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1. Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space Alaee, Aghil PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_0_j_idt606",{id:"formSmash:items:resultList:0:j_idt606",widgetVar:"widget_formSmash_items_resultList_0_j_idt606",onLabel:"Alaee, Aghil ",offLabel:"Alaee, Aghil ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_0_j_idt609",{id:"formSmash:items:resultList:0:j_idt609",widgetVar:"widget_formSmash_items_resultList_0_j_idt609",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Department of Mathematics and Computer Science, Clark University, Worcester, Massachusetts; Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:0:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Cabrera Pacheco, ArmandoDepartment of Mathematics, Universität Tübingen, 72076 Tübingen, Germany.McCormick, StephenMatematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:0:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space2021In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 374, no 5, p. 3535-3555Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_0_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:0:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_0_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown–York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown–York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense?

Here we consider a class of compact -manifolds with boundary that can be realized as graphs in , and establish the following. If the Brown–York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer–Fleming flat distance.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:0:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 2. Asymptotically hyperbolic extensions and an analogue of the Bartnik mass Cabrera Pacheco, Armando J. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_1_j_idt606",{id:"formSmash:items:resultList:1:j_idt606",widgetVar:"widget_formSmash_items_resultList_1_j_idt606",onLabel:"Cabrera Pacheco, Armando J. ",offLabel:"Cabrera Pacheco, Armando J. ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_1_j_idt609",{id:"formSmash:items:resultList:1:j_idt609",widgetVar:"widget_formSmash_items_resultList_1_j_idt609",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:1:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Cederbaum, CarlaDepartment of Mathematics, Universität Tübingen, 72076 Tübingen, Germany.McCormick, StephenInstitutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:1:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Asymptotically hyperbolic extensions and an analogue of the Bartnik mass2018In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 132, p. 338-357Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_1_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:1:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_1_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant.

Following the ideas of Mantoulidis and Schoen (2016), of Miao and Xie (2018), and of joint work of Miao and the authors (Cabrera Pacheco et al., 2017), we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:1:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 3. Asymptotically flat extensions of CMC Bartnik data Cabrera Pacheco, Armando J PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_2_j_idt606",{id:"formSmash:items:resultList:2:j_idt606",widgetVar:"widget_formSmash_items_resultList_2_j_idt606",onLabel:"Cabrera Pacheco, Armando J ",offLabel:"Cabrera Pacheco, Armando J ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_2_j_idt609",{id:"formSmash:items:resultList:2:j_idt609",widgetVar:"widget_formSmash_items_resultList_2_j_idt609",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Department of Mathematics, University of Connecticut, Storrs, CT 06269, United States of America.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:2:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Cederbaum, CarlaDepartment of Mathematics, Universität Tübingen, 72076 Tübingen, Germany.McCormick, StephenInstitutionen för Matematik, Kungliga Tekniska högskolan, 100 44 Stockholm, Sweden; School of Science and Technology, University of New England, Armidale, NSW 2351, Australia.Miao, PengziDepartment of Mathematics, University of Miami, Coral Gables, FL 33146, United States of America.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:2:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Asymptotically flat extensions of CMC Bartnik data2017In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 34, no 10, article id 105001Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_2_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:2:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_2_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Let

