Surfaces with $H^{2}=K$ along a given curve

Abstract

In the present paper, we handle the problem of construction of surfaces with
a given curve whose squared mean curvature is equal to the Gaussian
curvature, that is $H^{2}=K,$ along the given curve. We express surfaces
parametrically possessing the given curve and obtain conditions for the
coefficient functions to satisfy $H^{2}=K$ along the curve. The theory is
validated by illustrative examples.