Minimal surfaces and Quantization
The course of 2025.1 is going to be a 60/40 solution
of the default graduate course on Minimal Surfaces in three-dimensional Euclidean space
with research oriented Quantization agenda. The ratio 60/40 is a critical ratio
for the course to be classified as a regular course (as opposed to topics).
- Several equivalent definitions of minimal surfaces in Euclidean space
- Classic examples and their geometric characterizations
- The Weierstrass representation
- Curvature estimates and Bernstein’s theorem
- Schwarz reflection principle
- Conjugate and associate minimal surfaces
- Complete minimal surfaces of finite total curvature
- The maximum principle, Rado theorem and the half-space theorem
- Douglas–Rado solution to the Plateau problem
Basic Bibliography:
- Ulrich Dierkes,
Stefan Hildebrandt,
Friedrich Sauvigny:
Minimal surfaces. Springer, 2010.
688 pages
- Herbert Blaine Lawson:
Lectures on minimal submanifolds. Publish or Perish, 1980.
181 page
- Robert Osserman:
A survey of minimal surfaces. Dover, 1986.
224 pages
Additional Bibliography:
- Robert Osserman (Ed.):
Geometry V. Encyclopaedia of Mathematical Sciences, volume 90, Springer, 1997.
http://mi.mathnet.ru/book119
266 pages
- Tobias Holck Colding,
William Philip Minicozzi II:
A course in minimal surfaces, AMS, 2011.
313 pages
Extra bibliography:
- Robert Osserman: Minimal Varieties, Bulletin of the American Mathematical Society, 75, 6, 1092–1120 (1969).
A shorter survey that Osserman wrote soon after his survey [3] above,
with more details on higher-dimensional hypersurfaces, subvarieties of arbitrary dimension and codimension,
minimal cones and subvarieties in spheres.
It is a useful complement that one can skim before proceeding to [3].
References for Quantization