Embedded 𝐻-planes in hyperbolic 3-space

Source abstract

We show that for any C 0 \mathcal {C}^0 Jordan curve Γ \Gamma in S ∞ 2 ( H 3 ) S^2_{\infty }(\mathbf {H}^3) , there exists an embedded H H -plane P H \mathcal {P}_H in H 3 \mathbf {H}^3 with ∂ ∞ P H = Γ \partial _{\infty } \mathcal {P}_H =\Gamma for any H ∈ ( − 1 , 1 ) H\in (-1,1) . As a corollary, we prove that any quasi-Fuchsian hyperbolic 3 3 -manifold M ≃ Σ × R M\simeq \Sigma \times \mathbb {R} contains an H H -surface Σ H \Sigma _H in the homotopy class of the core surface Σ \Sigma for any H ∈ ( − 1 , 1 ) H\in (-1,1) . We also prove that for any C 1 C^1 Jordan curve in S ∞ 2 ( H 3 ) S^2_{\infty }(\mathbf {H}^3) , there exists a unique minimizing H H -plane P H \mathcal {P}_H with ∂ ∞ P H = Γ \partial _{\infty } \mathcal {P}_H =\Gamma for a generic H ∈ ( − 1 , 1 ) H\in (-1,1) .