INTRODUCTION TO MINIMAL SURFACES
Serre-Frenet formula for curves, Parametric surfaces, Isothermal parameters, Gauss Map, Gaussian Curvature, Mean curvature, Area functional etc.
Surfaces that locally minimise area in Euclidean space (minimal surfaces). Harmonic coordinates in isothermal parameters. Examples of minimal surfaces.
Minimal surfaces with boundary: Plateau's problem.
The gauss map for minimal surfaces with some examples.
The Weierstrass-Enneper representation of minimal surfaces. Many more examples of minimal surfaces.
Conjugate minimal surfaces. One parameter family of isometric minimal surfaces. the Bjorling problem and Schwartz's solution to it.
If time permits:
Surfaces that locally maximise area in Lorenztian space (maximal surfaces). A lot of examples and analogous results, as in minimal surface theory, for maximal surfaces.
Connection betwen minimal and maximal surfaces and Born Infeld solitions. Constant mean curvature surfaces of non-zero mean curvature (the optimization problem they solve)
Recommended books
Manfredo Do Carmo: Differential Geometry of curves and surfaces.
Robert Osserman: A survey of minimal surfaces.
Dierkes, Hildebrandt, Kuster, Wohlrab: Minimal Surfaces I.
Yi Fang: Lectures on Minimal Surfaces in R^{n}
K. Kenmotsu: Surfaces of constent mean curvature.