Existence theorems for regular, closed, convex surfaces

Abstract

With the help of C. Miranda's method, developed in RZh. Mat. 1972, IA 1121 and 2A 917, existence problems are studied for closed convex surfaces whose principal radii of curvatureR ₁(n) andR ₂(n) satisfy an equation of the form R₁R₂ + Φ(R₁ + R₂, R₁, R₂, n) + cn = ϕ(n), where c is a constant vector connected to the desired surface and the closure condition holds forϕ(n). Here, in contrast to C. Miranda's papers, it is not assumed thatФ ₁≥0. Instead, it is required that the first partial derivatives ofФ with respect toR ₁ andR ₂ be nonnegative. A special case of the proved general theorem is the theorem about the existence of an equation in whichФ is equal to the reciprocal of the mean curvature of the surface. The question of carrying over certain of Miranda's results to the case whereФ increases as (R₁R₂)^µ, where µ>1, is also considered.