In general relativity, in the absence of special symmetries, there is no reasonable, nontrivial notion of mass-energy density accounting not only for all source fields but also for gravity itself. Nevertheless there are good definitions, such as the ADM mass, of the mass of an entire isolated system, modelled as an asymptotically flat space-time. Numerous proposals have been made to define quasilocal mass: that is to associate a mass to a region bigger than a point but smaller than an isolated system. One natural regime to investigate for any such definition is its behavior on small regions shrinking to a point. Roughly thirty years ago Robert Bartnik suggested defining quasilocal mass as the infimum of the ADM mass over a class of asymptotically flat extensions of the region, satisfying certain (mostly) natural conditions. Bartnik conjectured that this infimum is realized, uniquely, by a stationary vacuum extension. After explaining Bartnik’s conjecture and summarizing its current status, I will present recent work that in particular yields a new estimate for the Bartnik mass of small metric balls in an initial data set, in terms of the ball’s radius and the geometry at its center.

Zoom invitation is sent to the geometry and topology seminar list. If you are not on the list, please, contact Peter McGrath host to get the link.