1.8. Surfaces with constant mean curvature 27 Theorem 1.35. (Constant mean curvature surfaces are isother- mic) Surfaces with constant mean curvature and without umbilic points are isothermic. Theorem 1.36. (Minimal surface is Christoffel dual to its Gauss map) A minimal surface without umbilic points f : R2 R3 is the Christof- fel dual f = n∗ of its Gauss map n : R2 S2. Theorem 1.37. (Parallel constant mean curvature surfaces) Let f : R2 R3 be a surface with constant mean curvature H0 = 0 and without umbilic points, and let n : R2 S2 be its Gauss map. Then (i) every parallel surface ft = f +tn is linear Weingarten, i.e., its mean and Gaussian curvature functions Ht, Kt satisfy a linear relation αHt + βKt = 1 with constant coefficients α, β (ii) the parallel surface f 1 H 0 = f + 1 H0 n is Christoffel dual to f and has constant mean curvature H0 (iii) the mid-surface f 1 2H 0 := f + 1 2H0 n has constant positive Gaussian curvature K0 = 4H2. 0 We summarize considerations of these chapter in the following table: Koenigs net f in RN Moutard net y in RN+1 A-surface f in R3 Moutard net n in R3 Orthogonal net f in RN conjugate net ˆ in QN 0 P(LN+1,1) Principally parametrized conjugate net ˆ in Rout 4,1 sphere congruence S in R3 K-surface f in R3 Moutard net n in S2 Isothermic surface f in RN Moutard net ˆ in LN+1,1 Minimal surface f in R3 Isothermic net n in S2 All these notions and relations will be discretized in the main text of the book. The actual list of examples treated in this book is even longer. We discretize some other classical examples including line congruences and con- stant mean curvature surfaces. In the context of Lie and Pl¨ ucker geometry, isotropic line congruences are interpreted as curvature and asymptotic line parametrized surfaces, respectively. A discrete version of this theory is also developed in the main text of the book.
Previous Page Next Page