# Area of a elliptic surface confined by a sphere

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

• 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 (2-dimensional 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.) – Joe Silverman Feb 15 at 23:54