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. 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 S {displaystyle S} is a compact star-shaped surface in R 3 {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 R 3 {displaystyle mathbb {R} ^{3}} with constant mean curvature H ≠ 0 {displaystyle H eq 0} must be a sphere. Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in R n {displaystyle mathbb {R} ^{n}} must be a standard embedded n − 1 {displaystyle n-1} sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in R 4 {displaystyle mathbb {R} ^{4}} . In 1984 Henry C. Wente constructed the Wente torus, an immersion into R 3 {displaystyle mathbb {R} ^{3}} of a torus with constant mean curvature. 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 R 3 {displaystyle mathbb {R} ^{3}} . Korevaar, Kusner and Solomon proved that a complete embedded CMC surface will have ends asymptotic to unduloids. Each end carries a n ( 2 π − n ) {displaystyle n(2pi -n)} 'force' along the asymptotic axis of the unduloid (where n is the circumference of the necks), the sum of which must be balanced for the surface to exist. Current work involves classification of families of embedded CMC surfaces in terms of their moduli spaces. In particular, for k ≥ 3 {displaystyle kgeq 3} coplanar k-unduloids of genus 0 satisfy ∑ i = 1 k n i ≤ ( k − 1 ) π {displaystyle sum _{i=1}^{k}n_{i}leq (k-1)pi } for odd k, and ∑ i = 1 k n i ≤ k π {displaystyle sum _{i=1}^{k}n_{i}leq kpi } for even k. At most k − 2 ends can be cylindrical. Like for minimal surfaces, there exist a close link to harmonic functions. An oriented surface S {displaystyle S} in R 3 {displaystyle mathbb {R} ^{3}} has constant mean curvature if and only if its Gauss map is a harmonic map. Kenmotsu’s representation formula is the counterpart to the Weierstrass–Enneper parameterization of minimal surfaces: Let V {displaystyle V} be an open simply connected subset of C {displaystyle mathbb {C} } and H {displaystyle H} be an arbitrary non-zero real constant. Suppose ϕ : V → C {displaystyle phi :V ightarrow mathbb {C} } is a harmonic function into the Riemann sphere. If ϕ z ¯ ≠ 0 {displaystyle phi _{ar {z}} eq 0} then X : V → R {displaystyle X:V ightarrow R} defined by