Extremal black hole initial data deformations

We study deformations of axially symmetric initial data for Einstein-Maxwell equations satisfying the time-rotation ($t$-$\varphi$) symmetry and containing one asymptotically cylindrical end and one asymptotically flat end. We find that the $t$-$\varphi$ symmetry implies the existence of a family of deformed data having the same horizon structure. This result allows us to measure how close solutions to Lichnerowicz equation are when arising from nearby free data.