Some uses of the second conformal structure on strictly convex surfaces

Abstract

1. An oriented surface 5 immersed smoothly in E3 has a conformai structure imposed upon it by the metric of the surrounding space. Thus S may be viewed as a Riemann surface Pi. But if S is strictly convex (and oriented so that mean curvature 77 >0) then the second fundamental form is positive definite and determines still another conformai structure on S. Thus S may be viewed as a second Riemann surface R2. In [4], Heinz Hopf uses the Pi structure on a surface of constant mean curvature as a basis for one of several proofs that a closed surface of genus zero with constant 77 must be a sphere. In passing, he derives a convenient formula for the index of an isolated umbilic on an arbitrary surface. This formula involves second derivatives of the functions describing the immersion in E3 of the surface in question. We imitate Hopfs procedures in this paper, restricting our attention to strictly convex surfaces and using R2 in place of Pi structure. We show that Gauss curvature A>0 plays a role relative to P2 structure which 77 plays relative to Pi structure. As a consequence we obtain a new proof of Liebmann's theorem that a closed surface with constant K must be a sphere. In addition we too get a formula for the index of an isolated umbilic (at which A>0). But our formula involves only first derivatives of the functions describing the immersion of the surface in E3. The paper closes with a brief comment on Caratheodory's conjecture.