2015/09/15 by Brian Weber, Weber, Brian
Mathematics · #53C55 (Primary) #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C55
paper · pdf · doi:10.48550/arxiv.1509.04585
54 pages, 22 figures
arxiv created 2015/09/15 · arxiv updated 2015/09/16
We study unbounded 2-dimensional metric polytopes such as those arising as Kähler quotients of complete Kähler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible metrics on on the corresponding Kähler 4-manifolds. If the polytope is the plane or half-plane then only flat metrics are possible, and if the polytope has one corner then the 2-parameter family of generalized Taub-NUTs (discovered by Donaldson) are indeed the only possible metrics. Polytopes with n≥3 edges admit an (n+2)-dimensional family of possible metrics.