No abstract uploaded!

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.

We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.

No abstract uploaded!

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth (n + 1)-dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen and Simon to any n.