Fu–Yau Hessian equations

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.