Existence of least area planes in hyperbolic 3-space with co-compact metric

Let r be a metric on the hyperbolic 3-space H3 induced from an arbitrary Riemannian metric on a closed hyperbolic 3-manifold. In this paper, we will show that any smooth simple loop in S∞2 spans a properly embedded r-least area plane in H3. This solves Gabaiʹs conjecture ((J. Amer. Math. Soc. 10 (1997) 37), Conjecture 3.12), affirmatively.