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On the Bartnik mass of two-spheres with non-negative constant mean curvature Martens, Adam


We establish three new upper bounds on the Bartnik quasi-local mass of triples (S²,g,H) where S² is a topological two sphere, g is a Riemannian metric on S², and H ≥ 0 is a specified (constant) value for the initial mean curvature. We use the initial data set approach under the additional assumptions of time-symmetry (TS) and the dominant energy condition (DEC) in which one first constructs a collar with initial boundary sphere isometric to (S² g) and then extends to an asymptotically flat (AF) 3-manifold with non-negative scalar curvature (which is the DEC under the TS setting). The first bound extends the main result in [Mantoulidis-Schoen "On the Bartnik mass of apparent horizons"] to include the boundary case. Precisely, we show that any metric g with non-negative first eigenvalue of the operator -Δ_g + K_g appears as an apparent horizon (in the TS/DEC/AF setting) and that its Bartnik mass is precisely the corresponding Hawking mass. The second bound establishes that the Bartnik mass of the triple (S²,g,H) is bounded above by r/2 whenever g has non-negative Gaussian curvature K_g and H > 0. This result was known when K_g is assumed to be strictly positive (see [Miao-Xie "Bartnik mass via vacuum extensions"]) though the methods used there do not apply when min K_g =0. For the last bound, given any metric g with K_g ≥ 0 and any H > 0, we give an explicit constant C (depending only on g and H) such that the Bartnik mass of the triple (S²,g,H) is bounded above by a quantity involving C which approaches the Hawking mass as C → 0, which happens as either H → 0 or as g becomes round. Moreover, C remains bounded if H → ∞ or r² min K_g → 0. This result can be extended to arbitrary metrics (that do not necessarily satisfy K_g ≥ 0) although the resulting bound in this case is only finite if H is sufficiently large depending on g.

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