*g*be a metric on the 2-sphere with positive Gaussian curvature and*H*be a positive constant. Under suitable conditions on (*g*,*H*), we construct smooth, asymptotically flat 3-manifolds*M*with non-negative scalar curvature, with outer-minimizing boundary isometric to and having mean curvature*H*, such that near infinity*M*is isometric to a spatial Schwarzschild manifold whose mass*m*can be made arbitrarily close to a constant multiple of the Hawking mass of . Moreover, this constant multiplicative factor depends only on (*g*,*H*) and tends to 1 as*H*tends to 0. The result provides a new upper bound of the Bartnik mass associated with such boundary data.PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:2:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 4. Quasi-Local Penrose Inequalities with Electric Charge Chen, Po-Ning PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_3_j_idt606",{id:"formSmash:items:resultList:3:j_idt606",widgetVar:"widget_formSmash_items_resultList_3_j_idt606",onLabel:"Chen, Po-Ning ",offLabel:"Chen, Po-Ning ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_3_j_idt609",{id:"formSmash:items:resultList:3:j_idt609",widgetVar:"widget_formSmash_items_resultList_3_j_idt609",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Department of Mathematics, University of California , Riverside 92507, USA.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:3:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); McCormick, StephenMatematiska Institutionen, Uppsala Universitet , 751 06 Uppsala, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:3:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Quasi-Local Penrose Inequalities with Electric Charge2021In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2022, no 22, p. 17333-17362Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_3_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:3:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_3_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with nonnegative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian Penrose inequality has also been established for the Einstein–Maxwell equations where the lower bound on mass depends also on electric charge, a charged Riemannian Penrose inequality. Here, we establish some quasi-local charged Penrose inequalities for surfaces isometric to closed surfaces in a suitable Reissner–Nordström manifold, which serves as a reference manifold for the quasi-local mass. In the case where the reference manifold has zero mass and nonzero electric charge, the lower bound on quasi-local mass is exactly the lower bound on the ADM mass given by the charged Penrose inequality.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:3:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 5. A note on mass-minimising extensions McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_4_j_idt606",{id:"formSmash:items:resultList:4:j_idt606",widgetVar:"widget_formSmash_items_resultList_4_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); School of Science and Technology, University of New England, Armidale, 2351, Australia.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:4:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:4:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); A note on mass-minimising extensions2015In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 47, no 12, article id 145Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_4_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:4:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_4_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); A conjecture related to the Bartnik quasilocal mass, is that the infimum of the ADM energy, over an appropriate space of extensions to a compact 3-manifold with boundary, is realised by a static metric. It was shown by Corvino (Commun Math Phys 214(1):137–189, 2000) that if the infimum is indeed achieved, then it is achieved by a static metric; however, the more difficult question of whether or not the infimum is achieved, is still an open problem. Bartnik (Commun Anal Geom 13(5):845–885, 2005) then proved that critical points of the ADM mass, over the space of solutions to the Einstein constraints on an asymptotically flat manifold without boundary, correspond to stationary solutions. In that article, he stated that it should be possible to use a similar construction to provide a more natural proof of Corvino’s result. In the first part of this note, we discuss the required modifications to Bartnik’s argument to adapt it to include a boundary. Assuming that certain results concerning a Hilbert manifold structure for the space of solutions carry over to the case considered here, we then demonstrate how Bartnik’s proof can be modified to consider the simpler case of scalar-flat extensions and obtain Corvino’s result. In the second part of this note, we consider a space of extensions in a fixed conformal class. Sufficient conditions are given to ensure that the infimum is realised within this class.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:4:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 6. An overview of Bartnik’s quasi-local mass McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_5_j_idt606",{id:"formSmash:items:resultList:5:j_idt606",widgetVar:"widget_formSmash_items_resultList_5_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:5:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:5:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); An overview of Bartnik’s quasi-local mass2024In: Beijing Journal of Pure and Applied Mathematics, ISSN 2994-8444, Vol. 1, no 2, p. 455-487Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_5_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:5:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_5_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); This article provides a concise introduction to Bartnik’s quasi-localmass, and surveys a selection of results pertaining to the understanding of it. The aim is to serve as both an entry point to thetopic, and a quick reference of results for those already familiarwith it.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:5:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Download full text (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_5_j_idt869_0_j_idt872",{id:"formSmash:items:resultList:5:j_idt869:0:j_idt872",widgetVar:"widget_formSmash_items_resultList_5_j_idt869_0_j_idt872",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:5:j_idt869:0:fullText"});}); 7. Fill-ins with scalar curvature lower bounds and applications to positive mass theorems McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_6_j_idt606",{id:"formSmash:items:resultList:6:j_idt606",widgetVar:"widget_formSmash_items_resultList_6_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:6:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:6:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fill-ins with scalar curvature lower bounds and applications to positive mass theorems2024In: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 65, no 4, article id 26Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_6_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:6:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_6_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Given a constant C and a smooth closed (𝑛−1)-dimensional Riemannian manifold (Σ,𝑔) equipped with a positive function H, a natural question to ask is whether this manifold can be realised as the boundary of a smooth n-dimensional Riemannian manifold with scalar curvature bounded below by C and boundary mean curvature H. That is, does there exist a fill-in of (Σ,𝑔,𝐻) with scalar curvature bounded below by C? We use variations of an argument due to Miao and the author (Int Math Res Not 7:2019, 2019) to explicitly construct fill-ins with different scalar curvature lower bounds, where we permit the fill-in to contain another boundary component provided it is a minimal surface. Our main focus is to illustrate the applications of such fill-ins to geometric inequalities in the context of general relativity. By filling in a manifold beyond a boundary, one is able to obtain lower bounds on the mass in terms of the boundary geometry through positive mass theorems and Penrose inequalities. We consider fill-ins with both positive and negative scalar curvature lower bounds, which from the perspective of general relativity corresponds to the sign of the cosmological constant, as well as a fill-in suitable for the inclusion of electric charge.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:6:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Download full text (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_6_j_idt869_0_j_idt872",{id:"formSmash:items:resultList:6:j_idt869:0:j_idt872",widgetVar:"widget_formSmash_items_resultList_6_j_idt869_0_j_idt872",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:6:j_idt869:0:fullText"});}); 8. First law of black hole mechanics as a condition for stationarity McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_7_j_idt606",{id:"formSmash:items:resultList:7:j_idt606",widgetVar:"widget_formSmash_items_resultList_7_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Department of Mathematics, School of Science and Technology, University of New England, Armidale, New South Wales 2351, Australia.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:7:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:7:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); First law of black hole mechanics as a condition for stationarity2014In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 90, no 10, article id 104034Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_7_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:7:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_7_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); In earlier work, we provided a Hilbert manifold structure for the phase space for the Einstein-Yang-Mills equations, and used this to prove a condition for initial data to be stationary [, Adv. Theor. Math. Phys. 18, 799 (2014)]. Here we use the same phase space to consider the evolution of initial data exterior to some closed 2-surface boundary, and establish a condition for stationarity in this case. It is shown that the differential relationship given in the first law of black hole mechanics is exactly the condition required for the initial data to be stationary; this was first argued nonrigorously by Sudarsky and Wald [Phys. Rev. D 46, 1453 (1992)]. Furthermore, we give evidence to suggest that if this differential relationship holds then the boundary surface is the bifurcation surface of a bifurcate Killing horizon.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:7:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 9. Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_8_j_idt606",{id:"formSmash:items:resultList:8:j_idt606",widgetVar:"widget_formSmash_items_resultList_8_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:8:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:8:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions2020In: Pacific Journal of Mathematics, ISSN 0030-8730, E-ISSN 1945-5844, Vol. 304, no 2, p. 629-653Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_8_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:8:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_8_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold (Ω,γ) with boundary. In one case, the extension is taken to be a manifold without boundary in which (Ω,γ) embeds isometrically, and in the other case the extension is taken to be a manifold with boundary where the boundary data is determined by ∂Ω.

