Work

A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow

Year: 2025

Type: article

Abstract: This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second-order elliptic operator. We discuss sufficient and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal mean curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

Source: Discrete and Continuous Dynamical Systems

DOI: 10.3934/dcds.2025029

Authors: Antoni Kijowski, Qing Liu, Ye Zhang, Xiaodan Zhou

Institution: Okinawa Institute of Science and Technology Graduate University

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Japan Society for the Promotion of Science

Awards: Weak notions of curvature-dimension conditions on step-two Carnot groups, Hamilton-Jacobi equations on metric measure spaces, Nonlinear partial differential equations on sub-Riemannian manifolds based on viscosity solution theory, Nonlinear Partial Differential Equations on Metric Spaces, Quasiconformal and Sobolev mappings on metric measure spaces