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We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface into a given closed manifold, we add to the area Lagrangian a term equal to the norm of the second fundamental form of the immersion times a “viscosity” parameter. This relaxation of the area functional satisfies the Palais–Smale condition for . This permits to construct critical points of the relaxed Lagrangian using classical min-max arguments such as the mountain pass lemma. The goal of this work is to describe the passage to the limit when the “viscosity” parameter tends to zero. Under some natural entropy condition, we establish a varifold convergence of these critical points towards a parametrized integer stationary varifold realizing the min-max value. It is proved in Pigati and Rivière (arXiv:1708.02211, 2017) that parametrized integer stationary varifold are given by smooth maps exclusively. As a consequence we conclude that every surface area minmax is realized by a smooth possibly branched minimal immersion.
@article{PMIHES_2017__126__177_0,
author = {Rivi\`ere, Tristan},
title = {A viscosity method in the min-max theory of minimal surfaces},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {177--246},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {126},
year = {2017},
doi = {10.1007/s10240-017-0094-z},
mrnumber = {3735867},
zbl = {1387.53084},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0094-z/}
}
TY - JOUR AU - Rivière, Tristan TI - A viscosity method in the min-max theory of minimal surfaces JO - Publications Mathématiques de l'IHÉS PY - 2017 SP - 177 EP - 246 VL - 126 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0094-z/ DO - 10.1007/s10240-017-0094-z LA - en ID - PMIHES_2017__126__177_0 ER -
%0 Journal Article %A Rivière, Tristan %T A viscosity method in the min-max theory of minimal surfaces %J Publications Mathématiques de l'IHÉS %D 2017 %P 177-246 %V 126 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0094-z/ %R 10.1007/s10240-017-0094-z %G en %F PMIHES_2017__126__177_0
Rivière, Tristan. A viscosity method in the min-max theory of minimal surfaces. Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 177-246. doi: 10.1007/s10240-017-0094-z
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