@article{AgmonDouglisNirenberg1959,
  author = {Agmon, Shmuel and Douglis, Avron and Nirenberg, Louis},
  doi = {10.1002/cpa.3160120405},
  journal = {Communications on Pure and Applied Mathematics},
  month = {November},
  number = {4},
  pages = {623--727},
  title = {Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. {I}},
  url = {https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160120405},
  volume = {12},
  year = {1959}
}

@article{Allard1972,
  author = {Allard, William K.},
  title = {On the First Variation of a Varifold},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {95},
  number = {3},
  pages = {417--491},
  year = {1972},
  doi = {10.2307/1970868},
  url = {https://www.jstor.org/stable/1970868}
}

@book{AmbrosioFuscoPallara2000,
  address = {Oxford},
  author = {Ambrosio, Luigi and Fusco, Nicola and Pallara, Diego},
  doi = {10.1093/oso/9780198502456.001.0001},
  isbn = {9780198502456},
  publisher = {Oxford University Press},
  series = {Oxford Mathematical Monographs},
  title = {Functions of Bounded Variation and Free Discontinuity Problems},
  url = {https://academic.oup.com/book/53762},
  year = {2000}
}

@article{Bartnik1986,
  author = {Bartnik, Robert},
  doi = {10.1002/cpa.3160390505},
  journal = {Communications on Pure and Applied Mathematics},
  number = {5},
  pages = {661--693},
  title = {The mass of an asymptotically flat manifold},
  url = {https://math.jhu.edu/~js/Math646/bartnik.mass.pdf},
  volume = {39},
  year = {1986}
}

@article{BartnikChrusciel2005,
  author = {Bartnik, Robert A. and Chru{\'s}ciel, Piotr T.},
  title = {Boundary value problems for {Dirac}-type equations},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {579},
  pages = {13--73},
  year = {2005},
  doi = {10.1515/crll.2005.2005.579.13},
  eprint = {math/0307278},
  archivePrefix = {arXiv},
  note = {Theorem numbering follows the author manuscript (July 21, 2003)},
  url = {https://homepage.univie.ac.at/piotr.chrusciel/papers/dirac/Dirac23.pdf}
}

@article{Bartnik2005PhaseSpace,
  author = {Bartnik, Robert},
  title = {Phase Space for the {Einstein} Equations},
  journal = {Communications in Analysis and Geometry},
  volume = {13},
  number = {5},
  pages = {845--885},
  year = {2005},
  doi = {10.4310/CAG.2005.v13.n5.a1},
  url = {https://archive.intlpress.com/site/pub/files/_fulltext/journals/cag/2005/0013/0005/CAG-2005-0013-0005-a001.pdf}
}

@article{BenDov2004,
  author = {Ben-Dov, Ishai},
  title = {The {Penrose} inequality and apparent horizons},
  journal = {Physical Review D},
  volume = {70},
  pages = {124031},
  year = {2004},
  doi = {10.1103/PhysRevD.70.124031},
  url = {https://arxiv.org/abs/gr-qc/0408066v2}
}

@article{BeigChrusciel1996,
  author = {Beig, Robert and Chru{\'s}ciel, Piotr T.},
  title = {Killing Vectors in Asymptotically Flat Space-Times. {I}. {Asymptotically} Translational {Killing} Vectors and the Rigid Positive Energy Theorem},
  journal = {Journal of Mathematical Physics},
  volume = {37},
  number = {4},
  pages = {1939--1961},
  year = {1996},
  doi = {10.1063/1.531497},
  url = {https://arxiv.org/abs/gr-qc/9510015v3}
}

@article{BeigChrusciel1997KID,
  author = {Beig, Robert and Chru{\'s}ciel, Piotr T.},
  title = {Killing Initial Data},
  journal = {Classical and Quantum Gravity},
  volume = {14},
  number = {1A},
  pages = {A83--A92},
  year = {1997},
  doi = {10.1088/0264-9381/14/1A/007},
  url = {https://arxiv.org/abs/gr-qc/9604040v1}
}

@article{BenedettiLisca2018,
  author = {Benedetti, Riccardo and Lisca, Paolo},
  title = {Framing {$3$}-manifolds with bare hands},
  journal = {L'Enseignement Math{\'e}matique},
  series = {2},
  volume = {64},
  number = {3--4},
  pages = {395--413},
  year = {2018},
  doi = {10.4171/LEM/64-3/4-9},
  url = {https://ems.press/journals/lem/articles/16378}
}

@misc{BiZhu2026,
 author={Bi, Yuchen and Zhu, Jintian},
 title={{Riemannian Penrose} inequality in all dimensions},
 year={2026}, howpublished={\url{https://arxiv.org/abs/2605.00680v1}},
 note={arXiv:2605.00680v1, 1 May 2026}
}

