Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If every point of S is elliptic, then must $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?

1$\begingroup$ Just thought I'd mention that the term elliptic surface has a very specific meaning in algebraic geometry, and it's not the one you're using! An elliptic surface is a surface (2dimensional algebraic variety) $S$ that admits an algebraic map $f:S\to C$ to a curve such that all but finitely many of the fibers are elliptic curves. (Usually it's assumed that there is a section, in which case the set of sections is a group.) $\endgroup$ – Joe Silverman Feb 15 at 23:54
I do not think so. Just imagine that you peel a large orange whose surface area is much bigger than that of a unit sphere. Then you "spiral" the peel to make it arbitrarily small and place it inside a unit sphere. Imagine a surface that looks more or less as this one:
It has positive curvature so very point is elliptic. Here you remove a small cylinder around the vertical axis so there is no problems with the curvature. Since you can have as many "turns" as you want, its surface area can be arbitrarily large while the surface occupies a small region in space.

$\begingroup$ Thanks. Originally I had a ray condition on S in mind. Your example shows I cannot dispense with it. I reformulated the question in mathoverflow.net/questions/352836/… $\endgroup$ – Thomas Feb 16 at 4:50