Canonical parameters on a surface in $\mathbb R^4$

TL;DR

This paper introduces canonical principal parameters for surfaces in four-dimensional space, generalizing previous concepts, and proves a fundamental theorem relating these parameters to the surface's geometry via PDEs.

Contribution

It defines and establishes the existence of canonical principal parameters on surfaces in , extending prior notions for minimal and special mean curvature surfaces, and provides a uniqueness theorem.

Findings

Canonical principal parameters exist locally on surfaces in .

Surfaces are determined by four functions satisfying PDEs.

A fundamental existence and uniqueness theorem is proved.

Abstract

In the present paper, we study surfaces in the four-dimensional Euclidean space $\mathbb{R}^4$. We define special principal parameters, which we call canonical, on each surface without minimal points, and prove that the surface admits (at least locally) canonical principal parameters. They can be considered as a generalization of the canonical parameters for minimal surfaces and the canonical parameters for surfaces with parallel normalized mean curvature vector field introduced before. We prove a fundamental existence and uniqueness theorem formulated in terms of canonical principal parameters, which states that the surfaces in $\mathbb{R}^4$ are determined up to a motion by four geometrically determined functions satisfying a system of partial differential equations.