Minimal Surface - Generalisations and Links To Other Fields

Generalisations and Links To Other Fields

Minimal surfaces can be defined in other manifolds than R3, such as hyperbolic space, higher-dimensional spaces or Riemannian manifolds.

The definition of minimal surfaces can be generalized/extended to cover constant-mean-curvature surfaces: surfaces with a constant mean curvature, which need not equal zero.

In Discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions. Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known.

Brownian motion on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces.

Minimal surfaces have become an area of intense scientific study, especially in the areas of molecular engineering and materials science, due to their anticipated applications in self-assembly of complex materials.

Minimal surfaces play a role in general relativity. The apparent horizon (marginally outer trapped surface) is a minimal hypersurface, linking the theory of black holes to minimal surfaces and the Plateau problem.

Minimal surfaces are part of the generative design toolbox used by modern designers. In architecture there has been much interest in tensile structures, which are closely related to minimal surfaces. A famous example is the Olympiapark in Münich by Frei Otto, inspired by soap surfaces.

In the art world, minimal surfaces have been extensively explored in the sculpture of Robert Engman (1927– ), Robert Longhurst (1949– ), and Charles O. Perry (1929–2011), among others.