On axisymmetric Helfrich surfaces (Q2630624)

A Helfrich surface \(\Sigma \subset {\mathbb R}^3\) is defined by an equation (\textit{Euler-Lagrange equation}) of the form \[ -2\Delta_\Sigma H - (2H-c)(2H^2 - 2K + cH) + 2\lambda_a H + \lambda_v = 0 \] where \(\Delta_\Sigma\) is the Laplace-Beltrami operator, \(K\), \(H\) denote the Gaussian and mean curvatures, resp., and \(c\), \(\lambda_a\), \(\lambda_v\), \(\lambda_u\) are constants. The authors prove three theorems for a special type of Helfrich surfaces with rotational symmetry. The first theorem states that the geodesic curvature \(k\) of the meridian curve \(\gamma(y) = (x(y), y)\) of such a surface \(\Sigma\) is a certain linear or fractional linear function of the distance \(y\) to the axis of revolution of \(\Sigma\) if a hyperbolic metric is used in the supporting plane of \(\gamma(y)\). From previous papers it is known that the Euler-Lagrange equation can be reduced to a certain non-linear differential equation for a function \(f(y)\) and that this equation can be solved by a particular power series. In their second theorem the authors prove that this power series converges in a neighborhood of \(y=0\). Their third theorem states that the inequality \[ \lambda_v R^3 +(c^2 + 2\lambda_a) R ^2 - 2c R + 1 \geq 0 \] holds for an axisymmetric Helfrich surface \(\Sigma\) homeomorphic to a sphere. Here, \(R\) denotes the maximal distance of a point \(X \in \Sigma\) and the rotation axis.