Quasi-local mass and 3D Riemannian manifolds with curvature lower bounds

This PhD project is in pure mathematics, with motivations coming from mathematical physics (mathematical general relativity) and aims to prove variational and rigidity properties of two types of quasi-local mass: Hawking mass and Bartnik mass. Though the thesis will restrict to the case of a three-dimensional, Riemannian ambient manifold, these concepts are fundamental in (mathematical) general relativity because they model the quasi-local mass contained inside a solid domain in a Riemannian space-like slice of a space-time.

The Hawking mass has been a key technical tool in modern mathematical general relativity, for instance the proof by Huisken and Ilmanen of the celebrated Penrose inequality (Riemannian case) is based in the monotonicity of the Hawking mass under inverse mean curvature flow.

The first goal of this project is to show that if for every point in a 3D Riemannian manifold there exists a small neighbourhood inside which the Hawking mass of any surface is zero (in fact non-positive), then the whole space must be flat. This can be seen as a local-to-global property of the Hawking mass. Then, using ideas from Huisken and Ilmanen, it relates the Hawking and Bartnik masses of certain surfaces. This enables a new proof of an existing rigidity theorem and a new result giving an asymptotic lower bound, both for the Bartnik mass.

The proofs will lie in the intersection of differential geometry, partial differential equations, and mathematical general relativity. Non-linear analysis and partial differential equations will play a key technical role to attack the Hawking mass theorem, because optimal (constrained critical points of the Willmore functional) surfaces will be found and exploited. A new Taylor expansion of the Hawking mass of these optimal surfaces will be computed, and a scalar curvature lower bound (corresponding to the Dominant Energy condition in general relativity) will then be necessary to conclude the rigidity.

The theory of sets of finite perimeter will be used to prove key properties of perturbed geodesic spheres (the optimal surfaces found earlier), which, combined with the expansion of the Hawking mass, will prove the Bartnik mass results.