A class of superconformal surfaces

TL;DR

This paper characterizes superconformal surfaces in Euclidean space, showing that pedal surfaces to 2-isotropic surfaces are superconformal, explicitly parametrized, and generally not conformally equivalent to minimal surfaces.

Contribution

It introduces a new class of superconformal surfaces derived from pedal surfaces of 2-isotropic surfaces, providing explicit parametrizations and contrasting with known minimal surface examples.

Findings

Pedal surfaces to 2-isotropic surfaces are superconformal.

Superconformal surfaces in this class are not conformally minimal.

Explicit parametric forms are obtained via Weierstrass representation.

Abstract

Superconformal surfaces in Euclidean space are the ones for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic Gauss curvature with the extrinsic normal and mean curvatures, due to Wintgen (\cite{Wi}) and Guadalupe-Rodr\'iguez (\cite{GR}) for any codimension, reaches equality at all points. In this paper, we show that any pedal surface to a $2$-isotropic Euclidean surface is superconformal. Opposed to almost all known examples, superconformal surfaces in this class are not conformally equivalent to minimal surfaces. Moreover, they can be given in an explicit parametric form since $2$-isotropic surfaces admit a Weierstrass type representation.