings of it in $\mathbb{R}^3$ are exactly the restrictions of the rigid motions from $\mathbb{R}^3$. Not all compact connected surfaces $S\subseteq\mathbb{R}^3$ satisfies this, as pointed out in the book Curves and Surfaces, by S. Montiel and A. Ros:

Anyway, it is possible to show that this isometry in particular cannot be obtained by a continuous family of isometries from the identity map to the latter, so it's not a counterexample to the conjecture.
Although this seems to me to be a very studied problem, and I imagine that there must be several approachs to it, I just couldn't find enough information about it on the internet (the Montiel & Ros book was recommended by Codá later in the same lecture linked above), for "rigidity" seems to be a very comprehensive term. So my actual question is:

Actual question: Where can I find more information about this problem and the current approaches going on?

 A: Not at all an answer: I once had a conversation about this question with a friend at IHES, and Gromov was hanging around, overheard us, and said: oh, that's a stupid problem!
We said (in one voice): why do you say that?
To which Gromov responded: because the question has been around for a century, and no interesting mathematics has come out of it.
Make of this what you will.
A: Not an answer, just a reference, and a quote from Robert Connelly's Rigidity article in the Handbook of Convex Geometry:

          


          

(P.M. Gruber, J.M. Wills, Eds. Handbook of Convex Geometry. Vol.2. North Holland, 1993. "Rigidity," p.231.)


Here's the reference: 

Han, Qing, and Marcus Khuri. "Rigidity in the class of orientable compact surfaces of minimal total absolute curvature." Differential Geometry and its Applications 29.4 (2011): 463-472. (Elsevier link to HTML version.) 

A: Also, not an answer but some comments. When one learns about the geometry of smooth surfaces in $\mathbb{R}^3$, the question of rigidity and flexibility arises quite naturally. And, at first sight, it is plausible that there should be some characterization of these properties in terms of geometric invariants, especially the second fundamental form.
However, a solution to this problem, except for a closed convex hypersurface, has proved to be quite elusive. This is closely related to the fact that the PDE associated with the isometric embedding is an extremely nasty one unless the Gauss curvature is positive, where it is elliptic. The next least nasty case is that of negative Gauss curvature, where the PDE is nonlinear hyperbolic. However, a closed surface with negative curvature cannot be isometrically embedded into $\mathbb{R}^3$. Even if it could be, then resulting PDE is nonlinear hyperbolic and difficult to analyze on a compact manifold, especially a non-periodic one as is the case here. If the Gauss curvature vanishes anywhere, the PDE's become even nastier, and even results about local isometric embeddings become quite difficult and technical. A good reference for this is the book Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, by Han and Hong.
So any results about rigidity and flexibility probably needs to circumvent the use of PDE's. As far as I know, nobody has a clue about how to do this.
If the isometric embedding is assumed to be only $C^1$, then the whole situation flips, due to the results of Nash and Kuiper. Any $C^1$ isometric embedding is flexible, and any closed orientable surface, including flat and negatively curved ones, can be globally isometrically embedded into $\mathbb{R}^3$. The proof of this is actually relatively elementary and worth learning. The techniques used were generalized by Gromov into what is now known as the $h$-principle and convex integration. More recently, the ideas of Nash and Kuiper have been adapted by De Lellis, Székelyhidi, and their collaborators to make dramatic progress on the Onsager conjecture for the Euler fluid flow equation.
ONE MORE COMMENT: As Andy Putman mentions, Bob Connelly found an explicit example of a flexible polyhedron. A metal model of this can be found in the IHES lounge. It is not obvious to me that this could not be turned into a flexible smooth isometric embedding of a smooth surface that approximates the polyhedron. The edges are easily modeled by a hinge. The question is whether one can create a smooth local model of the vertices. Note that these vertices are necessarily "negatively curved".