@article{Bray2001,
  author = {Bray, Hubert L.},
  title = {Proof of the {Riemannian Penrose} inequality using the positive mass theorem},
  journal = {Journal of Differential Geometry},
  volume = {59},
  number = {2},
  pages = {177--267},
  year = {2001},
  doi = {10.4310/jdg/1090349428}
}

@article{BrayLee2009,
  author = {Bray, Hubert L. and Lee, Dan A.},
  title = {On the {Riemannian Penrose} inequality in dimensions less than eight},
  journal = {Duke Mathematical Journal},
  volume = {148},
  number = {1},
  pages = {81--106},
  year = {2009},
  doi = {10.1215/00127094-2009-020}
}

@article{BrayKhuri2010,
  author = {Bray, Hubert L. and Khuri, Marcus A.},
  title = {A {Jang} equation approach to the {Penrose} inequality},
  journal = {Discrete and Continuous Dynamical Systems},
  volume = {27},
  number = {2},
  pages = {741--766},
  year = {2010},
  doi = {10.3934/dcds.2010.27.741},
  url = {https://arxiv.org/abs/0910.4785v1}
}

@article{BuntingMasoodUlAlam1987,
  author = {Bunting, Gary L. and {Masood-ul-Alam}, A. K. M.},
  title = {Nonexistence of multiple black holes in asymptotically {Euclidean} static vacuum space-time},
  journal = {General Relativity and Gravitation},
  volume = {19},
  number = {2},
  pages = {147--154},
  year = {1987},
  doi = {10.1007/BF00770326}
}

@book{BuragoBuragoIvanov2001,
  author = {Burago, Dmitri and Burago, Yuri and Ivanov, Sergei},
  title = {A Course in Metric Geometry},
  series = {Graduate Studies in Mathematics},
  volume = {33},
  publisher = {American Mathematical Society},
  address = {Providence, RI},
  year = {2001},
  url = {https://bookstore.ams.org/GSM/33}
}

@article{CavallettiManini,
 author={Cavalletti, Fabio and Manini, Davide},
 title={Rigidities of isoperimetric inequality under nonnegative {Ricci} curvature},
 journal={Journal of the European Mathematical Society},
 volume={28}, number={8}, pages={3235--3294}, year={2026}, doi={10.4171/JEMS/1532}
}

@article{CecchiniZeidler2024,
  archiveprefix = {arXiv},
  author = {Cecchini, Simone and Zeidler, Rudolf},
  doi = {10.1090/tran/8942},
  eprint = {2108.11972v2},
  journal = {Transactions of the American Mathematical Society},
  pages = {5271--5288},
  title = {The positive mass theorem and distance estimates in the spin setting},
  note = {Theorem numbering refers to arXiv:2108.11972v2},
  url = {https://arxiv.org/abs/2108.11972v2},
  volume = {377},
  year = {2024}
}

@article{CheegerGromoll1971,
 author={Cheeger, Jeff and Gromoll, Detlef},
 title={The splitting theorem for manifolds of nonnegative {Ricci} curvature},
 journal={Journal of Differential Geometry}, volume={6}, pages={119--128}, year={1971},
 doi={10.4310/jdg/1214430220}
}

@article{Chern1944,
  author = {Chern, Shiing-Shen},
  title = {A simple intrinsic proof of the {Gauss--Bonnet} formula for closed {Riemannian} manifolds},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {45},
  number = {4},
  pages = {747--752},
  year = {1944},
  month = oct,
  doi = {10.2307/1969302},
  url = {https://www.jstor.org/stable/1969302}
}

@misc{CompanionNumerical,
  author = {{OpenAI}},
  title = {{The spacetime Penrose inequality and enclosing area}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf}{OAI:The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026}},
  year = {2026}
}

@misc{CompanionRiemannian,
  author = {{OpenAI}},
  title = {{Conformal flow and the Riemannian Penrose inequality with minimizing frontiers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf}{OAI:Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026}},
  year = {2026}
}

@article{DeTurckKazdan1981,
 author={DeTurck, Dennis M. and Kazdan, Jerry L.},
 title={Some regularity theorems in {Riemannian} geometry},
 journal={Annales Scientifiques de l'\'Ecole Normale Sup\'erieure},
 series={4}, volume={14}, number={3}, pages={249--260}, year={1981},
 note={\url{https://www.numdam.org/item/ASENS_1981_4_14_3_249_0/}}
}

