Carla Cederbaum (Germany): On extensions of CMC‐Bartnik data

Bartnik data are a Riemannian 2‐sphere of positive Gaussian curvature equipped with a non‐negative function H to be thought of as its mean curvature in an ambient Riemannian 3‐manifold. Mantoulidis and Schoen suggested a construction of asymptotically flat Riemannian 3‐manifolds of non‐negative scalar curvature which allows to isometrically embed given Bartnik data of vanishing mean curvature, i.e. H=0.
They use their construction to explore — and disprove — stability of the Riemannian Penrose inequality. In collaboration with Cabrera Pacheco, McCormick, and Miao, we adapt their construction to constant mean curvature (CMC) Bartnik data, i.e. H=const.>0.
Moreover, with Cabrera Pacheco and McCormick, we extend their construction to the asymptotically hyperbolic setting both for H=0 and for H=const.>0 Bartnik data.
I will present the construction as well as the motivation for such a construction which is related to Bartnik’s quasi‐local capacity/mass functional and its minimizing properties.