@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{Bangert1982,
  author = {Bangert, Victor},
  title = {Sets with positive reach},
  journal = {Archiv der Mathematik},
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  pages = {54--57},
  year = {1982},
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}

@article{Bartnik1986,
 author={Bartnik, Robert}, title={The mass of an asymptotically flat manifold},
 journal={Communications on Pure and Applied Mathematics},
 volume={39}, number={5}, pages={661--693}, year={1986}, doi={10.1002/cpa.3160390505}
}

@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}
}

@misc{BiZhu2026v2,
 author={Bi, Yuchen and Zhu, Jintian},
 title={{Riemannian Penrose} inequality in all dimensions},
 year={2026}, howpublished={\url{https://arxiv.org/abs/2605.00680v2}},
 note={arXiv:2605.00680v2, 15 September 2026}
}

@misc{BiZhuSmoothing2026,
 author={Bi, Yuchen and Zhu, Jintian},
 title={Curvature-free effects from volume growth and ends-counting and their applications},
 year={2026}, howpublished={\url{https://arxiv.org/abs/2605.12403v2}},
 note={arXiv:2605.12403v2, 15 July 2026}
}

@article{BombieriGiusti1972Harnack,
  author = {Bombieri, E. and Giusti, E.},
  title = {{Harnack}'s inequality for elliptic differential equations on minimal surfaces},
  journal = {Inventiones Mathematicae},
  volume = {15},
  pages = {24--46},
  year = {1972},
  doi = {10.1007/BF01418640}
}

@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}
}

@misc{BrendleWang2026,
 author={Brendle, Simon and Wang, Yipeng},
 title={A dimension descent scheme for the positive mass theorem in arbitrary dimension},
 year={2026}, howpublished={\url{https://arxiv.org/abs/2604.08473v2}},
 note={arXiv:2604.08473v2, 20 April 2026}
}

@article{DeSilvaJerison2011Gradient,
  author = {De Silva, Daniela and Jerison, David},
  title = {A gradient bound for free boundary graphs},
  journal = {Communications on Pure and Applied Mathematics},
  volume = {64},
  number = {4},
  pages = {538--555},
  year = {2011},
  doi = {10.1002/cpa.20354},
  eprint = {1009.4694},
  archivePrefix = {arXiv},
  primaryClass = {math.AP},
  url = {https://math.mit.edu/~jerison/gbf.finaldraft.pdf},
  note = {Theorem and page locators refer to the author manuscript, \url{https://math.mit.edu/~jerison/gbf.finaldraft.pdf}}
}

@book{GilbargTrudinger,
 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}
}

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

@misc{Lee2007NearEquality,
 author={Lee, Dan A.},
 title={On the near-equality case of the Positive Mass Theorem},
 year={2007},
 howpublished={\url{https://arxiv.org/abs/0705.0677v1}},
 note={arXiv:0705.0677v1, 4 May 2007. Theorem 1.3 is the fixed-radius-factor version used here}
}

@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{Miao2002,
 author={Miao, Pengzi}, title={Positive mass theorem on manifolds admitting corners along a hypersurface},
 journal={Advances in Theoretical and Mathematical Physics},
 volume={6}, number={6}, pages={1163--1182}, year={2002},
 note={\url{https://arxiv.org/abs/math-ph/0212025v3}}
}

@article{NaberValtorta2020,
  author = {Naber, Aaron and Valtorta, Daniele},
  title = {The singular structure and regularity of stationary varifolds},
  journal = {Journal of the European Mathematical Society},
  volume = {22},
  number = {10},
  pages = {3305--3382},
  year = {2020},
  doi = {10.4171/JEMS/987},
  eprint = {1505.03428v4},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1505.03428v4},
  note = {Theorem and page locators refer to arXiv:1505.03428v4, 28 July 2017}
}

@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}
}

@article{Rifford2004,
  author = {Rifford, Ludovic},
  title = {A {Morse--Sard} theorem for the distance function on {Riemannian} manifolds},
  journal = {Manuscripta Mathematica},
  volume = {113},
  pages = {251--265},
  year = {2004},
  doi = {10.1007/s00229-003-0436-7},
  url = {https://math.univ-cotedazur.fr/~rifford/Papiers_en_ligne/RiffordSard.pdf},
  note = {Author manuscript: \url{https://math.univ-cotedazur.fr/~rifford/Papiers_en_ligne/RiffordSard.pdf}}
}

