Constant-mean-curvature surface

Builds on Regular surface Mean curvature

Definition

Let $S \subset \R^3$ be an orientable regular surface with a chosen unit normal, and let $H \colon S \to \R$ be its mean curvature. The surface $S$ is a constant-mean-curvature surface if there is a real number $H_0$ with $H(p) = H_0$ for every $p \in S$ .

Remarks

The sphere of radius $r$ is a CMC surface: for the inward normal both principal curvatures are $1/r$ , so $H = 1/r$ everywhere. The paraboloid $z = x^2 + y^2$ is not: its mean curvature is $2$ at the origin and tends to $0$ far from the origin.

The case $H_0 = 0$ gives the minimal surfaces. Changing the normal changes the sign of $H$ , so the condition does not depend on the choice of normal. The same definition applies to immersed surfaces, where self-intersections are allowed. By a theorem of Alexandrov, a compact embedded CMC surface is a round sphere, but compact immersed CMC tori exist, such as the Wente torus.

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Mean curvature concept
Regular surface concept
Constant-mean-curvature surface definition

Tag	`87EFGXB17V`
URI	`m2://87EFGXB17V`
Kind	definition
Writing	W0 stub
MSC	53A10
Defines	Constant-mean-curvature surface
Uses	Regular surface, Mean curvature
Changed	2 November 2026

Constant-mean-curvature surface

Definition

Remarks

Reading path

Further reading

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