Closed Surfaces with Constant Gauss Curvature (Hilbert’s Method)— Generalizations and Problems — General Remarks on Weingarten Surfaces

Abstract

Our aim in this section is to prove that the spheres are 1) the only closed surfaces with constant Gauss curvature K, and 2) the only ovaloids with constant mean curvature H. We will actually prove the stronger result that if the principle curvatures k₁ and k₂ of an ovaloid satisfy a relationship k₂ = f(k₁) where f is a decreasing function, then the ovaloid is a sphere. Since K = k₁k₂ and $H = \frac{1}{2}\left( {{k_1} + {k_2}} \right)$ , the two results, 1) and 2) stated above will follows Irom this theorem. The difference in the formulation of 1) and 2) is 3ue to the fact that on any closed surface there are points where K>O (See II, 4.2). Therefore if K is constant, then K is a positive constant and hence by IV, 1.4, the surface already is an ovaloid. The problem of characterizing arbitrary closed surfaces for which H is constant is much more difficult. It will be considered in Chapter VI and VII.