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Canonical Weierstrass representations for minimal surfaces in Euclidean 4-space | | Georgi Ganchev
; Krasimir Kanchev
; | | Date: |
6 Sep 2016 | | Abstract: | Minimal surfaces of general type in Euclidean 4-space are characterized with
the conditions that the ellipse of curvature at any point is centered at this
point and has two different principal axes. Any minimal surface of general type
locally admits geometrically determined parameters - canonical parameters. In
such parameters the Gauss curvature and the normal curvature satisfy a system
of two natural partial differential equations and determine the surface up to a
motion. For any minimal surface parameterized by canonical parameters we obtain
Weierstrass representations - canonical Weierstrass representations. These
Weierstrass formulas allow us to solve explicitly the system of natural partial
differential equations and to establish geometric correspondence between
minimal surfaces of general type, the solutions to the system of natural
equations and pairs of holomorphic functions in the Gauss plane. On the base of
these correspondences we obtain that any minimal surface of general type in
Euclidean 4-space determines locally a pair of two minimal surfaces in
Euclidean 3-space and vice versa. Finally some applications of this phenomenon
are given. | | Source: | arXiv, 1609.1606 | | Services: | Forum | Review | PDF | Favorites | |
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