The capillary Gauss curvature flow

TL;DR

This paper introduces a new capillary Gauss curvature flow for hypersurfaces, proves finite-time shrinking, and shows normalized flow convergence to a soliton, extending classical curvature flow results to capillary boundary conditions.

Contribution

It develops a novel capillary Gauss curvature flow, analyzes its finite-time behavior, and establishes convergence to solitons, extending existing curvature flow theories to capillary hypersurfaces.

Findings

Flow shrinks to a point in finite time.

Normalized flow converges to a soliton.

Classification of solitons remains open.

Abstract

In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.