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.

In a previous paper we have constructed an invariant of fourdimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four- manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.

In a symmetric space of noncompact type X = G=K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homogeneous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G=P for maximal parabolic subgroups P. The side lengths of polygons in Euclidean buildings are studied in the related paper [KLM2]. Applications of the geomet- ric results in both papers to algebraic group theory are given in[KLM3].

We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern–Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

For Riemannian manifolds with a measure (M, g, e.fdvolg) we prove mean curvature and volume comparison results when the亣- Bakry-Emery Ricci tensor is bounded from below and f or |f| is bounded, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.