Harmonic mean curvature flow on surfaces of negative Gaussian curvature
The \textit{harmonic mean curvature flow} is the geometric evolution equation \[ \frac{\partial P}{\partial t}=\frac{K}{H} \nu, \] in which each point of a surface evolves in the normal direction \(\nu\) with velocity equal to the quotient of the Gaussian curvature \(K\) by the mean curvature \(H\). The convex and weakly convex cases have been studied in [1]-[4]. In this paper, the authors consider a surface of revolution \(f(x,t)\) with boundary, whose principal eigenvalues have opposite signs, and whose mean curvature is negative at each point. The function \(f\) satisfies the fully nonlinear equation \[ \frac{\partial f}{\partial t}=\frac{f_{xx}}{-ff_{xx}+f_x^2+1}. \] The principal coefficient of the linearization is non-negative, independent of the signs of \(H\) and \(K\). Assume that the initial surface satisfies \(f >0\), \(f_{xx} >0\), and \(\tilde H := \;-f f_{xx}+f_x^2+1 <0\) on an interval \(I=[0,1]\), as well as the boundary growth condition \(c\leq x^{2-p}(1-x)^{2-p}f_{xx}\leq C,\;\;0 <x <1\) for some \(0 <p <1\), and constants \(c >0\) and \(C< \infty\). The main result is that there is a time \(T_0\) up to which there exists a unique solution of the flow, so that the mean curvature is negative for \(t< T_0\). Sharp a priori derivative bounds are derived to show smoothness of the flow and positivity of \(\tilde H\) up to \(T_0\). At \(T_0\), \(\tilde H\) vanishes on an interval whose length depends on the initial surface. The area of the surface is preserved under the flow, and the boundary of the surface moves by the curve shortening flow.
scientific article; zbMATH DE number 5141317