Minimal surface
This definition makes minimal surfaces a 2-dimensional analogue to geodesics, which are analogously defined as critical points of the length functional.
The partial differential equation in this definition was originally found in 1762 by Lagrange,[2] and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.
A direct implication of this definition and the maximum principle for harmonic functions is that there are no compact complete minimal surfaces in
This definition uses that the mean curvature is half of the trace of the shape operator, which is linked to the derivatives of the Gauss map.
If the projected Gauss map obeys the Cauchy–Riemann equations then either the trace vanishes or every point of M is umbilic, in which case it is a piece of a sphere.
The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than
By expanding Lagrange's equation to Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces.
Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods.
Weierstrass and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and harmonic functions.
The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone.
Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important.
Another cause was the verification by H. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist.
[8] The endoplasmic reticulum, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.
[9] In the fields of general relativity and Lorentzian geometry, certain extensions and modifications of the notion of minimal surface, known as apparent horizons, are significant.
[10] In contrast to the event horizon, they represent a curvature-based approach to understanding black hole boundaries.
