Classification of polytope metrics and complete scalar-flat K\"ahler 4-Manifolds with two symmetries

TL;DR

This paper classifies complete scalar-flat K"ahler 4-manifolds with two symmetries by analyzing their associated unbounded 2D polytopes, revealing the structure of possible metrics based on polytope shape and edges.

Contribution

It provides a complete classification of metrics on unbounded 2D polytopes arising from K"ahler quotients of 4-manifolds with two symmetries, extending known families and identifying new possibilities.

Findings

Flat metrics on plane or half-plane polytopes.

Generalized Taub-NUT metrics for polytopes with one corner.

Existence of an (n+2)-dimensional family of metrics for polytopes with n≥3 edges.

Abstract

We study unbounded 2-dimensional metric polytopes such as those arising as K\"ahler quotients of complete K\"ahler 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 K\"ahler 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 $n\ge3$ edges admit an $(n+2)$-dimensional family of possible metrics.