Mathematical Research Letters

Volume 29 (2022)

Number 5

Existence and uniqueness of stationary solutions in $5$-dimensional minimal supergravity

Pages: 1279 – 1346



Aghil Alaee (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.; and Department of Mathematics and Computer Science, Clark University, Worcester, Massachusetts, U.S.A.)

Marcus Khuri (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Hari Kunduri (Department of Mathematics and Statistics, Memorial University of Newfoundland, NF, Canada)


We study the problem of stationary bi-axially symmetric solutions of the $5$-dimensional minimal supergravity equations. Essentially all possible solutions with nondegenerate horizons are produced, having the allowed horizon cross-sectional topologies of the sphere $S^3$, ring $S^1 \times S^2$, and lens $L(p, q)$, as well as the three different types of asymptotics. The solutions are smooth apart from possible conical singularities at the fixed point sets of the axial symmetry. This analysis also includes the solutions known as solitons in which horizons are not present but are rather replaced by nontrivial topology called bubbles which are sustained by dipole fluxes. Uniqueness results are also presented which show that the solutions are completely determined by their angular momenta, electric and dipole charges, and rod structure which fixes the topology. Consequently we are able to identify the finite number of parameters that govern a solution. In addition, a generalization of these results is given where the spacetime is allowed to have orbifold singularities.

A. Alaee acknowledges the support of a NSERC Postdoctoral Fellowship. M. Khuri acknowledges the support of NSF Grant DMS-2104229, and Simons Foundation Fellowship 681443. H. Kunduri acknowledges the support of NSERC Discovery Grant RGPIN-2018-04887.

Received 29 April 2019

Received revised 12 May 2022

Accepted 25 June 2022

Published 21 April 2023