Readings

Readings are from the following textbooks.

Amazon logo Osserman, Robert. A Survey of Minimal Surfaces. Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.

Amazon logo Do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Englewood Cliffs, N. J.: Prentice Hall, February 1, 1976. ISBN: 0132125897.

Amazon logo Spivak, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. New York: Westview Press, June 1, 1965. ISBN: 0805390219.

Amazon logo Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Vol 3. 3rd ed. Boston: Publish or Perish, January 1, 1999. ISBN: 0914098721.

Amazon logo Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Vol 4. 3rd ed. Berkeley: Publish or Perish, January 1, 1999. ISBN: 091409873X.

Amazon logo Ahlfors, Lars. Complex Analysis. 3rd ed. New York: McGraw-Hill Science/Engineering/Math, January 1, 1979. ISBN: 0070006571.

Amazon logo 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

Do Carmo 5.3, exercise 11, p.337 

21 Gauss Maps and Minimal Surfaces Osserman pp. 70-78, 81-83