Zero Mean Curvature Surfaces in Euclidean and Lorentz-Minkowski 3-Space - Lecture 1

Explore the fascinating world of zero mean curvature surfaces in this comprehensive lecture from the International Centre for Theoretical Sciences' online discussion meeting. Delve into topics such as volume of hypersurfaces, stability of minimal surfaces, the Bernstein problem, and Weierstrass representation. Examine various examples of minimal surfaces, including Scherk's surfaces, singly, doubly, and triply periodic surfaces, and nonorientable surfaces like the Möbius strip and Klein bottle. Learn about the Gauss map, the Osserman inequality, and engage with complex mathematical concepts in this in-depth exploration of geometric surfaces in Euclidean and Lorentz-Minkowski 3-space.

Zero mean curvature surfaces in Euclidean and Lorentz-Minkowski 3-space Lecture 1
Volume of hypersurfaces
The first variation of the volume
The second variation of the volume
Stability
Stability of the catenoid
Stable minimal surface v.s. area-minimizing surface
Graph hypersurface
The Bernstein problem
Examples of minimal surface
Scherk' minimal surfaces before Weierstrass
Weierstrass representation
The first and second fundamental forms, the Gauss map
The period problem
Period problem
Symmetry
Example minimal surfaces of finite total curvature
Example Singly periodic minimal surfaces
Example Doubly periodic minimal surfaces
Example Triply periodic minimal surfaces
New triply periodic minimal surfaces
Schwarz P, Schwarz
Limit of Schwarz P, Schwarz D: a -
Limit of Schwarz P, Schwarz D: a - 1
Minimal surfaces of finite total curvature
The Osserman inequality
Examples which satisfy the equality n 3
The case n 2
Nonorientable minimal surfaces
The Gauss map
deg
Example: Mobius strip degg = 3
Example: Klein bottle-{1 pt} degg = 4 M = 2, W
Q&A