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McCormick, Stephenorcid.org/0000-0001-9536-9908

Open this publication in new window or tab >>Quasi-Local Penrose Inequalities with Electric Charge### Chen, Po-Ning

### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_some",{id:"formSmash:j_idt204:0:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_otherAuthors",{id:"formSmash:j_idt204:0:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_otherAuthors",multiple:true}); 2021 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2022, no 22, p. 17333-17362Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Oxford University Press, 2021
##### National Category

Geometry
##### Identifiers

urn:nbn:se:ltu:diva-95218 (URN)10.1093/imrn/rnab215 (DOI)000756410100001 ()
#####

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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-01-16Bibliographically approved

Department of Mathematics, University of California , Riverside 92507, USA.

Matematiska Institutionen, Uppsala Universitet , 751 06 Uppsala, Sweden.

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.

Open this publication in new window or tab >>Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space### Alaee, Aghil

### Cabrera Pacheco, Armando

### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2021 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 374, no 5, p. 3535-3555Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Mathematical Society (AMS), 2021
##### National Category

Geometry Mathematical Analysis
##### Identifiers

urn:nbn:se:ltu:diva-95214 (URN)10.1090/tran/8297 (DOI)000636650500015 ()2-s2.0-85104237271 (Scopus ID)
#####

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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Department of Mathematics and Computer Science, Clark University, Worcester, Massachusetts; Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts.

Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany.

Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden.

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.

Open this publication in new window or tab >>The Hilbert manifold of asymptotically flat metric extensions### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_some",{id:"formSmash:j_idt204:2:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_otherAuthors",{id:"formSmash:j_idt204:2:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_otherAuthors",multiple:true}); 2021 (English)In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 53, no 1, article id 14Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2021
##### Keywords

Bartnik mass, Constraint equations, Hamiltonian formulation
##### National Category

Geometry Mathematical Analysis
##### Identifiers

urn:nbn:se:ltu:diva-95217 (URN)10.1007/s10714-021-02785-4 (DOI)000612839800001 ()2-s2.0-85099904169 (Scopus ID)
#####

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##### Funder

Uppsala University
Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Matematiska institutionen, Uppsala universitet, Uppsala, Sweden.

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.

Open this publication in new window or tab >>Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_some",{id:"formSmash:j_idt204:3:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_otherAuthors",{id:"formSmash:j_idt204:3:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: Pacific Journal of Mathematics, ISSN 0030-8730, E-ISSN 1945-5844, Vol. 304, no 2, p. 629-653Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Mathematical Sciences Publishers (MSP), 2020
##### Keywords

quasilocal mass, Bartnik mass, gluing construction
##### National Category

Mathematical Analysis Geometry
##### Identifiers

urn:nbn:se:ltu:diva-95216 (URN)10.2140/pjm.2020.304.629 (DOI)000514173500012 ()2-s2.0-85079762014 (Scopus ID)
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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden.

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.

Open this publication in new window or tab >>On the charged Riemannian Penrose inequality with charged matter### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_some",{id:"formSmash:j_idt204:4:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_otherAuthors",{id:"formSmash:j_idt204:4:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_otherAuthors",multiple:true}); 2019 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 37, no 1, article id 015007Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Institute of Physics (IOP), 2019
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:ltu:diva-95215 (URN)10.1088/1361-6382/ab50a8 (DOI)000505740800001 ()2-s2.0-85078405747 (Scopus ID)
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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden.

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.

Open this publication in new window or tab >>On the evolution of the spacetime Bartnik mass### McCormick, Stephen

### Miao, Pengzi

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_some",{id:"formSmash:j_idt204:5:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_otherAuthors",{id:"formSmash:j_idt204:5:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_otherAuthors",multiple:true}); 2019 (English)In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 15, no 3, p. 897-920Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

International Press of Boston, Inc., 2019
##### Keywords

quasi-local mass, initial data
##### National Category

Geometry
##### Identifiers

urn:nbn:se:ltu:diva-95212 (URN)10.4310/pamq.2019.v15.n3.a6 (DOI)000505800800006 ()2-s2.0-85077688341 (Scopus ID)
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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Matematiska institutionen, Uppsala universitet, Uppsala, Sweden.

Department of Mathematics, University of Miami, Coral Gables, Florida, US.A..

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.

Open this publication in new window or tab >>Asymptotically hyperbolic extensions and an analogue of the Bartnik mass### Cabrera Pacheco, Armando J.

### Cederbaum, Carla

### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_some",{id:"formSmash:j_idt204:6:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_otherAuthors",{id:"formSmash:j_idt204:6:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_otherAuthors",multiple:true}); 2018 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 132, p. 338-357Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2018
##### Keywords

Quasi-local mass, Asymptotically hyperbolic manifolds, Bounded scalar curvature
##### National Category

Mathematical Analysis Geometry
##### Identifiers

urn:nbn:se:ltu:diva-95211 (URN)10.1016/j.geomphys.2018.06.010 (DOI)000442066100022 ()2-s2.0-85049837601 (Scopus ID)
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##### Funder

Knut and Alice Wallenberg Foundation
Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA.

Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany.

Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden.

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.

Open this publication in new window or tab >>On a Minkowski-like inequality for asymptotically flat static manifolds### McCormick, Stephen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_some",{id:"formSmash:j_idt204:7:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_otherAuthors",{id:"formSmash:j_idt204:7:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_otherAuthors",multiple:true}); 2018 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 146, no 9, p. 4039-4046Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Mathematical Society (AMS), 2018
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:ltu:diva-95210 (URN)10.1090/proc/14047 (DOI)000438582900037 ()2-s2.0-85049929419 (Scopus ID)
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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Institutionen för Matematik, Kungliga Tekniska högskolan, 100 44 Stockholm, Sweden.

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.

Open this publication in new window or tab >>Asymptotically flat extensions of CMC Bartnik data### Cabrera Pacheco, Armando J

### Cederbaum, Carla

### McCormick, Stephen

### Miao, Pengzi

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_some",{id:"formSmash:j_idt204:8:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_otherAuthors",{id:"formSmash:j_idt204:8:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 34, no 10, article id 105001Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Institute of Physics (IOP), 2017
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:ltu:diva-95207 (URN)10.1088/1361-6382/aa6921 (DOI)000413778300001 ()2-s2.0-85018988148 (Scopus ID)
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Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Department of Mathematics, University of Connecticut, Storrs, CT 06269, United States of America.

Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany.

Institutionen för Matematik, Kungliga Tekniska högskolan, 100 44 Stockholm, Sweden; School of Science and Technology, University of New England, Armidale, NSW 2351, Australia.

Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States of America.

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.

Open this publication in new window or tab >>On a Penrose-Like Inequality in Dimensions Less than Eight### McCormick, Stephen

### Miao, Pengzi

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_some",{id:"formSmash:j_idt204:9:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_otherAuthors",{id:"formSmash:j_idt204:9:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2019, no 7, p. 2069-2084Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Oxford University Press, 2017
##### National Category

Geometry
##### Identifiers

urn:nbn:se:ltu:diva-95213 (URN)10.1093/imrn/rnx181 (DOI)000472803500007 ()2-s2.0-85072225401 (Scopus ID)
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##### Funder

Knut and Alice Wallenberg Foundation
Available from: 2023-01-10 Created: 2023-01-10 Last updated: 2023-05-08Bibliographically approved

Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden.

Department of Mathematics, University of Miami, Coral Gables, FL, USA.

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 of