Minimal surfaces in finite volume noncompact hyperbolic 3-manifolds

Abstract

In this note, we explain how a mistake made in [2] can be corrected. Actually this mistake appears in the proof of Proposition 8 (the second maximum principle) and was brought to our attention by A. Song [4]. Let us notice that unfortunately, we did not find an alternative proof of this proposition but we found an alternative proposition. The new proposition does not change the subsequent applications we made of the original proposition. At the end of the note we explain which modifications should be done where the original Proposition 8 is applied.The difference with the original Proposition 8 is that here we have to assume a control on the index of the minimal surface Σ.The new proposition. Let (T, ds 2 T ) be a flat 2 torus of diameter 1 and consider M = T×[− 1 2 , +∞) endowed with the metric g = e −2h(t) Λ 2 ds 2 T +dt 2 ; here Λ is a positive constant and h satisfies the following assumptions:•The main example is h(t) = t. We denote by T s = T × {s}.Proposition 1. Let i 0 ∈ N. There is a Λ 0 such that the following is true. For any Λ ≤ Λ 0 and any compact embedded minimal surface Σ ⊂ M with ∂Σ ⊂ T − 1 2 and index less than i 0 , we have Σ ⊂ T × [− 1 2 , 0]Proof. If the proposition is not true there is a sequence Λ n → 0 and Σ n a minimal surface in M with index less than i 0 , boundary in T − 1 2 and the maximum of the function t on Σ n is t n > 0.In order to study this sequence, we translate in the t direction by −t n and make a homothety by λ n = e h(tn) Λn → +∞. We thus obtain a compact minimal surface S n in T × [−( 1 2 + t n )λ n , 0] endowed with the metric g n = e −2hn(t) ds 2 T + dt 2 (where h n (t) = h( t λn + t n ) − h(t n )) with boundary in T −( 1 2 +tn)λn and containing a point in T 0 . We notice that h n → 0 uniformly on any compact. So the ambient space smoothly converges to X = T × R − endowed with the flat metric ds 2 T × dt 2 . Let ε be positive. Since S n has index at most i 0 , there is a set E n (ε) of at most i 0 points in S n such that, for any p ∈ X with d(p, E n (ε)) > 2ε, 1

Drilling short geodesics in hyperbolic 3-manifolds

“…Actually these two results are very similar to results we obtained with Collin and Hauswirth in [5] concerning the geometry of minimal surfaces in hyperbolic cusps. In both cases, the argument is based on the fact that the tubular neighborhoods are foliated by equidistant tori whose diameter are small.…”

“…The aim of this section is to understand the behaviour of a minimal surface in a tubular end when we know a priori an upper bound on its curvature. A similar study was made for cusp ends in [5].…”

Minimal hypersurfaces of least area

“…We notice that in this case the estimates π|χ(Σ)| ≤ |Σ| ≤ 2π|χ(Σ)| are still valid for properly embedded stable minimal surfaces (see [4]). …”

“…If the hypothesis on the Heegaard genus is dropped, the monotonicity formula and the thin-thick decomposition of M tells us that any minimal surface in a closed hyperbolic 3-manifold has area at least some c > 0 that does not depend on M (see [4]) (this is also true for closed immersed Hsurfaces with H < 1). So this leads us to ask: what is a closed orientable hyperbolic 3-manifold M that minimizes A 1 (M ) among such 3−manifolds?…”

“…In this section we extend Theorem C to the case where M is a complete non-compact hyperbolic 3-manifold with finite volume. Notice that such a manifold has closed minimal surfaces (see [4]). …”