@article{CorvinoHuang2020,
  author = {Corvino, Justin and Huang, Lan-Hsuan},
  title = {Localized deformation for initial data sets with the dominant energy condition},
  journal = {Calculus of Variations and Partial Differential Equations},
  volume = {59},
  number = {1},
  year = {2020},
  doi = {10.1007/s00526-019-1679-9},
  note = {Article 42. Section numbering follows the linked author manuscript},
  url = {https://lhhuang-math.media.uconn.edu/wp-content/uploads/sites/1744/2019/04/2017_01_18-gluing_DEC.pdf}
}

@article{GibbonsIdaShiromizu2002,
  author = {Gibbons, Gary W. and Ida, Daisuke and Shiromizu, Tetsuya},
  title = {Uniqueness and non-uniqueness of static vacuum black holes in higher dimensions},
  journal = {Progress of Theoretical Physics Supplement},
  volume = {148},
  pages = {284--290},
  year = {2002},
  doi = {10.1143/PTPS.148.284},
  url = {https://academic.oup.com/ptps/article/doi/10.1143/PTPS.148.284/1881057}
}

@book{GilbargTrudinger2001,
 author={Gilbarg, David and Trudinger, Neil S.},
 title={Elliptic Partial Differential Equations of Second Order},
 edition={Second}, publisher={Springer}, address={Berlin},
 year={2001}, series={Classics in Mathematics},
 note={Reprint of the 1998 edition}, doi={10.1007/978-3-642-61798-0}
}

@misc{HirschHuang2025,
  archiveprefix = {arXiv},
  author = {Hirsch, Sven and Huang, Lan-Hsuan},
  eprint = {2510.10306v1},
  note = {arXiv:2510.10306v1, 11 October 2025; Theorems 4 and 7},
  primaryclass = {gr-qc},
  title = {Monotonicity of Causal {Killing} Vectors and Geometry of {ADM} Mass Minimizers},
  url = {https://arxiv.org/abs/2510.10306v1},
  year = {2025}
}

@article{HuangLee2024Bartnik,
  author = {Huang, Lan-Hsuan and Lee, Dan A.},
  title = {{Bartnik} Mass Minimizing Initial Data Sets and Improvability of the Dominant Energy Scalar},
  journal = {Journal of Differential Geometry},
  volume = {126},
  number = {2},
  pages = {741--800},
  year = {2024},
  doi = {10.4310/jdg/1712344222},
  note = {Section numbering agrees with arXiv:2007.00593v2},
  url = {https://arxiv.org/abs/2007.00593v2}
}

@article{HuiskenIlmanen2001,
  author = {Huisken, Gerhard and Ilmanen, Tom},
  doi = {10.4310/jdg/1090349447},
  journal = {Journal of Differential Geometry},
  number = {3},
  pages = {353--437},
  title = {The inverse mean curvature flow and the {Riemannian Penrose} inequality},
  url = {https://doi.org/10.4310/jdg/1090349447},
  volume = {59},
  year = {2001}
}

@article{Hwang1998,
 author={Hwang, Seungsu},
 title={A Rigidity Theorem for {Ricci} Flat Metrics},
 journal={Geometriae Dedicata},
 volume={71}, pages={5--17}, year={1998},
 doi={10.1023/A:1005094911005}
}

@article{Jaracz2025StrictDEC,
  author = {Jaracz, Jaroslaw S.},
  title = {Conformal deformations of initial data sets to the strict dominant energy condition using a spacetime {Poisson} equation},
  journal = {International Journal of Geometric Methods in Modern Physics},
  pages = {2550198},
  year = {2025},
  doi = {10.1142/S0219887825501981},
  eprint = {2405.12249v1},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2405.12249v1},
  note = {Theorem and equation locators refer to preprint version 1, May 19, 2024}
}

@article{JulinLiimatainenSalo2017,
 author={Julin, Vesa and Liimatainen, Tony and Salo, Mikko},
 title={{$p$}-harmonic coordinates for {H\"older} metrics and applications},
 journal={Communications in Analysis and Geometry},
 volume={25}, number={2}, pages={395--430}, year={2017},
 note={\url{https://users.jyu.fi/~salomi/pub/pharmonic_holder_v27.pdf}}
}

@book{LawsonMichelsohn1989,
  author = {Lawson, Jr., H. Blaine and Michelsohn, Marie-Louise},
  title = {Spin Geometry},
  series = {Princeton Mathematical Series},
  volume = {38},
  publisher = {Princeton University Press},
  address = {Princeton, NJ},
  year = {1989},
  isbn = {9780691085425},
  url = {https://www.jstor.org/stable/j.ctt1bpmb28}
}

@article{LesourdUngerYau2021,
  author = {Lesourd, Martin and Unger, Ryan and Yau, Shing-Tung},
  title = {The positive mass theorem with arbitrary ends},
  journal = {Journal of Differential Geometry},
  volume = {128},
  number = {1},
  pages = {257--293},
  year = {2024},
  doi = {10.4310/jdg/1721075263},
  eprint = {2103.02744v1},
  archiveprefix = {arXiv},
  primaryclass = {math.DG},
  url = {https://arxiv.org/abs/2103.02744v1},
  note = {Theorem and definition numbering refers to arXiv:2103.02744v1}
}

