Explicit minimizing sequences for Bartnik's mass functional

01/10/2017 to 30/09/2019

This project deals with questions related to the denition and computation of the "mass" of a gravitating object such as a star, a galaxy, or a black hole. While defining
and computing the mass of an object is straightforward in Newton's classical theory of
gravity, it has proven elusive in Einstein's general theory of relativity. In 1968, Arnowitt--
Deser--Misner succeeded in defining a notion of mass for an "isolated" object (i.e. an
object not gravitationally interacting with any other objects), but we still do not know
how to properly capture the mass of an object that is not isolated.
Many suggestions have been made, e.g. by Hawking. They are called quasi-local (QL)
masses, in contrast to the total mass assigned to an isolated object. While certainly
mathematically very useful (e.g. for proving "Penrose-style" inequalities relating physical
and geometric properties of black holes), most QL masses are in conflict with physical
intuition: Hawking's QL mass, e.g., will be negative in almost all cases!

In 1989, Bartnik dened a QL mass that satises all physical criteria (as listed by
Christodoulou--Yau). However, it is not known whether Bartnik's mass is indeed "computable",
meaning that one can explicitly or approximately compute it for many examples
of non-isolated objects.

Associated with this are two famous conjectures by Bartnik that will be investigated in this project.
We will do so by explicitly constructing sequences of "asymptotically flat" Riemannian
3-manifolds with certain delicate features, generalizing work by Mantoulidis--Schoen and
Cabrera Pacheco--Miao. They will be used to test Bartnik's conjecture and to investigate
stability of the seminal (generalized) Penrose inequality in geometrically special situations.