Gravitating vortices with positive curvature

Abstract: In this talk I will overview recent joint work with Vamsi Pingali and Chengjian Yao in arXiv:1911.09616 about gravitating vortices. These equations couple a K\"ahler metric on a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations.

In our work we give a complete solution to the existence problem for gravitating vortices on the Riemann sphere with positive topological constant c > 0. Our main result establishes the existence of solutions provided that a GIT stability condition for an effective divisor on CP^1 is satisfied. To this end, we use a continuity path starting from Yang's solution with c = 0. A salient feature of our argument is a new bound S \geq c for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

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CRM - Séminaire du CIRGET / Géométrie et Topologie

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