Embedded 𝐻-planes in hyperbolic 3-space

Coskunuzer, Baris

doi:10.1090/tran/7286

Embedded $H$-planes in hyperbolic $3$-space
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Trans. Amer. Math. Soc. 371 (2019), 1253-1269 Request permission

Abstract:

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

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