@book{Maggi2012,
  author = {Maggi, Francesco},
  title = {Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {135},
  publisher = {Cambridge University Press},
  address = {Cambridge},
  year = {2012},
  doi = {10.1017/CBO9781139108133},
  url = {https://doi.org/10.1017/CBO9781139108133}
}

@article{MichaelSimon1973,
 author={Michael, J. H. and Simon, L. M.},
 title={Sobolev and mean-value inequalities on generalized submanifolds of {$\mathbb R^n$}},
 journal={Communications on Pure and Applied Mathematics},
 volume={26}, number={3}, pages={361--379}, year={1973}, doi={10.1002/cpa.3160260305}
}

@article{Penrose1973,
  author = {Penrose, Roger},
  title = {Naked singularities},
  journal = {Annals of the New York Academy of Sciences},
  volume = {224},
  number = {1},
  pages = {125--134},
  year = {1973},
  doi = {10.1111/j.1749-6632.1973.tb41447.x}
}

@book{Petersen2016,
  author = {Petersen, Peter},
  title = {{Riemannian} Geometry},
  edition = {Third},
  series = {Graduate Texts in Mathematics},
  volume = {171},
  publisher = {Springer},
  address = {Cham},
  year = {2016},
  doi = {10.1007/978-3-319-26654-1},
  url = {https://doi.org/10.1007/978-3-319-26654-1}
}

@article{Reshetnyak1968,
 author={Reshetnyak, Yu. G.},
 title={The weak convergence of completely additive vector-valued set functions},
 journal={Siberian Mathematical Journal}, volume={9}, pages={1039--1045},
 year={1968}, doi={10.1007/BF02196453}}

@misc{Simon2014,
  author = {Simon, Leon},
  howpublished = {Stanford University lecture notes for the Tsinghua lectures},
  note = {Author's notes: \url{https://web.stanford.edu/class/math285/ts-gmt.pdf}},
  title = {Introduction to Geometric Measure Theory},
  url = {https://web.stanford.edu/class/math285/ts-gmt.pdf},
  year = {2014}
}

@misc{Simon2018,
 author={Simon, Leon}, title={Introduction to Geometric Measure Theory},
 year={2018}, howpublished={\url{https://math.stanford.edu/~lms/ntu-gmt-text.pdf}},
 note={NTU lecture notes, version updated 7 March 2018}
}

@article{Spector2011,
  author = {Spector, Daniel},
  title = {Simple Proofs of Some Results of {Reshetnyak}},
  journal = {Proceedings of the American Mathematical Society},
  volume = {139},
  number = {5},
  pages = {1681--1690},
  year = {2011},
  doi = {10.1090/S0002-9939-2010-10593-2},
  note = {Theorem numbering refers to the author manuscript revised 18 May 2010},
  url = {https://cvgmt.sns.it/media/doc/paper/505/reshetnyakcontinuityrevised.pdf}
}

@article{Tangherlini1963,
  author = {Tangherlini, F. R.},
  doi = {10.1007/BF02784569},
  journal = {Il Nuovo Cimento},
  pages = {636--651},
  title = {{Schwarzschild} field in {$n$} dimensions and the dimensionality of space problem},
  url = {https://link.springer.com/article/10.1007/BF02784569},
  volume = {27},
  year = {1963}
}

@article{Witten1981,
  author = {Witten, Edward},
  doi = {10.1007/BF01208277},
  journal = {Communications in Mathematical Physics},
  number = {3},
  pages = {381--402},
  title = {A new proof of the positive energy theorem},
  url = {https://www.ias.edu/sites/default/files/sns/%5B32%5DCMP_80_1981.pdf},
  volume = {80},
  year = {1981}
}

@book{RudinFunctionalAnalysis1991,
  author = {Rudin, Walter},
  title = {Functional Analysis},
  edition = {Second},
  publisher = {McGraw--Hill},
  year = {1991}
}

@book{RudinRealComplex1987,
  author = {Rudin, Walter},
  title = {Real and Complex Analysis},
  edition = {Third},
  publisher = {McGraw--Hill},
  year = {1987}
}

@misc{MelroseEllipticRegularity,
  author = {Melrose, Richard B.},
  title = {Elliptic Regularity},
  howpublished = {MIT lecture notes, Chapter 4},
  year = {n.d.},
  note = {Section 3, Theorem 3.1, p.~127},
  url = {https://math.mit.edu/~rbm/18-155-F13/Chapter4.pdf}
}