2016/09/06 by Georgi Ganchev, Ganchev, Georgi, Krasimir Kanchev +1 · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1609.01606
Minimal surfaces of general type in Euclidean 4-space are characterized with\nthe conditions that the ellipse of curvature at any point is centered at this\npoint and has two different principal axes. Any minimal surface of general type\nlocally admits geometrically determined parameters - canonical parameters. In\nsuch parameters the Gauss curvature and the normal curvature satisfy a system\nof two natural partial differential equations and determine the surface up to a\nmotion. For any minimal surface parameterized by canonical parameters we obtain\nWeierstrass representations - canonical Weierstrass representations. These\nWeierstrass formulas allow us to solve explicitly the system of natural partial\ndifferential equations and to establish geometric correspondence between\nminimal surfaces of general type, the solutions to the system of natural\nequations and pairs of holomorphic functions in the Gauss plane. On the base of\nthese correspondences we obtain that any minimal surface of general type in\nEuclidean 4-space determines locally a pair of two minimal surfaces in\nEuclidean 3-space and vice versa. Finally some applications of this phenomenon\nare given.\n