Cyclic Surfaces With Prescribed Mean Curvature

In this thesis, the geometry properties of cyclic surfaces in Eucilidean 3-space is mainly studied. Let n = n(s) be the smooth unit vector field of each plane including s-circle. Then cyclic surface is parametrized bywhere a = n(s), b = n'(s), c = n(s)∧n'(s),ρ=ρ(s) and r = r(s) denote respectively the radius and center of each s-circle, and s is the arc parameter of n(s), t is the radian of each s-circle.Let T = a, N = bcost+csint, B =-bsint+ccost, choose moving frame {X; T, N, B}. Then use this moving frame to compute the coefficients of the first fundamental form E, F, G and the coefficients of the second fundamental form L, M, N. Let us denote by [,, ] the mixed product in R³ and put W = EG - F², then the mean curvature H can write aswhere H₁ = G[X_s,X_t,X_ss] - 2F[X_s,X_t,X_st] + E[X_s, X_t,X_tt].It is intended to study the cyclic surfaces that satisfy (?)H/(?)t = 0. It is easy to check that 2H_1tW - 3H₁W_t=0 is equivalent to (?)H/(?)t = 0. So it is sufficient to study the cyclic surfaces that satisfy 2H_1tW -3H₁W_t= 0. Expand 2H_1tW-3H₁W_t into Fourier expansion about t. ThenAll coefficients E_i, F_i are smooth function on s. So 2H_1tW - 3H₁W_t = 0 holds if and only if E₀= E_i = F_i = 0, i = 1,2,3,4. The main results of this paper are:Proposition 1 If the cyclic surfaces S satisfy the type (?)H/(?)t = 0, then S is a sphere or rank(Φ₂) = 3.Remark: The representation ofΦ₂ can be refered to page 17(3.3.2) of this paper.Proposition 2 If cyclic surface S is foliated by pieces of circles lying in parallel planes, and (?)H/(?)t = 0, then S must be a surface of revolution or a Riemann minimal surface.Furthermore, after choosing a suitable frame, we give a representation of cyclic surface. Then under this representation, we give a method to look for the line of striction of cyclic surface.