[Flot par l'inverse de la courbure moyenne dans l'espace hyperbolique complexe]
Nous considérons l'évolution par l'inverse de la courbure moyenne d'une surface étoilée, fermée et à courbure moyenne positive dans l'espace hyperbolique complexe. Nous montrons que le flot est défini pour tout temps positif et que la surface reste étoilée et à courbure moyenne positive. De plus, la métrique induite, après un changement d'échelle, converge vers un multiple conforme de la métrique sous-riemannienne standard sur la sphère de dimension impaire. Nous allons montrer l'existence d'exemples de données initiales telles que cette limite sous-riemannienne n'a pas courbure de Webster constante.
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub-Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.
DOI : 10.24033/asens.2404
Keywords: Inverse mean curvature flow, complex hyperbolic space, sub-Riemannian geometry.
Mot clés : Flot par l'inverse de la courbure moyenne, espace hyperbolique complexe, géométrie sous-riemannienne.
@article{ASENS_2019__52_5_1107_0,
author = {Pipoli, Giuseppe},
title = {Inverse mean curvature flow in complex hyperbolic space},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {1107--1135},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 52},
number = {5},
year = {2019},
doi = {10.24033/asens.2404},
mrnumber = {4057778},
zbl = {1462.53087},
language = {en},
url = {http://www.numdam.org/articles/10.24033/asens.2404/}
}
TY - JOUR AU - Pipoli, Giuseppe TI - Inverse mean curvature flow in complex hyperbolic space JO - Annales scientifiques de l'École Normale Supérieure PY - 2019 SP - 1107 EP - 1135 VL - 52 IS - 5 PB - Société Mathématique de France. Tous droits réservés UR - http://www.numdam.org/articles/10.24033/asens.2404/ DO - 10.24033/asens.2404 LA - en ID - ASENS_2019__52_5_1107_0 ER -
%0 Journal Article %A Pipoli, Giuseppe %T Inverse mean curvature flow in complex hyperbolic space %J Annales scientifiques de l'École Normale Supérieure %D 2019 %P 1107-1135 %V 52 %N 5 %I Société Mathématique de France. Tous droits réservés %U http://www.numdam.org/articles/10.24033/asens.2404/ %R 10.24033/asens.2404 %G en %F ASENS_2019__52_5_1107_0
Pipoli, Giuseppe. Inverse mean curvature flow in complex hyperbolic space. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1107-1135. doi : 10.24033/asens.2404. http://www.numdam.org/articles/10.24033/asens.2404/
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