In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a counterexample to a weaker form of the speculation DEF = DIFF of R. Friedman and J. Morgan, i.e., in the case where (by M. Freedman’s theorem) the topological type is completely determined by the numerical invariants of the surface. We hope that the methods of proof may turn out to be quite useful to show diffeomorphism and indeed symplectic equivalence for many important classes of algebraic surfaces and symplectic 4-manifolds.

A hyperbolic 3-manifold M which fibers over the circle admits a flow called the suspension flow. We show that such a flow can be isotoped to be uniformly quasigeodesic in the hyperbolic metric on M; i.e., the flow lines lifted to hyperbolic space are K-bilipschitz embeddings of {$\Bbb R$} $K$ > fixed.

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in $\mathbb{R}^3$. We prove a general local existence result for a large class of Monge-Ampère equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.

We calculate the dimension of the locus of Jacobian elliptic surfaces over $\mathbf{P}^1$with a given Picard number, in the corresponding moduli space.