On the number of optimal surfaces

Let X be a closed oriented Riemann surface of genus > 1 of constant negative curvature -1. A surface containing a disk of maximal radius is an optimal surface. This paper gives exact formulae for the number of optimal surfaces of genus > 3 up to orientation-preserving isometry. We show that the automorphism group of such a surface is always cyclic of order 1,2,3 or 6. We also describe a combinatorial structure of nonorientable hyperbolic optimal surfaces.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

Gorenstein flat dimension of complexes

We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.Comment: to appear in J. Math. Kyoto Uni