Pointed k -surfaces

Graham Smith

Bulletin de la Société Mathématique de France (2006)

  • Volume: 134, Issue: 4, page 509-557
  • ISSN: 0037-9484

Abstract

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Let S be a Riemann surface. Let ℍ 3 be the 3 -dimensional hyperbolic space and let ∂ ∞ ℍ 3 be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping ϕ : S → ∂ ∞ ℍ 3 = ℂ ^ . If i : S → ℍ 3 is a convex immersion, and if N is its exterior normal vector field, we define the Gauss lifting, ı ^ , of i by ı ^ = N . Let n → : U ℍ 3 → ∂ ∞ ℍ 3 be the Gauss-Minkowski mapping. A solution to the Plateau problem ( S , ϕ ) is a convex immersion i of constant Gaussian curvature equal to k ∈ ( 0 , 1 ) such that the Gauss lifting ( S , ı ^ ) is complete and n → ∘ ı ^ = ϕ . In this paper, we show that, if S is a compact Riemann surface, if 𝒫 is a discrete subset of S and if ϕ : S → ℂ ^ is a ramified covering, then, for all p 0 ∈ 𝒫 , the solution ( S ∖ 𝒫 , i ) to the Plateau problem ( S ∖ 𝒫 , ϕ ) converges asymptotically as one tends to p 0 to a cylinder wrapping a finite number, k , of times about a geodesic terminating at ϕ ( p 0 ) . Moreover, k is equal to the order of ramification of ϕ at p 0 . We also obtain a converse of this result, thus completely describing complete, constant Gaussian curvature, immersed hypersurfaces in ℍ 3 with cylindrical ends.

How to cite

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Smith, Graham. "Pointed $k$-surfaces." Bulletin de la Société Mathématique de France 134.4 (2006): 509-557. <http://eudml.org/doc/272442>.

@article{Smith2006,
abstract = {Let $S$ be a Riemann surface. Let $\mathbb \{H\}^3$ be the $3$-dimensional hyperbolic space and let $\partial _\infty \mathbb \{H\}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\rightarrow \partial _\infty \mathbb \{H\}^3=\widehat\{\mathbb \{C\}\}$. If $i:S\rightarrow \mathbb \{H\}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\hat\{\imath \}$, of $i$ by $\hat\{\imath \}=N$. Let $\overrightarrow\{n\}:U\mathbb \{H\}^3\rightarrow \partial _\infty \mathbb \{H\}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\hat\{\imath \})$ is complete and $\overrightarrow\{n\}\circ \hat\{\imath \}=\varphi $. In this paper, we show that, if $S$ is a compact Riemann surface, if $\mathcal \{P\}$ is a discrete subset of $S$ and if $\varphi :S\rightarrow \widehat\{\mathbb \{C\}\}$ is a ramified covering, then, for all $p_0\in \mathcal \{P\}$, the solution $(S\setminus \mathcal \{P\},i)$ to the Plateau problem $(S\setminus \mathcal \{P\},\varphi )$ converges asymptotically as one tends to $p_0$ to a cylinder wrapping a finite number, $k$, of times about a geodesic terminating at $\varphi (p_0)$. Moreover, $k$ is equal to the order of ramification of $\varphi $ at $p_0$. We also obtain a converse of this result, thus completely describing complete, constant Gaussian curvature, immersed hypersurfaces in $\mathbb \{H\}^3$ with cylindrical ends.},
author = {Smith, Graham},
journal = {Bulletin de la Société Mathématique de France},
keywords = {immersed hypersurfaces; pseudo-holomorphic curves; contact geometry; plateau problem; gaussian curvature; hyperbolic space; moduli spaces; teichmüller theory},
language = {eng},
number = {4},
pages = {509-557},
publisher = {Société mathématique de France},
title = {Pointed $k$-surfaces},
url = {http://eudml.org/doc/272442},
volume = {134},
year = {2006},
}

TY - JOUR
AU - Smith, Graham
TI - Pointed $k$-surfaces
JO - Bulletin de la Société Mathématique de France
PY - 2006
PB - Société mathématique de France
VL - 134
IS - 4
SP - 509
EP - 557
AB - Let $S$ be a Riemann surface. Let $\mathbb {H}^3$ be the $3$-dimensional hyperbolic space and let $\partial _\infty \mathbb {H}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\rightarrow \partial _\infty \mathbb {H}^3=\widehat{\mathbb {C}}$. If $i:S\rightarrow \mathbb {H}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\hat{\imath }$, of $i$ by $\hat{\imath }=N$. Let $\overrightarrow{n}:U\mathbb {H}^3\rightarrow \partial _\infty \mathbb {H}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\hat{\imath })$ is complete and $\overrightarrow{n}\circ \hat{\imath }=\varphi $. In this paper, we show that, if $S$ is a compact Riemann surface, if $\mathcal {P}$ is a discrete subset of $S$ and if $\varphi :S\rightarrow \widehat{\mathbb {C}}$ is a ramified covering, then, for all $p_0\in \mathcal {P}$, the solution $(S\setminus \mathcal {P},i)$ to the Plateau problem $(S\setminus \mathcal {P},\varphi )$ converges asymptotically as one tends to $p_0$ to a cylinder wrapping a finite number, $k$, of times about a geodesic terminating at $\varphi (p_0)$. Moreover, $k$ is equal to the order of ramification of $\varphi $ at $p_0$. We also obtain a converse of this result, thus completely describing complete, constant Gaussian curvature, immersed hypersurfaces in $\mathbb {H}^3$ with cylindrical ends.
LA - eng
KW - immersed hypersurfaces; pseudo-holomorphic curves; contact geometry; plateau problem; gaussian curvature; hyperbolic space; moduli spaces; teichmüller theory
UR - http://eudml.org/doc/272442
ER -

References

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