We give a type of convexity condition under which we can say both of these types of extensions indeed yield the same value for the Bartnik mass. Under the same hypotheses we prove that the Bartnik mass varies continuously with respect to the boundary data. This also provides a method to use estimates for the Bartnik mass of constant mean curvature (CMC) Bartnik data, to obtain estimates for the Bartnik mass of non-CMC Bartnik data. The key idea for these results is a method for gluing Bartnik extensions of given Bartnik data to other nearby Bartnik data.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:8:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 10. On a Minkowski-like inequality for asymptotically flat static manifolds McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_9_j_idt606",{id:"formSmash:items:resultList:9:j_idt606",widgetVar:"widget_formSmash_items_resultList_9_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Institutionen för Matematik, Kungliga Tekniska högskolan, 100 44 Stockholm, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:9:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:9:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); On a Minkowski-like inequality for asymptotically flat static manifolds2018In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 146, no 9, p. 4039-4046Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_9_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:9:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_9_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); The Minkowski inequality is a classical inequality in differential geometry giving a bound from below on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than for example, such an inequality holds for surfaces in spatial Schwarzschild and AdS-Schwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:9:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 11. On the charged Riemannian Penrose inequality with charged matter McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_10_j_idt606",{id:"formSmash:items:resultList:10:j_idt606",widgetVar:"widget_formSmash_items_resultList_10_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:10:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:10:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); On the charged Riemannian Penrose inequality with charged matter2019In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 37, no 1, article id 015007Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_10_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:10:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_10_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Throughout the literature on the charged Riemannian Penrose inequality, it is generally assumed that there is no charged matter present; that is, the electric field is divergence-free. The aim of this article is to clarify when the charged Riemannian Penrose inequality holds in the presence of charged matter, and when it does not.

First we revisit Jang's proof of the charged Riemannian Penrose inequality to show that under suitable conditions on the charged matter, this argument still carries though. In particular, a charged Riemannian Penrose inequality is obtained from this argument when charged matter is present provided that the charge density does not change sign. Moreover, we show that such hypotheses on the sign of the charge are in fact required by constructing counterexamples to the charged Riemannian Penrose inequality when these conditions are violated. We conclude by noting that one of these counterexamples contradicts a version of the charged Penrose inequality existing in the literature, and explain how this existing result can be repaired.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:10:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 12. The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_11_j_idt606",{id:"formSmash:items:resultList:11:j_idt606",widgetVar:"widget_formSmash_items_resultList_11_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); School of Science and Technology, University of New England, Armidale, 2351, Australia; Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44, Stockholm, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:11:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:11:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary2017In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 27, no 3, p. 2269-2277Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_11_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:11:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_11_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension n≥3. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric

*g*, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with*g*on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: given a function*f*on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to*g*whose boundary mean curvature is given by*f*. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:11:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 13. The Hilbert manifold of asymptotically flat metric extensions McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_12_j_idt606",{id:"formSmash:items:resultList:12:j_idt606",widgetVar:"widget_formSmash_items_resultList_12_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Matematiska institutionen, Uppsala universitet, Uppsala, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:12:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:12:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); The Hilbert manifold of asymptotically flat metric extensions2021In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 53, no 1, article id 14Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_12_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:12:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_12_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); In [Commun Anal Geom 13(5):845–885, 2005], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity (𝑔,𝜋)∈𝐻

