From the Kähler-Ricci flow to moving free boundaries and shocks

Berman, Robert J.; Lu, Chinh H.

doi:10.5802/jep.77

From the Kähler-Ricci flow to moving free boundaries and shocks
[Du flot de Kähler-Ricci au flot des bords libres mobiles et aux chocs]

Berman, Robert J.¹ ; Lu, Chinh H.²

¹ Mathematical Sciences, Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden
² Scuola Normale Superiore di Pisa Piazza dei Cavalieri 3, 56126 Pisa, Italy

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 519-563.

Résumé
Abstract

Nous montrons que le flot de Kähler-Ricci tordu sur une variété kählérienne compacte $X$ 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 $X$ .

We show that the twisted Kähler-Ricci flow on a compact Kähler manifold $X$ 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 $X$ .

Reçu le : 2016-06-24
Accepté le : 2018-06-14
Publié le : 2018-07-02

Zbl

DOI : 10.5802/jep.77

Classification : 53C44, 53C55, 76D27, 35F21
Keywords: Kähler-Ricci flow, Kähler manifold, free boundary, Hele-Shaw flow, Hamilton-Jacobi equation
Mot clés : Flot de Kähler-Ricci, variété kählérienne, flot de Hele-Shaw, équation de Hamilton-Jacobi

Affiliations des auteurs :

Berman, Robert J. ¹ ; Lu, Chinh H. ²

¹ Mathematical Sciences, Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden
² Scuola Normale Superiore di Pisa Piazza dei Cavalieri 3, 56126 Pisa, Italy

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Berman, Robert J.; Lu, Chinh H. 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. http://archive.numdam.org/articles/10.5802/jep.77/

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