On rotational surfaces with circular boundary and almost constant mean curvature

We analyze compact rotationally symmetric surfaces of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^3$$\end{document} with circular boundary and almost constant mean curvature, showing that they must nearly spherical caps.


Introduction
Given a surface in R 3 , we define its mean curvature H at point p as the average of principal curvatures of at p. In this paper, we will be interested in the case H is constant and also almost constant.
The interest in constant mean curvature (CMC) surfaces of R 3 comes from the fact they are local solutions to an important classical problem in geometry: minimize the area of surfaces with enclosed volume. These surfaces have been studied since the 18th century yielding application to many other areas, including physics and material sciences.
Around 1950, a breakthrough happened in the theory of constant mean curvature surfaces. First, Hopf [4] proved that the round sphere is the only closed CMC surface of genus 0 in R 3 . Next, Alexandrov [1,2] proved that the round sphere is the only closed embedded CMC surface of R 3 . After their work, the question of whether Hopf's result could be generalized to higher genus surface was open until the work of Wente [7] who showed the existence of immersed tori in R 3 with constant mean curvature.
Analogous questions can also be asked for surfaces with boundary but much less is known, including the case of CMC surfaces with circular boundary. In particular, it is open question whether spherical caps are the only immersed CMC disks in R 3 with circular boundary. It is also unknown whether a compact embedded CMC surface with circular boundary must be a This article is part of the section "Applications of PDEs" edited by Hyeonbae Kang.
B Jason Minghan Dong jasonminghandong@gmail.com 1 Association Cooperative du College Jean-de-Brebeuf, Montreal, QC, Canada spherical cap. Some partial results have been obtained under additional hypothesis, but there general cases of these questions remain largely open, we recommend the interested reader to the recent book by López [6]. In this paper, we will investigate this and similar questions under the additional hypothesis that the surface is rotationally symmetric. We will first revisit the following result which can be traced back to Delaunay [3].

Theorem 1.1 Suppose is a surface of revolution with constant mean curvature H and with circular boundary ∂ . Then must be a spherical cap.
To understand better how coercive the mean curvature function of a surface must be, our main result analyzes the case of surfaces with almost constant mean curvature. We show, in an appropriate sense, that these must be close to a spherical cap: Theorem 1.2 A rotational surface with almost constant mean curvature must be near spherical in the following sense: There exist numbers δ, ε > 0 such that if γ (s) = (x(s), y(s)) is any curve parametrized by arc length and the mean curvature of the surface (s) of rotation of γ along the x-axis has mean curvature satisfying where H > 0 is a real number, then H must be H r ,d and γ must be δ-close to the arc r ,d , 1 that is, every γ (s), s ∈ [0, L], is at a distance at most δ from r ,d .

Remark
In both theorems, we do not assume that the curves are embedded.

Preliminaries
In this section, we set up the differential equations that appear when one tries to prescribe the mean curvature of a surface obtained by the rotation of a curve. Suppose is a compact surface obtained by rotating a curve γ (s) = (x(s), y(s)) from the x y-plane around x-axis. Assume the boundary of is the circle y 2 + z 2 = r 2 on the yz-plane. Moreover, suppose γ is parametrized by unit speed and has total length L. Note that by smoothness of we must have x (L) = 0, y(L) = 0. The first and second fundamental forms can be calculated: Moreover, the mean curvature H (s) of satisfies: In order to solve the differential equation (1) where i = √ −1, which by (1), must satisfy: This first-order linear differential equation can then be solved to find: where the two functions F and G are: and the constants c 1 and c 2 comes from integration. The solutions (x(s), y(s)) would satisfy:

Proof of Theorem 1.1
To prove Theorem 1.1, assume has constant mean curvature H > 0. In this case, the functions F and G would be given by: We then find the solutions x (s) and y(s): We next solve for the constants c 1 and c 2 using the boundary conditions. Since ∂ is the circle y 2 + z 2 = r 2 , we have that x(0) = 0, y(0) = r , z(0) = 0. Moreover, x (L) = 0, y(L) = 0. Plugging in y(L) = 0 and squaring both sides of the Eq. (4), we get: Because c 1 and c 2 are real numbers, we can safely assume that the two squares must be equal to zero. Hence, the constants would be: and thus: Using standard trigonometric identities, this can be reduced to Now, using trigonometric identities, the expression for x (s) can be reduced to: x (s) = sin (H (L − s)). Therefore: where c 3 is another constant of integration. The solution to the equation would be: This a spherical cap with radius 1 H and center (c 3 , 0, 0), where c 3 = − 1 H cos(H L).

Proof of Theorem 1.2
We will prove this by contradiction: Suppose that there exists a family of curves γ n (s) = (x n (s), y n (s)) defined on [0, L n ] and satisfying condition (1) and such that but γ n is not δ-close to a circle, for some fixed δ > 0. We will show that this is impossible by proving these curves must converge to the circle r ,d . A key obstacle to proving this convergence is that fact that despite H n converging to H and using Eqs. (2), (3) to find (x n (s), y n (s)) based on H n (s), these equations are defined on intervals [0, L n ] and we don't even know a priori that L n stays bounded as n → ∞. Namely, for s ∈ [0, L n ], if one defines for some constants c n 1 , c n 2 . Thus, showing that such F n , G n , x n , y n converge to their respective limits, without knowing whether L n are uniformly bounded is a delicate issue since, in fact, there are curves whose surface of rotation has constant mean curvature equal to H and infinite length. These surfaces are called unduloids, see Kenmotsu [5,Page 41].
Thus, by previous section, γ (s) = (x(s), y(s)) must be a piece of circle which when rotated has mean curvature exactly H and such that γ (0) = (0, r ). But since there are at most two circles like that, but only one of which passes through the endpoint (d, 0) of the curves γ n (s), namely the circle r ,d , the curves must be converging to r ,d but this is a contradiction.