In differential geometry, **constant-mean-curvature (CMC) surfaces** are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.

## Contents

- History
- Representation formula
- Conjugate cousin method
- Discrete numerical methods
- Applications
- References

Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere.

## History

In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.

In 1853 J. H. Jellet showed that if

Up until this point it had seemed that CMC surfaces were rare; new techniques produced a plethora of examples. In particular gluing methods appear to allow combining CMC surfaces fairly arbitrarily. Delaunay surfaces can also be combined with immersed "bubbles", retaining their CMC properties.

Meeks showed that there are no embedded CMC surfaces with just one end in *k*-unduloids of genus 0 satisfy *k*, and *k*. At most *k* − 2 ends can be cylindrical.

## Representation formula

Like for minimal surfaces, there exist a close link to harmonic functions. An oriented surface

Let

with

for

For

## Conjugate cousin method

Lawson showed 1970 that each CMC surface in

## Discrete numerical methods

Discrete differential geometry can be used to produce approximations to CMC surfaces (or discrete counterparts), typically by minimizing a suitable energy functional.

## Applications

CMC surfaces are natural for representations of soap bubbles, since they have the curvature corresponding to a nonzero pressure difference.

Besides macroscopic bubble surfaces CMC surfaces are relevant for the shape of the gas–liquid interface on a superhydrophobic surface.

Like triply periodic minimal surfaces there has been interest in periodic CMC surfaces as models for block copolymers where the different components have a nonzero interfacial energy or tension. CMC analogs to the periodic minimal surfaces have been constructed, producing unequal partitions of space. CMC structures have been observed in ABC triblock copolymers.

In architecture CMC surfaces are relevant for air-supported structures such as inflatable domes and enclosures, as well as a source of flowing organic shapes.