The Global Theory of Minimal Surfaces in Flat Spaces: Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Martina Franca, Italy July 7-14, 1999

by Meeks, William; Ros, A.; Rosenberg, H.; Pirola, Gian Pietro; Meeks, W. H., III; Pirola, Gian Pietro

• ISBN13: 9783540431206
• ISBN10: 3540431209
• Format: Paperback
• Copyright: 2002-05-01
• Publisher: Springer Verlag

Summary

In the second half of the twentieth century the global theory of minimal surface in flat space had an unexpected and rapid blossoming. Some of the classical problems were solved and new classes of minimal surfaces found. Minimal surfaces are now studied from several different viewpoints using methods and techniques from analysis (real and complex), topology and geometry. In this lecture course, Meeks, Ros and Rosenberg, three of the main architects of the modern edifice, present some of the more recent methods and developments of the theory. The topics include moduli, asymptotic geometry and surfaces of constant mean curvature in the hyperbolic space.

Table of Contents

Minimal Surfaces in Flat Three - Dimensional Spaces	1	(14)
William H. Meeks III
Introduction	1	(1)
The maximum principle at infinity conjecture and the stable minimal surface conjecture	1	(3)
The geometric Dehn's lemma and related barrier constructions	4	(1)
Triply periodic minimal surfaces	5	(2)
Doubly periodic minimal surfaces	7	(1)
Singly periodic minimal surfaces	8	(2)
The geometry of minimal surfaces with more than one end	10	(5)
References	13	(2)
Properly embedded minimal surfaces with finite total curvature	15	(52)
Joaquin Perez
Antonio Ros
Introduction	15	(1)
Background	15	(7)
Weierstrass Representation	16	(1)
Finite Total Curvature	17	(2)
Maximum Principle	19	(1)
Monotonicity Formula	20	(1)
Stability	21	(1)
The Plateau Problem	21	(1)
Minimal Surfaces with Vertical Forces I	22	(14)
Basic Properties of Forces	22	(2)
Vertical Forces	24	(4)
Other Results on Vertical Forces	28	(8)
Minimal Surfaces with Vertical Forces II	36	(10)
Immersed 3-manifolds	36	(2)
Topological Uniqueness	38	(5)
Related Results	43	(3)
Limits of Minimal Surfaces	46	(9)
Minimal Graphs	46	(3)
Sequences with Uniform Curvature Bounds	49	(3)
Sequences with Total Curvature Bounds	52	(3)
Compactness of the Moduli Space of Minimal Surfaces	55	(12)
Weak Compactness	56	(4)
Strong Compactness	60	(3)
References	63	(4)
Bryant Surfaces	67
Harold Rosenberg
Introduction	67
Existence and unicity problems	67
The cousin relation	69
CMC's in R3	70
Some problems	70
H-surfaces in H3	72
Properly embedded minimal surfaces in R3	73
Bryants' Representation	74
Moving frames	76
The structure equation of H3 and S	79
Surfaces in H3 and the Structure equations of adapted frames	80
Constructing explicit examples of Bryant surfaces starting with a minimal surface in R3	81
Properly embedded Bryant annular ends	92
Non-density at infinity	103
Some applications of the annular end theorem	108
References	109