The dissolving limit and large volume limit of Einstein-Bogomol'nyi metrics

TL;DR

This paper investigates the limits of Einstein-Bogomol'nyi metrics on the Riemann sphere, revealing how vortices dissolve or concentrate as the volume approaches bounds, and recovers known solutions in the large volume limit.

Contribution

It introduces a detailed analysis of the dissolving and large volume limits of Einstein-Bogomol'nyi metrics, connecting these regimes to vortex behavior and known solutions.

Findings

Vortices dissolve near the lower volume bound, similar to Bradlow limit.

Magnetic fields concentrate around Higgs field zeros in the large volume limit.

Recovered Einstein-Bogomol'nyi metrics on  with exponential asymptotic behavior.

Abstract

We study the limits of Einstein-Bogomol'nyi metrics on $\mathbf{P}^1$, which is the solution to a dimensional reduction of Einstein-Maxwell-Higgs system in dimension four, in two regimes. In one regime called the "dissolving limit" where the volume of the metrics is approaching the admissible lower bound, it exhibits a pattern that all the vortices are dissolving similar to the Bradlow limit in the study of vortices on Riemann surfaces. In another regime called the "large volume limit" where the volume of of the metrics is approaching infinity, the magnetic field is concentrating around the zeros of the Higgs field. In the meantime, the volume-normalized underlying metric is approaching the Euclidean cone metric determined by the Higgs field in the case of stable Higgs field. Moreover, by studying the large volume limit of Yang's solution for a strictly polystable Higgs field, for each natural number $N'$ we recover the Einstein-Bogomol'nyi metrics on $\mathbf{C}$ which is asymptotically cylindrical at exponential rate and with total string number $N'$ firstly discovered by Linet and Yang.