^{2}×𝐻^{1}. In particular, it was established that the space of solutions to the constraints form a Hilbert submanifold of this phase space. The motivation for this work was to study the quasi-local mass functional now bearing his name. However, the phase space considered there was over a manifold without boundary. Here we demonstrate that analogous results hold in the case where the manifold has an interior compact boundary, and the metric is prescribed on the boundary. Then, still following Bartnik’s work, we demonstrate the critical points of the mass functional over this space of extensions correspond to stationary solutions with vanishing Killing vector on the boundary. Furthermore, if this solution is smooth then it is in fact a static black hole solution. In particular, in the vacuum case, critical points only occur at exterior Schwarzschild solutions; that is, critical points of the mass over this space do not exist generically. Finally, we briefly discuss a version of the result when the boundary data is related to Bartnik’s geometric boundary data. In particular, by imposing different boundary conditions on the Killing vector, we show that stationary solutions in this case correspond to critical points of an energy resembling the difference between the ADM mass and the Brown–York mass of the boundary.PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:12:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 14. The phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamics McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_13_j_idt606",{id:"formSmash:items:resultList:13:j_idt606",widgetVar:"widget_formSmash_items_resultList_13_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); School of Science and Technology, University of New England, Armidale, Australia.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:13:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:13:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); The phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamics2014In: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753, Vol. 18, no 4, p. 799-825Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_13_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:13:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_13_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); We use the techniques of Bartnik [5] to show that the space of solutions to the Einstein-Yang-Mills constraint equations on an asymptotically flat manifold with one end and zero boundary components, has a Hilbert manifold structure; the Einstein-Maxwell system can be considered as a special case. This is equivalent to the property of linearisation stability, which was studied in depth throughout the 70s [1, 2, 9, 11, 13, 18, 19].

This framework allows us to prove a conjecture of Sudarsky and Wald [22], namely that the validity of the first law of black hole thermodynamics is a suitable condition for stationarity. Since we work with a single end and no boundary conditions, this is equivalent to critical points of the ADM mass subject to variations fixing the Yang-Mills charge corresponding exactly to stationary solutions. The natural extension to this work is to prove the second conjecture from [22], which is the case where an interior boundary is present; this will be addressed in future work.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:13:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 15. On a Penrose-Like Inequality in Dimensions Less than Eight McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_14_j_idt606",{id:"formSmash:items:resultList:14:j_idt606",widgetVar:"widget_formSmash_items_resultList_14_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_14_j_idt609",{id:"formSmash:items:resultList:14:j_idt609",widgetVar:"widget_formSmash_items_resultList_14_j_idt609",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:14:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Miao, PengziDepartment of Mathematics, University of Miami, Coral Gables, FL, USA.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:14:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); On a Penrose-Like Inequality in Dimensions Less than Eight2017In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2019, no 7, p. 2069-2084Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_14_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:14:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_14_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); On an asymptotically flat manifold M

^{n}with nonnegative scalar curvature, with outer minimizing boundary we prove a Penrose-like inequality in dimensions n<8 under suitable assumptions on the mean curvature and the scalar curvature ofPrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:14:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 16. On the evolution of the spacetime Bartnik mass McCormick, Stephen PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_15_j_idt606",{id:"formSmash:items:resultList:15:j_idt606",widgetVar:"widget_formSmash_items_resultList_15_j_idt606",onLabel:"McCormick, Stephen ",offLabel:"McCormick, Stephen ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_15_j_idt609",{id:"formSmash:items:resultList:15:j_idt609",widgetVar:"widget_formSmash_items_resultList_15_j_idt609",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Matematiska institutionen, Uppsala universitet, Uppsala, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:15:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Miao, PengziDepartment of Mathematics, University of Miami, Coral Gables, Florida, US.A..PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:15:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); On the evolution of the spacetime Bartnik mass2019In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 15, no 3, p. 897-920Article in journal (Refereed)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_15_j_idt644_0_j_idt645",{id:"formSmash:items:resultList:15:j_idt644:0:j_idt645",widgetVar:"widget_formSmash_items_resultList_15_j_idt644_0_j_idt645",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); It is conjectured that the full (spacetime) Bartnik mass of a surface Σ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by Σ. Assuming this holds true for a 1-parameter family of surfaces Σ

_{t}evolving in an initial data set with the dominant energy condition, we compute an expression for the derivative of the Bartnik mass along these surfaces. An immediate consequence of this formula is that the Bartnik mass of Σ_{t }is monotone non-decreasing whenever Σ_{t }flows outward.PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:15:j_idt644:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500});

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