Contracting an axially symmetric torus by its harmonic mean curvature

We consider the harmonic mean curvature flow of an axially symmetric torus whose axis is a closed geodesic, where the ambient space is a hyperbolic three-manifold. Assuming the initial surface is strictly convex and its harmonic mean curvature is less than 1/2, we show that the evolving surface satisfies a curvature condition comparable to that of a perfectly symmetric torus evolving under harmonic mean curvature flow. In other words, we prove that λ₁ ≈ e^-t, λ₂ ≈ e^t and λ₁λ₂ ≈ 1, where λ₁ and λ₂ are the principal curvatures of the evolving torus.