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On the Bartnik mass of twospheres with nonnegative constant mean curvature Martens, Adam
Description
We establish three new upper bounds on the Bartnik quasilocal 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 timesymmetry (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) 3manifold with nonnegative scalar curvature (which is the DEC under the TS setting). The first bound extends the main result in [MantoulidisSchoen "On the Bartnik mass of apparent horizons"] to include the boundary case. Precisely, we show that any metric g with nonnegative 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 nonnegative Gaussian curvature K_g and H > 0. This result was known when K_g is assumed to be strictly positive (see [MiaoXie "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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Title 
On the Bartnik mass of twospheres with nonnegative constant mean curvature

Creator  
Publisher 
University of British Columbia

Date Issued 
2021

Description 
We establish three new upper bounds on the Bartnik quasilocal 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 timesymmetry (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) 3manifold with nonnegative scalar curvature (which is the DEC under the TS setting).
The first bound extends the main result in [MantoulidisSchoen "On the Bartnik mass of apparent horizons"] to include the boundary case. Precisely, we show that any metric g with nonnegative 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 nonnegative Gaussian curvature K_g and H > 0. This result was known when K_g is assumed to be strictly positive (see [MiaoXie "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.

Genre  
Type  
Language 
eng

Date Available 
20210401

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0396543

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
202105

Campus  
Scholarly Level 
Graduate

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Aggregated Source Repository 
DSpace

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Rights
AttributionNonCommercialNoDerivatives 4.0 International