LetMbe a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1, 2, . . . , n, is defined as the k-th ele- mentary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a con- formal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.

We study isometric Lie group actions on symmetric spaces admitting a section, i.e., a submanifold that meets all orbits orthogonally at every intersection point. We classify such actions on the compact symmetric spaces with simple isometry group and rank greater than one. In particular, we show that these actions are hyperpolar, i.e., the sections are flat.

We derive large time upper bounds for heat kernels on vector bundles of differential forms on a class of non-compact Riemannian manifolds under certain curvature conditions.

Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a Q-variation of Hodge structures of weight one on Y with Higgs bundle E1,0 ⊕ E0,1, coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the slopes satisfy μ(E1,0) − μ(E0,1) ≤ μ(­1 Y ). We prove a similar inequality in the higher dimensional case. If the latter is an equality, and if the discriminant of E1,0 or the one of E0,1 is zero, one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank> 1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds U.

Continuing the program of [DS] and [U1], we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes invariant is easy to define, our invariants agree with those analogues. We also prove a vanishing result for some of the invariants that count nodal curves.