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May 2021 Fu–Yau Hessian equations
Duong H. Phong, Sebastien Picard, Xiangwen Zhang
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J. Differential Geom. 118(1): 147-187 (May 2021). DOI: 10.4310/jdg/1620272943

Abstract

We solve the Fu–Yau equation for arbitrary dimension and arbitrary slope $\alpha^\prime$. Actually we obtain at the same time a solution of the open case $\alpha^\prime \gt 0$, an improved solution of the known case $\alpha^\prime \lt 0$, and solutions for a family of Hessian equations which includes the Fu–Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

Funding Statement

Work supported in part by the National Science Foundation Grants DMS-12-66033 and the Simons Collaboration Grant for Mathematicians: 523313.

Citation

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Duong H. Phong. Sebastien Picard. Xiangwen Zhang. "Fu–Yau Hessian equations." J. Differential Geom. 118 (1) 147 - 187, May 2021. https://doi.org/10.4310/jdg/1620272943

Information

Received: 1 February 2018; Published: May 2021
First available in Project Euclid: 7 May 2021

Digital Object Identifier: 10.4310/jdg/1620272943

Keywords: $C^3$ estimate , a priori estimates with scale , ellipticity condition , Fu–Yau Hessian equation

Rights: Copyright © 2021 Lehigh University

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Vol.118 • No. 1 • May 2021
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