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This paper is concerned with a PDE approach to horizontally quasiconvex(h-quasiconvex) functions in the Heisenberg group based on a nonlinear secondorder elliptic operator. We discuss sufficient conditions and necessaryconditions for upper semicontinuous, h-quasiconvex functions in terms of theviscosity subsolution to the associated elliptic equation. Since the notion ofh-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of thefunction's sublevel sets, we further adopt these conditions to study theh-convexity preserving property for horizontal curvature flow in the Heisenberggroup. Under the comparison principle, we show that the curvature flow startingfrom a star-shaped h-convex set preserves the h-convexity during the evolution.

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