ementas

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).

Regular Syllabus for Minimal Surfaces

• 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:

1. Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny: Minimal surfaces. Springer, 2010. 688 pages
2. Herbert Blaine Lawson: Lectures on minimal submanifolds. Publish or Perish, 1980. 181 page
3. Robert Osserman: A survey of minimal surfaces. Dover, 1986. 224 pages

Additional Bibliography:

1. Robert Osserman (Ed.): Geometry V. Encyclopaedia of Mathematical Sciences, volume 90, Springer, 1997. http://mi.mathnet.ru/book119 266 pages
2. 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].

Some references for Lagrangian mechanics and classical field theory:

• Vladimir Arnold: Mathematical methods of classical mechanics. A classic textbook on classical mechanics, that introduces most of relevant concepts of Lagrangian mechanics, Hamiltonian mechanics, Legendre transform, and relevant mathematics, including Euler–Lagrange equations in Lagrangian formulation, canonical coordinates and Hamilton’s equations in Hamiltonian formulation, symplectic geometry and Lagrangian submanifolds, contact geometry and Legendrian submanifolds (and even things like the simplest derivation of the fundamental theorem of projective duality from properties of Legendrian submanifolds and Legendrian lifts).

References for Quantization

References for string theory:

• David Tong: Lectures on String Theory. 210 pages, 2009. Available online
• Barton Zwiebach: A First Course in String Theory. 2004. Available in DMAT library.
• Joseph Polchinski: String Theory. 1998, two volumes. Volume 1 is “Introduction to Bosonic String”, our discussion is not far from first chapters of Vol 1. Vol 2 is superstrings and beyond.
• M. Green, J. Schwarz, E. Witten: Superstring Theory. 1989, maybe? The first textbook, written soon after first superstring revolution. Quite a good source to learn various topics in complex geometry, e.g. how to deform a tangent bundle on a hypersurface.