Concentration of Small Hawking Type Surfaces

TL;DR

This paper introduces Hawking type functionals as a generalization of Willmore functionals to analyze small surfaces in spacetime, revealing their minimizers and concentration behavior without symmetry assumptions.

Contribution

It extends the theory of Willmore functionals to Hawking type functionals, characterizing small surface minimizers and their concentration points in general spacetimes.

Findings

Hawking type functionals are generalized Willmore functionals.

Small area constrained critical surfaces are smooth spheres.

Concentration points and expansions of Hawking energy are characterized.

Abstract

We investigate the Hawking energy of small surfaces in space times without symmetry assumptions by introducing the notion of Hawking type functionals. In particular, we find that Hawking type functionals are generalized Willmore functionals which allows us to find area constrained, minimizing, immersed, haunted bubble trees. These bubble trees are smooth spheres provided their area is small enough. Following a similar analysis of the Willmore functional conducted by T. Lamm and J. Metzger we characterize the concentration points of area constrained, critical surfaces for Hawking type functionals and the Hawking energy. Moreover, we determine their expansion on small surfaces.