Comparison principles and dynamical stability for a logarithmic inverse-trace flow on compact Hermitian manifolds

Abstract

We study a logarithmic inverse-trace flow on compact connected Hermitian manifolds. We first establish a parabolic comparison principle and the resulting oscillation nonexpansiveness for admissible potentials modulo constants. We then introduce common-drift sub- and supersolutions. Their combination yields a uniform drift-corrected zero-order estimate for arbitrary admissible initial data, while the lower barrier alone implies the strict cone condition required in the second-order estimate. Combining these observations with Sun's Hermitian second-order estimate, parabolic Evans-Krylov regularity, Schauder estimates, and a uniform Harnack inequality, we obtain long-time existence and exponential smooth convergence. In particular, whenever the associated elliptic equation admits a smooth solution, its class is the unique globally attracting fixed point of the induced semiflow, and any two trajectories synchronize exponentially modulo constants. We also derive a Hermitian cone-supersolution convergence criterion and an affine-renormalization criterion that uses Sun's convergence theorem to construct one elliptic solution and then propagates global dynamical stability to every admissible initial potential.