Gravitating vortices with positive curvature

Authors: Mario Garcia-Fernandez (ICMAT-UAM), Vamsi Pritham Pingali and Chengjian Yao

Source: Advances in Mathematics vol. 388

Date of publication: 17 September 2021



A combination of ideas from cosmology and particle physics suggests the existence of symmetry-breaking phase transitions in the very early universe, as it expanded and cooled down. Like the transitions in condensed matter, they may have led to the formation of cosmic strings, exceedingly narrow filaments of primordial material left over from the early moments of the universe. The basic mathematical structure of a cosmic string is given by a special instance of Einstein’s field equations, known as the self-dual Einstein-Maxwell-Higgs equations, coupling gravity with an electromagnetic field and a Higgs field. Solutions to these equations saturate a suitable energy bound (often called Bogomol’nyi bound) and typically have a self-similar behaviour, as solitons in the theory of parabolic partial differential equations. Geometrically, the strings are located at the zeros of the Higgs field (a complex scalar), and cut transversally a surface which is embedded in a space-like hypersurface in space-time.

A surprising relation between solutions of the self-dual Einstein-Maxwell-Higgs equations and algebraic geometry was found by L. Álvarez-Cónsul (ICMAT-CSIC), M. Garcia-Fernandez (ICMAT-UAM) and O. García-Prada (ICMAT-CSIC). The basic idea is that the Higgs field determines an effective divisor on a Riemann surface (that is, a distribution of delta sources with masses, concentrated at points), which must satisfy a balancing condition with origins in the Geometric Invariant Theory (GIT), as introduced by D. Mumford in the 1960s to construct quotients by group actions in algebraic geometry. This has led in recent years, jointly with V. Pingali (IISc, Bangalore), to the discovery of new obstructions to the existence of cosmic string solutions and also to a natural generalization of the theory for a more general system of PDE known as the gravitating vortex equations.

Building on these works, in this publication in Advances in Mathematics, M. Garcia-Fernandez (ICMAT-UAM), V. Pingali (IISc, Bangalore) and C. Yao (Shanghai Tech) give a complete solution to the existence problem for gravitating vortices with non-negative topological constant c⩾0 in a compact surface. Their first main result builds on previous results by Yang and establishes the existence of solutions to the self-dual Einstein-Maxwell-Higgs equations, corresponding to c = 0, for all admissible volumes (Kähler classes). Their second main result completely solves the existence problem for c>0. Both results are proved by the continuity method and require that a GIT stability condition, for an effective divisor on the Riemann sphere, is satisfied. For the former, the continuity path starts from a given solution with c=0 and deforms the total volume of the solution. For the latter result the authors start from the established solution in any fixed admissible volume and deform the gravitational constant α towards 0. A salient feature of their argument is a new bound Sc for the Gauss curvature of gravitating vortices, which is applied to construct a limiting solution along the path via Cheeger-Gromov theory.