LAUSR

In this paper, we study gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics (both metrics are called Ricci-harmonic). First, we prove that all ends of $$\tau $$τ-quasi Ricci-harmonic metrics with $$\tau >1$$τ>1 should be f-non-parabolic if $$\lambda =0,\mu >0$$λ=0,μ>0, or $$\lambda <0, \mu \ge 0$$λ<0,μ≥0. For the case that $$\lambda<0, \mu < 0$$λ<0,μ<0, we can also arrive at the f-non-parabolic property if we add a condition about the scalar curvature. Furthermore, we discuss the connectivity at infinity for quasi Ricci-harmonic metrics. We also conclude that all ends of steady or expanding gradient Ricci-harmonic solitons should be f-non-parabolic, based on which we establish structure theorems for these two solitons.