Parabolic evolution with boundary to the Bishop family of holomorphic discs

TL;DR

This paper proves that certain rotationally symmetric holomorphic discs evolving under mean curvature flow in a neutral Kähler space converge to maximal or holomorphic discs, extending to Bishop fillings of complex points.

Contribution

It establishes convergence results for mean curvature flow of holomorphic discs with boundary conditions, including Bishop fillings at complex points.

Findings

Flow converges to maximal surfaces under Dirichlet and Neumann conditions.

Holomorphic boundary conditions lead to convergence to holomorphic discs.

Flow of discs with boundary on symmetric line congruences converges to a Bishop filling.

Abstract

It is proven that a definite graphical rotationally symmetric line congruence evolving under mean curvature flow with respect to the neutral Kaehler metric in the space of oriented lines of Euclidean 3-space, subject to suitable Dirichlet and Neumann boundary conditions, converges to a maximal surface. When the Neumann condition implemented is that the flowing disc be holomorphic at the boundary, it is proven that the flow converges to a holomorphic disc. This is extended to the flow of a family of discs with boundary lying on a fixed rotationally symmetric line congruence, which is shown to converge to a filling by maximal surfaces. Moreover, if the family is required to be holomorphic at the boundary, it is shown that the flow converges to the Bishop filling by holomorphic discs of an isolated complex point of Maslov index 2.