Reference request for Minimal Surfaces.

Question

I need books or articles based on minimal surfaces. By minimal surface, I mean a surface with 0 mean curvature.

More specifically, I wish to explore the Plateau's Problem: There exists a minimal surface with a given boundary.

I would also like to see a proof of the fact that a surface of revolution that is minimal is either a plane, helicoid or a catenoid.

As a supplementary text, could I also have a reference for calculus of variations? (Unimportant, but why is calculus of variations not taught as a course in universities?)

Thank you for your time.

Answer 1

I guess the books Calculus of variations and the Plateau problem and Variational methods by M. Struwe might be relevant to your question. Also, calculus of variations is teached in some universities, but usually only to advanced students. I happened to follow a course by professor Struwe on the material of the second book mentioned above during my MSc.

Answer 2

some nice texts are the following

• Geometric Measure Theory and Minimal Surfaces - Bombieri, E.
• Lectures on Geometric Measure Theory - Simon, L.
• Minimal Surfaces and Functions of Bounded Variation - Giusti, E.
• Sets of Finite Perimeter and Geometric Variational Problems - Maggi, F.
• Calculus of Variations and PDEs - Ambrosio, L. & Norman, D.
• Gamma Convergence for Beginners - Braides, A.