Constant-mean-curvature surface

Like for minimal surfaces, there exist a close link to harmonic functions. An oriented surface ${\displaystyle S}$ in ${\displaystyle \mathbb {R} ^{3}}$ has constant mean curvature if and only if its Gauss map is a harmonic map.[14] Kenmotsu’s representation formula[15] is the counterpart to the Weierstrass–Enneper parameterization of minimal surfaces:

Let ${\displaystyle V}$ be an open simply connected subset of ${\displaystyle \mathbb {C} }$ and ${\displaystyle H}$ be an arbitrary non-zero real constant. Suppose ${\displaystyle \phi :V\rightarrow \mathbb {C} }$ is a harmonic function into the Riemann sphere. If ${\displaystyle \phi _{\bar {z}}\neq 0}$ then ${\displaystyle X:V\rightarrow R}$ defined by

${\displaystyle X(z)=\Re \int _{z_{0}}^{z}X_{z}(z')\,dz'}$

with

${\displaystyle X_{z}(z)={\frac {-1}{H(1+\phi (z){\bar {\phi }}(z))^{2}}}\left\{(1-\phi (z)^{2},i(1+\phi (z)^{2}),2\phi (z)){\frac {\bar {\partial \phi }}{\partial {\bar {z}}}}(z)\right\}}$

for ${\displaystyle z\in V}$ is a regular surface having ${\displaystyle \phi }$ as Gauss map and mean curvature ${\displaystyle H}$.

For ${\displaystyle \phi (z)=-1/{\bar {z}}}$ and ${\displaystyle H=1}$ this produces the sphere. ${\displaystyle \phi (z)=-e^{ix}}$ and ${\displaystyle H=1/2}$ gives a cylinder where ${\displaystyle z=x+iv}$.

Lawson showed in 1970 that each CMC surface in ${\displaystyle \mathbb {R} ^{3}}$has an isometric "cousin" minimal surface in ${\displaystyle \mathbb {S} ^{3}}$.[16][17] This allows constructions starting from geodesic polygons in ${\displaystyle \mathbb {S} ^{3}}$, which are spanned by a minimal patch that can be extended into a complete surface by reflection, and then turned into a CMC surface.

Hitchin, Pinkall, Sterling and Bobenko showed that all constant mean curvature immersions of a 2-torus into the space forms ${\displaystyle \mathbb {R} ^{3},\mathbb {S} ^{3}}$ and ${\displaystyle \mathbb {H} ^{3}}$ can be described in purely algebro-geometric data. This can be extended to a subset of CMC immersions of the plane which are of finite type. More precisely there is an explicit bijection between CMC immersions of ${\displaystyle \mathbb {R} ^{2}}$ into ${\displaystyle \mathbb {R} ^{3},\mathbb {S} ^{3}}$ and ${\displaystyle \mathbb {H} ^{3}}$, and spectral data of the form ${\displaystyle (\Sigma ,\lambda ,\rho ,\lambda _{1},\lambda _{2},L)}$ where ${\displaystyle \Sigma }$ is a hyperelliptic curve called the spectral curve, ${\displaystyle \lambda }$ is a meromorphic function on ${\displaystyle \Sigma }$, ${\displaystyle \lambda _{1}}$ and ${\displaystyle \lambda _{2}}$ are points on ${\displaystyle \mathbb {C} \setminus \{0\}}$, ${\displaystyle \rho }$ is an antiholomorphic involution and ${\displaystyle L}$ is a line bundle on ${\displaystyle \Sigma }$ obeying certain conditions.[18][19][20]

Discrete differential geometry can be used to produce approximations to CMC surfaces (or discrete counterparts), typically by minimizing a suitable energy functional.[21][22]