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Readings are from the following textbooks.
Osserman, Robert. A Survey of Minimal Surfaces. Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.
Do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Englewood Cliffs, N. J.: Prentice Hall, February 1, 1976. ISBN: 0132125897.
Spivak, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. New York: Westview Press, June 1, 1965. ISBN: 0805390219.
Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Vol 3. 3rd ed. Boston: Publish or Perish, January 1, 1999. ISBN: 0914098721.
Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Vol 4. 3rd ed. Berkeley: Publish or Perish, January 1, 1999. ISBN: 091409873X.
Ahlfors, Lars. Complex Analysis. 3rd ed. New York: McGraw-Hill Science/Engineering/Math, January 1, 1979. ISBN: 0070006571.
Ahlfors, L. V., and L. Sario. Riemann Surfaces. Princeton, N.J.: Princeton University Press, 1960. ISBN: 0691080275.
Readings
| 1 |
Introduction |
|
| 2 |
A Review on Differentiation |
Spivak ISBN: 0914098721 pp. 15-31 or 33 |
| 3 |
Inverse Function Theorem |
Spivak ISBN: 0805390219 pp. 31-33
same Spivak pp. 34-39 (Rudin also a good reference)
Osserman pp. 3-4 |
| 4 |
Implicit Function Theorem |
Spivak ISBN: 0805390219 pp. 40-43
Osserman p. 3 - top of p. 6
See examples of parametrised surfaces: Do Carmo, Spivak ISBN: 0914098721 |
| 5 |
First Fundamental Form |
Osserman p. 10
Follow Do Carmo 2.5 for more details and examples
Osserman pp. 6, 7 |
| 6 |
Curves |
Osserman pp. 8-11
Do Carmo 1.2-1.4, 1.5 up to p. 18 (don't discuss torsion), prop 1 of 2.4 |
| 7 |
Gauss Map I: Background and Definition |
Do Carmo 2.4, props 2-3, 3.2: definition of Gauss map, p. 135, appendix to chapter 3, p. 214 |
| 8 |
Gauss Map II: Geometric Interpretation |
Osserman p. 12
Do Carmo 3.2, definitions 1-4, 6, examples |
| 9 |
Gauss Map III: Local Coordinates |
Do Carmo 3.2, definitions 5, 7, 9 |
| 10 |
Introduction to Minimal Surfaces I |
Do Carmo 3.3 up to p. 156
Osserman pp. 13-14
Osserman p. 18, Do Carmo 3.3, example 3 (Catenoid)
Refer to Do Carmo 3.5B and Spivak ISBN: 0914098721 chapter 3 for more details |
| 11 |
Introduction to Minimal Surfaces II |
Osserman p. 18, 20-23
Refer to Do Carmo 3.5B and Spivak ISBN: 0914098721 chapter 3 for more details
Ahlfors ISBN: 0070006571 chapter 2, 1.1: just definition 4, example of real function of a complex variable, 1.2 |
| 12 |
Review on Complex Analysis I |
Ahlfors ISBN: 0070006571 chapter 2, 2.4, chapter 5, 1.2, p. 179, chapter 2, 3.1-3.4, chapter 3, 2.2, 2.3, chapter 4, 3.2 pp. 126-127 |
| 13 |
Review on Complex Analysis II |
Ahlfors ISBN: 0070006571 chapter 4, pp.127-129 |
| 14 |
Isothermal Parameters |
Osserman pp. 15-19, pp. 27-30 |
| 15 |
Bernstein's Theorem |
Osserman pp. 23-26, 31-32 |
| 16 |
Manifolds and Geodesics I |
Osserman pp. 40-44
Do Carmo 2.2, p. 52, p. 55, section 5.10, p. 425 |
| 17 |
Manifolds and Geodesics II |
Osserman pp. 44-47 Observe (Do Carmo 5.10), p. 49
Do Carmo p. 430, p. 441, 2.6 p. 103, p. 427, p. 436 |
| 18 |
Complete Minimal Surfaces I |
Do Carmo 4.4, 5.3 pp. 238-248, pp. 325-328
|
| 19 |
Complete Minimal Surfaces II |
Do Carmo 4.4, 5.3 pp. 238-248, pp. 325-328 |
| 20 |
Weierstrass-Enneper Representations |
Osserman pp. 63-68
Do Carmo 5.3, exercise 11, p.337
|
| 21 |
Gauss Maps and Minimal Surfaces |
Osserman pp. 70-78, 81-83 |