Minimal Surfaces in the Four-Dimensional Euclidean Space

Details Minimal Surfaces in the Four-Dimensional Euclidean Space Minimal Surfaces in the Four-Dimensional Euclidean Space

2008-06-20

arxiv.org/abs/0806.3334v1

Mathematics

Differential Geometry

19 pages

Scientific paper

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.

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