MA 383: Introduction to Minimal Surfaces
Credits: 3:0
- Serre-Frenet formula for curves, Parametric surfaces, Isothermal parameters,
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Gauss Map, Gaussian Curvature, Mean curvature, Area functional etc.
- Surfaces that locally minimise area in Euclidean space (minimal surfaces).
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Harmonic coordinates in isothermal parameters. Examples of minimal surfaces.
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Minimal surfaces with boundary: Plateau’s problem.
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The gauss map for minimal surfaces with some examples.
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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:
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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)
Suggested books and references:
- Dierkes, Hildebrandt, Kuster, Wohlrab, Minimal Surfaces I
- Manfredo Do Carmo, Differential Geometry of curves and surfaces
- Robert Osserman, A survey of minimal surfaces
- Yi Fang, Lectures on Minimal Surfaces in $\R^n$
- K. Kenmotsu, Surfaces of constant mean curvature