Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential

TL;DR

This paper introduces a family of deformations between minimal and non-minimal constant mean curvature surfaces in Euclidean space that preserve the Hopf differential and symmetries, providing explicit formulas and applications to surface transformations.

Contribution

It defines a new deformation family of CMC surfaces preserving the Hopf differential and symmetries, with explicit formulas linking minimal and non-minimal surfaces, and introduces a dressing action for minimal surfaces.

Findings

Deformation family preserves Hopf differential and symmetries.

Explicit formulas relate minimal and non-minimal surfaces.

Application to dressing actions and new surface examples.

Abstract

We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space $E^3$ which preserve the Hopf differential. We prove that, given a CMC $H$ surface $f$, either minimal or not, and a fixed basepoint $z_0$ on this surface, there is a naturally defined family $f_h$, for all real $h$, of CMC $h$ surfaces that are tangent to $f$ at $z_0$, and which have the same Hopf differential. Given the classical Weierstrass data for a minimal surface, we give an explicit formula for the generalized Weierstrass data for the non-minimal surfaces $f_h$, and vice versa. As an application, we use this to give a well-defined dressing action on the class of minimal surfaces. In addition, we show that symmetries of certain types associated with the basepoint are preserved under the deformation, and this gives a canonical choice of basepoint for surfaces with symmetries. We use this to define new examples of non-minimal CMC surfaces naturally associated to known minimal surfaces with symmetries.