Minimal Surfaces in the Four-Dimensional Euclidean Space

Abstract

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for strongly regular minimal surfaces of general type in terms of their invariants. We introduce canonical parameters on strongly regular minimal surfaces of general type and prove that any such a surface is determined up to a motion by two invariant functions satisfying a system of two natural partial differential equations. On any minimal surface of the basic class of non strongly regular minimal surfaces we define canonical parameters and prove that any such a surface is determined up to a motion by two invariant functions of one variable satisfying a system of two natural ordinary differential equations. We find a geometric description of this class of non strongly regular minimal surfaces.

19 pages

References in corpus (2)

• On the Invariant Theory of Weingarten Surfaces in Euclidean Space
• All superconformal surfaces in \R^4 in terms of minimal surfaces

Cited by in corpus (6)

• Invariants and bonnet-type theorem for surfaces in $\r^4$
• General Rotational Surfaces in ${\mathbb{R}}^4$ with Meridians Lying in Two-Dimensional Planes
• Geometric interpretation of the invariants of a surface in R^4 via the tangent indicatrix and the normal curvature ellipse
• Minimal Surfaces in the Three-Dimensional Sphere and Minimal Hypersurfaces of Type Number Two
• Canonical Weierstrass representations for minimal surfaces in Euclidean 4-space
• On rotational surfaces with zero mean curvature in the pseudo-Euclidean space ${\mathbb{E}}_2^4$