Pointed k -surfaces

Graham Smith

Bulletin de la Société Mathématique de France (2006)

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

Abstract

top
Let S be a Riemann surface. Let 3 be the 3 -dimensional hyperbolic space and let 3 be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping ϕ : S 3 = ^ . If i : S 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 3 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 ^ 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 3 with cylindrical ends.

How to cite

top

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

top
  1. [1] W. Ballman, M. Gromov & V. Schroeder – Manifolds of nonpositive curvature, Progress in Math., vol. 61, Birkhäuser, Boston, 1985. Zbl0591.53001MR823981
  2. [2] M. Gromov – « Foliated Plateau problem I. Minimal varieties », Geom. Funct. Anal.1 (1991), p. 14–79. Zbl0768.53011MR1091610
  3. [3] F. Labourie – « Problèmes de Monge-Ampère, courbes holomorphes et laminations », Geom. Funct. Anal.7 (1997), p. 496–534. Zbl0885.32013MR1466336
  4. [4] —, « Un lemme de Morse pour les surfaces convexes », Invent. Math.141 (2000), p. 239–297. Zbl0981.52002MR1775215
  5. [5] O. Lehto & K. I. Virtanen – Quasiconformal mappings in the plane, Grundlehren Math. Wiss., vol. 126, Springer-Verlag, New York-Heidelberg, 1973. Zbl0267.30016MR344463
  6. [6] M. P. Muller – « Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves », Holomorphic curves in symplectic geometry, Progress in Math., vol. 117, Birkhäuser, Basel, 1994, p. 217–231. MR1274931
  7. [7] H. Rosenberg & J. Spruck – « On the existence of convex hyperspheres of constant Gauss curvature in hyperbolic space », J. Diff. Geom.40 (1994), p. 379–409. Zbl0823.53047MR1293658
  8. [8] G. Smith – « Problèmes elliptiques pour des sous-variétés riemanniennes », Thèse, Orsay, 2004. 
  9. [9] —, « Hyperbolic Plateau problems », http://arxiv.org/abs/math/0506231v1, 2005. 
  10. [10] —, « Positive special Legendrian structures and Weingarten problems », http://arxiv.org/abs/math/0506230v1, 2005. 

NotesEmbed ?

top

You must be logged in to post comments.