Minimal Surface - Definitions

Definitions

Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.

Local least area definition: A surface M ⊂ R3 is minimal if and only if every point p ∈ M has a neighbourhood with least-area relative to its boundary.

Note that this property is local: there might exist other surfaces that minimize area better with the same global boundary.

Variational definition:A surface M ⊂ R3 is minimal if and only if it is a critical point of the area functional for all compactly supported variations.

This definition makes minimal surfaces a 2-dimensional analogue to geodesics.

Soap film definition: A surface M ⊂ R3 is minimal if and only if every point p ∈ M has a neighbourhood D_p which is equal to the unique idealized soap film with boundary ∂D_p

By the Young–Laplace equation the curvature of a soap film is proportional to the difference in pressure between the sides: if it is zero, the membrane has zero mean curvature. Note that bubbles are not minimal surfaces as per this definition: while they minimize total area subject to a constraint on internal volume, they have a positive pressure.

Mean curvature definition: A surface M ⊂ R3 is minimal if and only if its mean curvature vanishes identically.

A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures.

Differential equation definition: A surface M ⊂ R3 is minimal if and only if it can be locally expressed as the graph of a solution of

The partial differential equation in this definition was originally found in 1762 by Lagrange, and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.

Energy definition: A conformal immersion X: M → R3 is minimal if and only if it is a critical point of the Dirichlet energy for all compactly supported variations, or equivalently if any point p ∈ M has a neighbourhood with least energy relative to its boundary.

This definition ties minimal surfaces to harmonic functions and potential theory.

Harmonic definition: If X = (x₁, x₂, x₃): M → R3 is an isometric immersion of a Riemann surface into space X is said to be minimal if x_i is a harmonic function on M for each i.

A direct implication of this definition and the maximum principle for harmonic functions is that there are no compact complete minimal surfaces in R3.

Gauss map definition: A surface M ⊂ R3 is minimal if and only if its stereographically projected Gauss map g: M → C ∪ {∞} is meromorphic with respect to the underlying Riemann surface structure.

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 the trace vanishes.

Mean curvature flow definition: Minimal surfaces are the critical points for the mean curvature flow.

The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3.