This is a very good question with lots of possible answers.

From a mathematics perspective, a good place to start in answering this question is

Article Applications of a completeness lemma in minimal surface theo...

and

Article Minimal surface system in Euclidean four-space

and

Article Applications of Quaternionic Holomorphic Geometry to minimal surfaces

I requested the full-text for these papers.

Here are applications:

Black holes: The minimal surface of a black hole is considered in

Article Extending the scope of holographic mutual information and ch...

Convective patterns in a 2D flow field:

Chapter Flow Visualization via Partial Differential Equations

Computer graphics:

Conference Paper Bézier Surfaces of Minimal Area.

We try to add something to interesting prof James F Peters's

answer:

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.

See

Chrusciel, Piotr T.; Galloway, Gregory J.; Pollack, Daniel (2010). "Mathematical general relativity: a sampler". Bull. Amer. Math. Soc. 47: 567–638. MR 2721040. arXiv:1004.1016?Freely accessible. doi:10.1090/S0273-0979-2010-01304-5.

Eichmair, Michael (2009). "The Plateau problem for marginally outer trapped surfaces". Journal of Differential Geometry. 83 (3): 551–584. MR 2581357. arXiv:0711.4139?Freely accessible.

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.

1. Conformal geometry.

Thomsen, Schadow 1923, Blaschke 1929, Willmore 1965.

2. Mechanic-elasticity. Friesecke, James, Muller 2001. Non-linear

plate theory.

3. Cell Biology. Helfrich 1973. Spontaneous curvature

model for bio-membranes, vesicles and smectic A-liquid crystals.

4. General relativity. Hawking 1968 mass of 2 spheres

There is a nice article on applications of minimal surfaces by F. C. Marques [Minimal surfaces - variational theory and applications, arXivL1409.7648, 2014].

On https://www2.cose.isu.edu/~palmbenn/indx/index.html various papers can be found of Bennett Palmer and Miyuki Koiso, on anisotropic energies and the associated surfaces (including catenoids).

"Anisotropic surface energies are used to model surface energies which depend on the direction of the surface normal. Equilibria of such energies are characterized as surfaces with constant anisotropic mean curvature. The surface of a crystal and certain interfaces of liquid crystals with an isotropic substrate give physical examples of such equilibria".

The study of minimal surfaces constitutes a central area of research in mathematics, with applications in both geometry and theoretical physics. There are so many nice applicatione related to minimal surfaces in variational theory and you can find so many papers online on variational theory. As one of them is given in the answer by Prof. B. Y. Chen

nice dear

I think symmetry is the most important property of the minimal surfaces.

When you look at a minimal curve or a minimal surface, you will see the BEAUTY of the SYMMETRY.

Erhan