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.
| lec # | TOPICS | 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 |
| 21 | Gauss Maps and Minimal Surfaces | Osserman pp. 70-78, 81-83 |