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  "body": "\n## Definition\n\nLet $S \\subset \\R^3$ be an orientable [[regular surface|6080X3FAD4]] with a chosen unit normal, and let $H \\colon S \\to \\R$ be its [[mean curvature|PA78NFEPCF]].\nThe surface $S$ is a *constant-mean-curvature surface* if there is a real number $H_0$ with $H(p) = H_0$ for every $p \\in S$.\n\n## Remarks\n\nThe sphere of radius $r$ is a CMC surface: for the inward normal both principal curvatures are $1/r$, so $H = 1/r$ everywhere.\nThe paraboloid $z = x^2 + y^2$ is not: its mean curvature is $2$ at the origin and tends to $0$ far from the origin.\n\nThe case $H_0 = 0$ gives the [[minimal surfaces|41X0YW2R9N]].\nChanging the normal changes the sign of $H$, so the condition does not depend on the choice of normal.\nThe same definition applies to immersed surfaces, where self-intersections are allowed.\nBy a theorem of Alexandrov, a compact embedded CMC surface is a round sphere, but compact immersed CMC tori exist, such as the [[Wente torus|E5NNY7XM44]].\n\n## Explanations\n\n### L1 human\n\nA CMC surface is a surface whose mean curvature is the same number at every point.\n",
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