Bartolucci, Daniele; Gui, Changfeng; Jevnikar, Aleks; Moradifam, Amir A singular sphere covering inequality: uniqueness and symmetry of solutions to singular Liouville-type equations. (English) Zbl 1426.35127 Math. Ann. 374, No. 3-4, 1883-1922 (2019). The authors consider the Louiville type equation \(\Delta u + h(x)e^u=f(x)\) on a smooth bounded domain \(\Omega\) in \(\mathbb{R}^2\) where \(h(x)>0\). They derive a singular version of the sphere covering inequality (see [C. Gui and A. Moradifam, Invent. Math. 214, No. 3, 1169–1204 (2018; Zbl 1410.53037)]) and use this result to derive uniqueness results for solutions to the singular mean field equations. They also give a new proof of the uniqueness of spherical convex polygons (see [F. Luo and G. Tian, Proc. Am. Math. Soc. 116, No. 4, 1119–1129 (1992; Zbl 0806.53012)]).The first section contains an introduction to the matter at hand. The second section treats the singular sphere covering inequality. The third section deals with the singular sphere covering equality with the same total mass. The fourth section studies uniqueness of solutions of the singular Liouville equation on the 2 dimensional sphere. The fifth section discusses symmetry for the spherical Onsager vortex equation. Reviewer: Peter B. Gilkey (Eugene) Cited in 8 Documents MSC: 35J61 Semilinear elliptic equations 35R01 PDEs on manifolds 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35B06 Symmetries, invariants, etc. in context of PDEs Keywords:spherical Onsager vortex equation; uniqueness of spherical convex polytopes; singular mean field equation; electro weak and Chern-Simons self-dual vortices Citations:Zbl 1410.53037; Zbl 0806.53012 PDFBibTeX XMLCite \textit{D. Bartolucci} et al., Math. Ann. 374, No. 3--4, 1883--1922 (2019; Zbl 1426.35127) Full Text: DOI arXiv References: [1] Bandle, C.: On a differential Inequality and its applications to geometry. Math. 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