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

@article{Wickramasekera2014Maximum,
  author = {Wickramasekera, Neshan},
  title = {A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces},
  journal = {Calculus of Variations and Partial Differential Equations},
  volume = {51},
  pages = {799--812},
  year = {2014},
  doi = {10.1007/s00526-013-0695-4},
  eprint = {1310.4031v1},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  url = {https://arxiv.org/abs/1310.4031v1},
  note = {Theorem and page locators refer to arXiv:1310.4031v1}
}

@book{Federer1969,
  author = {Federer, Herbert},
  title = {Geometric Measure Theory},
  series = {Die Grundlehren der mathematischen Wissenschaften},
  volume = {153},
  publisher = {Springer-Verlag},
  address = {New York},
  year = {1969}
}

@article{Federer1970,
  author = {Federer, Herbert},
  title = {The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension},
  journal = {Bulletin of the American Mathematical Society},
  volume = {76},
  number = {4},
  pages = {767--771},
  year = {1970},
  doi = {10.1090/S0002-9904-1970-12542-3},
  url = {https://www.math.stonybrook.edu/~bishop/classes/math638.F20/Federer_Regularity.pdf}
}


@article{Hawking1968,
  author = {Hawking, S. W.},
  title = {Gravitational radiation in an expanding universe},
  journal = {Journal of Mathematical Physics},
  volume = {9},
  number = {4},
  pages = {598--604},
  year = {1968},
  doi = {10.1063/1.1664615}
}

@article{Geroch1973,
  author = {Geroch, Robert},
  title = {Energy extraction},
  journal = {Annals of the New York Academy of Sciences},
  volume = {224},
  number = {1},
  pages = {108--117},
  year = {1973},
  doi = {10.1111/j.1749-6632.1973.tb41445.x}
}

@article{JangWald1977,
  author = {Jang, Pong Soo and Wald, Robert M.},
  title = {The positive energy conjecture and the cosmic censor hypothesis},
  journal = {Journal of Mathematical Physics},
  volume = {18},
  number = {1},
  pages = {41--44},
  year = {1977},
  doi = {10.1063/1.523134}
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@article{MalecOMurchadha1994,
  author = {Malec, Edward and {\'O} Murchadha, Niall},
  title = {Trapped surfaces and the {Penrose} inequality in spherically symmetric geometries},
  journal = {Physical Review D},
  volume = {49},
  number = {12},
  pages = {6931--6934},
  year = {1994},
  doi = {10.1103/PhysRevD.49.6931}
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@article{Herzlich1997,
  author = {Herzlich, Marc},
  title = {A {Penrose}-like inequality for the mass of {Riemannian} asymptotically flat manifolds},
  journal = {Communications in Mathematical Physics},
  volume = {188},
  pages = {121--133},
  year = {1997},
  doi = {10.1007/s002200050159}
}

@article{Simon1987Maximum,
  author = {Simon, Leon},
  title = {A strict maximum principle for area minimizing hypersurfaces},
  journal = {Journal of Differential Geometry},
  volume = {26},
  number = {2},
  pages = {327--335},
  year = {1987},
  doi = {10.4310/jdg/1214441373}
}

@article{Ilmanen1996Maximum,
  author = {Ilmanen, Tom},
  title = {A strong maximum principle for singular minimal hypersurfaces},
  journal = {Calculus of Variations and Partial Differential Equations},
  volume = {4},
  pages = {443--467},
  year = {1996},
  doi = {10.1007/BF01246151}
}

@misc{CompanionBoundaryGraphs,
  author = {{OpenAI}},
  title = {{Boundary graph deformations for the spacetime Penrose inequality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf}{OAI:Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026}},
  year = {2026}
}

@misc{CompanionSpacetimeNumerical,
  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{CompanionSpacetimeRigidity,
  author = {{OpenAI}},
  title = {{Equality and rigidity in the spacetime Penrose inequality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf}{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026}},
  year = {2026}
}


@misc{Lam2010Graphs,
  author = {Lam, Mau-Kwong George},
  title = {The Graph Cases of the {Riemannian Positive Mass and Penrose Inequalities} in All Dimensions},
  year = {2010},
  howpublished = {\url{https://arxiv.org/abs/1010.4256v1}},
  note = {arXiv:1010.4256v1, 20 October 2010}
}

@article{deLimaGirao2015,
  author = {de Lima, Levi Lopes and Gir{\~a}o, Frederico},
  title = {The {ADM} mass of asymptotically flat hypersurfaces},
  journal = {Transactions of the American Mathematical Society},
  volume = {367},
  number = {9},
  pages = {6247--6266},
  year = {2015},
  eprint = {1108.5474v3},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1108.5474v3},
  note = {Theorem locators refer to arXiv:1108.5474v3, 3 July 2012}
}