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Constant scalar curvature metrics on toric surfaces. (English) Zbl 1177.53067
The results from this paper are a continuation of author’s series of papers in which there are studied the scalar curvature of Kähler metrics on toric varieties and relations with the analysis of convex functions on polytopes in Euclidean space. The author studies the equation $$\sum_{i,j}{{\partial^2 u^{ij}}\over {\partial x_i\partial x_j}}= -A$$ in the case where the dimension $$n$$ is $$2$$ and the function $$A$$ is constant. The main result is Theorem 1: Suppose $$P\subset {\mathbb{R}}^2$$ is a polygon and $$\sigma$$ a measure on $$\partial P$$ with the property that the mass and moments of $$(\partial P, \sigma)$$ and $$(P,Ad\mu)$$ are equal to some constant $$A$$. Then either there is a solution of the considered equation satisfying Guillemin’s boundary condition, or there is a convex function $$f$$, not affine linear, with $$L_{A,\sigma}(f)\leq 0$$. In the last section the author studies an explicit family of zero scalar curvature metrics, generalizing the Taub-NUT metric, and explains that these can be expected to arise as blow-up limits of solution.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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