Non-properly embedded $H$-planes in $\mathbb{H}^3$

Abstract

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Non-properly embedded H-planes in $${\mathbb H}^2\times {\mathbb R}$$ H 2 × R

“…Proof The proof is almost identical to the proof of Lemma 4.1 in [3], and for the sake of completeness, we give it here. Let T α be the isometry of which is a translation by α in the θ direction, i.e.,…”

“…Moreover, n separates n into two regions. Similarly to Lemma 4.1 in [3], in the following lemma we show that for any such n , the minimizer surface n is a θ-graph.…”

“…Recall that the necksize of C 2 is η 2 = b 2 (0). Let C 3 = ϕ η 2 (C 2 ), see equation (3) for the definition of ϕ η 2 . Then, C 3 is a rotationally invariant catenoid whose rotational axis is the line (η 2 , 0) × R (Fig.…”

“…When H = 0, Rodríguez and Tinaglia [10] constructed non-proper, complete minimal planes embedded in H 2 × R. However, their construction does not generalize to produce complete, non-proper planes embedded in H 2 × R with non-zero constant mean curvature. Instead, the construction presented in this paper is related to the techniques developed by the authors in [3] to obtain examples of non-proper, stable, complete planes embedded in H 3 with constant mean curvature H , for any H ∈ [0, 1).…”

Asymptotic Plateau Problem in H2xR

“…In the last decade, minimal surfaces in H 2 × R have been studied extensively, and many important results have been obtained on the existence of many different types of minimal surfaces in H 2 × R and their properties, e.g. [NR,CR,CMT,MMR,MoR,MRR,PR,RT,ST1,ST2].…”