Constant-mean-curvature surface

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.

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

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 ${\displaystyle S}$ is a compact star-shaped surface in ${\displaystyle \mathbb {R} ^{3}}$ with constant mean curvature, then it is the standard sphere. Subsequently A. D. Alexandrov proved that a compact embedded surface in ${\displaystyle \mathbb {R} ^{3}}$ with constant mean curvature ${\displaystyle H\neq 0}$ must be a sphere. Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in ${\displaystyle \mathbb {R} ^{n}}$must be a standard embedded ${\displaystyle n-1}$ sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in ${\displaystyle \mathbb {R} ^{4}}$. In 1984 Henry C. Wente constructed the Wente torus, an immersion into ${\displaystyle \mathbb {R} ^{3}}$ of a torus with constant mean curvature.

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