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Annales mathématiques du Québec

, Volume 42, Issue 2, pp 169–189 | Cite as

On the existence of constant scalar curvature Kähler metric: a new perspective

  • Xiuxiong ChenEmail author
Article

Abstract

In this note, we introduce a new continuity path of fourth order nonlinear equations connecting the cscK equation to a second order elliptic equation, which is the critical point equation of the J-flow introduced by Donaldson (Asian J Math 3(1):1–16, 1999) and the author (Commun Anal Geom 12(4):837–852, 2004). This is a generalization of the classical Aubin continuity path for Kähler–Einstein metrics. The aim of this new path is to attack the existence problem of cscK metric. The “openness” along this continuity path is proved and a set of open problems associated with this new path is proposed.

Keywords

New continuity path Twisted cscK metric Monge-Ampere equation 

Résumé

Dans cette note, nous introduisons un nouveau chemin de continuité d’équations non linéaires du quatrième ordre connectant l’équation cscK à une équation elliptique du deuxième ordre, qui est l’équation point critique du J-flot introduit par Donaldson (Asian J Math 3(1):1–16, 1999) et l’auteur (Commun Anal Geom 12(4):837–852, 2004). Il s’agit d’une généralisation du chemin de continuité classique de Aubin pour les métriques de Kähler-Einstein. Le but de ce nouveau chemin est de s’attaquer au problème d’existence des métriques cscK. L’«ouverture» au long de ce chemin de continuité est prouvée et on propose un nombre de problèmes ouverts liés à ce nouveau chemin.

Mathematics Subject Classification

53C21 53C55 58J05 58J60 

Notes

Acknowledgements

The author wishes to thank his students Yu Zeng and Chengjian Yao for critical help through the preparation of this paper. Very recently, Yoshinori Hashimoto informed us that he can solve Question 1.6 with an outline of proof, and his proof can also be extended to the case \(t\in (0,1)\).

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Copyright information

© Fondation Carl-Herz and Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Stony Brook UniversityStony BrookUSA
  2. 2.School of MathematicsUniversity of Science and Technology of ChinaAnhuiPeople’s Republic of China

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