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The subject of this paper is properly embedded H − H- surfaces in Riemannian three manifolds of the form M 2 × R M^2\times \mathbf {R} , where M 2 M^2 is a complete Riemannian surface. When M 2 = R 2 M^2={\mathbf R}^2 , we are in the classical domain of H − H- surfaces in R 3 {\mathbf R}^3 . In general, we will make some assumptions about M 2 M^2 in order to prove stronger results, or to show the effects of curvature bounds in M 2 M^2 on the behavior of H − H- surfaces in M 2 × R M^2\times \mathbf {R} .

There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one.

While advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in R3 [3, 5, 8, 9, 11, 12, 13, 23], progress has depended, in an essential manner, on the special structure of such surfaces with finite total curvature. A properly immersed minimal surface with finite total curvature is, conformally, a compact Riemann surface punctured in a finite number of points, and its Gauss map extends to the compact surface as a meromorphic function [22]. In particular, all the topological ends are conformally equivalent to a punctured disk, and there is a well-defined limit tangent plane at each end. Outside of a sufficiently large compact set, such an end is a multisheeted graph over the limit tangent plane, and if the end is embedded, it is asymptotic to either a plane or a half-catenoid. (See, e.g., Schoen [24]. For a survey of results on properly immersed surfaces of finite total curvature, see [9, ? 1].) Without the assumption of finite total curvature, these results are not true in general. As an indication of the relative lack of information, consider that it is unknown whether or not the helicoid is the only simply-connected minimal surface that is properly embedded and nonplanar (a question raised by Osserman). The helicoid has infinite total curvature and a single annular end.' If a surface has finite topology, then all of its ends are annular. Any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends. In this paper, we prove

The diameter of orbits of a compact isometry group G of a Riemannian manifold M cannot be uniformly small. If the sectional curvature of M is bounded above by b 2 {b^2} (b real or pure imaginary), then explicit bounds are found for D ( M ) D(M) , where D ( M ) D(M) is defined to be the largest number such that: If every orbit G has diameter less than D ( M ) D(M) , then G acts trivially on M. These bounds depend only on b and the injectivity radius of M. The proofs involve an investigation of various types of convex sets and an estimate for distance contraction of the exponential map on a manifold with bounded curvature.

M. H. A. Newman proved that if M is a connected topological manifold with metric d, there exists a number E > 0, depending only upon M and d, such that every compact Lie group acting effectively on M has at least one orbit of diameter at least e. In this paper the authors consider the case where M is a Riemannian manifold and d is the distance function on M arising from the Riemannian metric. They obtain estimates for e in terms of convexity and curvature invariants of M.