Classical Minimal Surfaces in Euclidean Space by Examples-03--Weber-p69--pIRX.pdf

Classical Minimal Surfaces in Euclidean Space
by Examples
Geometric putational Aspects of the Weierstraß representation
January 8, 2003
Matthias Weber
Contents
1 Introduction 6
2 Basic Differential Geometry Of Surfaces 7
................. 7
.................. 8
ßMap................ 9
3 The Weierstrass Representation 11
TheGaußmap ......................... 11
oordinatefunctions ................ 12
Weierstrassdata ........................ 13
........................ 15
................. 17
Theassociatefamily ...................... 19
............. 20
4 Minimal surfaces on Punctured Spheres 22
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A surface with a planar end and an Enneper end . . . . . . . 23
A sphere with one planar and two catenoid ends . . . . . . . . 24
The k-Noids of Jorge and Meeks . . . . . ........... 25
5 The Scherk Surfaces 28
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3
...................... 29
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............... 31
Singly periodic Scherk surfaces with higher dihedral symmetry 34
The twist deformation of Scherk’s singly periodic surface . . . 36
6 Minimal Surfaces Defined On Punctured Tori 38
........................... 38
Algebraicequations....................... 39
.......................... 40
Elliptic functions via Schwarz-Christoffel maps ......... 41
-Gackstattersurface.................. 44
’sminimalsurface.................... 46
...................... 47
’sminimalsurface..................... 49
The Jorge-Meeks k-noidswithanadditionalhandle...... 52
...