Fu-Yau Hessian Equations

Abstract

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\hat{I}{\pm }^{\prime }$. Actually we obtain at the same time a solution of the open case $\hat{I}{\pm }^{\prime }>0$, an improved solution of the known case $\hat{I}{\pm }^{\prime }<0$, 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 $\hat{I}{“}_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

36 pages