Nous montrons que le flot de Kähler-Ricci tordu sur une variété kählérienne compacte converge vers le flot des bords libres mobiles, dans une certaine limite d’échelle. Ceci conduit à un nouveau phénomène de formation de singularités et de changement topologique qui peut être vu comme une généralisation complexe du phénomène abondamment étudié de la formation de chocs dans la théorie des équations de Jacobi et des lois de conservation hyperboliques (notamment dans le modèle d’adhésion en cosmologie). En particulier, nous montrons comment retrouver le flot de Hele-Shaw (croissance du laplacien) dans des domaines 2D croissants à partir du flot de Ricci. Comme il sera expliqué ailleurs, la limite d’échelle en question apparaît comme la limite à température nulle de certains systèmes multi-particules sur .
We show that the twisted Kähler-Ricci flow on a compact Kähler manifold converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilton-Jacobi equations and hyperbolic conservation laws (notably, in the adhesion model in cosmology). In particular we show how to recover the Hele-Shaw flow (Laplacian growth) of growing 2D domains from the Ricci flow. As will be explained elsewhere the scaling limit in question arises as the zero-temperature limit of a certain many particle system on .
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@article{JEP_2018__5__519_0,
author = {Robert J. Berman and Chinh H. Lu},
title = {From the {K\"ahler-Ricci} flow to~moving~free~boundaries and shocks},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {519--563},
publisher = {\'Ecole polytechnique},
volume = {5},
year = {2018},
doi = {10.5802/jep.77},
zbl = {06988587},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.77/}
}
TY - JOUR AU - Robert J. Berman AU - Chinh H. Lu TI - From the Kähler-Ricci flow to moving free boundaries and shocks JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 519 EP - 563 VL - 5 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.77/ DO - 10.5802/jep.77 LA - en ID - JEP_2018__5__519_0 ER -
%0 Journal Article %A Robert J. Berman %A Chinh H. Lu %T From the Kähler-Ricci flow to moving free boundaries and shocks %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 519-563 %V 5 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.77/ %R 10.5802/jep.77 %G en %F JEP_2018__5__519_0
Robert J. Berman; Chinh H. Lu. From the Kähler-Ricci flow to moving free boundaries and shocks. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 519-563. doi : 10.5802/jep.77. https://jep.centre-mersenne.org/articles/10.5802/jep.77/
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