"C^1" isometric deformations of S^2 in R^3

[Daniel Asimov]

It's known that the standard sphere S^2 in R^3 is rigid with respect
to C^2 deformations through C^2 surfaces. (See, e.g., Edgar Kann, "A
new method for infinitesimal rigidity of surfaces with K > 0," J.
Diff. Geom., 1970 pp. 5-12.)

But have there been results about isometric deformation of the
standard S^2 in R^3, through surfaces that need be only C^1
embeddings? The deformation itself need be only C^0.

And what about the same question but with C^0 replacing C^1 ?

Any pointers to the literature or to mathematicians who may be
specialists in this area will be gratefully accepted.