Fu-Yau Hessian Equations

TL;DR

This paper develops a new method to solve the Fu-Yau Hessian equations across all dimensions and slopes, providing solutions for previously open cases and a broader family of Hessian equations.

Contribution

It introduces a novel ellipticity condition that ensures solutions exist for the Fu-Yau equation in all cases, including open and improved known cases.

Findings

Solutions obtained for all dimensions and slopes

New ellipticity condition preserves solution estimates

Includes a family of Hessian equations as special cases

Abstract

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, 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.