Families of surfaces with constant ratio of principal curvatures and Plateau's problem

Abstract

This work is on surfaces with a constant ratio of principal curvatures. These CRPC surfaces generalize minimal surfaces but are much more challenging to construct. We propose a construction of a family of such surfaces containing a given minimal surface without flat points. This leads to a partial solution of Plateau's problem for CRPC surfaces. We obtain analogous results in isotropic geometry. This work illustrates a general approach to solving Euclidean problems by starting with their isotropic analogs. Besides, we apply the method of successive approximations and analytic majorization.