Singular surfaces of revolution with prescribed unbounded mean curvature

We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.


Introduction
In this note, we study singular surfaces of revolution. Let I ⊂ R be a subset, and γ : I → R 2 a C ∞ plane curve. We set γ(t) = (x(t), y(t)) (y > 0), and set the revolution surface s(t, θ) = x(t), y(t) cos θ, y(t) sin θ (0.1) of γ. The curve γ is called the profile curve of s. We denote by H(t) the mean curvature of s(t, θ). Given a C ∞ function H(t) on I, it is given by Kenmotsu [9] that the concrete solution of the profile curve (x(t), y(t)) satisfying the revolution surface s(t, θ) has the mean curvature H(t). Moreover, the periodicity of s is also studied [10].
On the other hand, in the recent decades, there are several articles concerning differential geometry of singular curves and surfaces, namely, curves and surfaces with singular points, in the 2 and 3 dimensional Euclidean spaces [2][3][4][5][6][7][11][12][13][14][15]. If the profile curve γ is regular, then the mean curvature H is differentiable on I, but if γ has a singularity, then H may diverge [14] (see also [11]). Given a C ∞ function H defined on I \ P , where P is a discrete set, we give a concrete solution γ = (x, y) such that the mean curvature of revolution surface of γ is H. Moreover, we give conditions for the fundamental singularities of γ. We also discuss the periodicity of the surface. This condition is equivalent to saying that γ is a frontal (see Section 2 for detail). We choose the unit normal vector of the revolution surface s by ν(t, θ) = sin ϕ(t), − cos ϕ(t) cos θ, − cos ϕ(t) sin θ . (1.1) Then the mean curvature H can be given on the regular set of s. We have the following lemma.
Lemma 1.1. The function Hl can be extended to a C ∞ function on I.
Proof. By a direct computation, Hl with respect to the unit normal vector (1.1) can be calculated by where ′ = d/dt. Since y > 0, this proves the assertion.
See [11,Proposition 3.8] for more detailed behavior of the mean curvature for the case of cuspidal edges. We remark that the case y = 0 is already considered in [9].
Conversely, given a C ∞ function H : I \ P → R, where P is a discrete set, and function l : I → R satisfying that Hl is a C ∞ function on I and l −1 (0) = P , we look for a surface of revolution with the profile curve γ whose mean curvature with respect to (1.1) is H and γ ′ = l(cos ϕ, sin ϕ). Then x, y satisfy the differential equation: (1.2) Following Kenmotsu [9], we solve this equation together with (x ′ (t), y ′ (t)) = l(t)(cos ϕ(t), sin ϕ(t)).
We set z(t) = y(t) sin ϕ(t) + √ −1y(t) cos ϕ(t). Then (1.2) can be modified into and the general solution of this equation is where c 1 , c 2 ∈ R, and By y(t) 2 = |z(t)| 2 and x ′ (t) = l(t) cos ϕ(t) = l(t)(z(t) −z(t))/(2 √ −1y(t)), we have (1.5) We take the initial values c 1 , c 2 satisfying that (F (t) − c 1 ) 2 + (G(t) − c 2 ) 2 > 0 on the considering domain. It should be mentioned that on the set of regular points, there is Kenmotsu's result [9], and singularities can be considered by taking the limits of regular parts. However, we will see the class of singularities of γ in Section 2, which cannot be investigated just looking at limits of regular points. Furthermore, we believe that the formula (1.5), (1.4), which is able to pass though the singularities, can extend the treatment of singular surfaces of revolution. We remark that there is a representation formula [8,Theorem 4] for surfaces which have prescribed H and the unit normal vector.

Singularities of profile curves
In this section, we study conditions for singularities of profile curves and revolution surfaces. A singular point p of a map γ is called a ordinary cusp (3,4), (3,5). It is known that the singularity of (R, 0) → (R 2 , 0) which are determined by its 5-jet with respect to A-equivalence are only these cusps. Criteria for these singularities are known. See [1] for example.

Periodicity
In this section, we study the condition for periodicity of surfaces when H and l are periodic, where the condition for regular case is obtained by Kenmotsu [10]. We define the profile curve (x, y) of the surface of revolution given by (0.1) being periodic with the period L if there exists T > 0 such that x(s + L) = x(s) + T and y(s + L) = y(s). Then we have the following theorem. where (x ′ (0), y ′ (0)) = l(0)(cos ϕ(0), sin ϕ(0)).
Kenmotsu gave the condition for the case of the profile curve is regular [10, Theorem 1]. If the profile curve is regular, the above condition is the same as Kenmotsu's condition. In fact, for regular case, since one can take t = 0 giving the minimum of y, we can assume that ϕ(0) = 0. However, in our case, the profile curve may have singularities, the existence of t 0 such that ϕ(t 0 ) = 0 fails in general. One can show