@incollection{gromov1996,
  author = {Gromov, M.},
  title = {Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures},
  booktitle = {Functional Analysis on the Eve of the 21st Century, Volume II},
  editor = {Gindikin, Simon and Lepowsky, James and Wilson, Robert L.},
  series = {Progress in Mathematics},
  volume = {132},
  pages = {1--213},
  publisher = {Birkh{\"a}user Boston},
  year = {1996},
  doi = {10.1007/978-1-4612-4098-3_1},
  url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/positivecurvature.pdf}
}

@article{dranishnikov2014,
 author={Dranishnikov, Alexander},
 title={On macroscopic dimension of universal coverings of closed manifolds},
 journal={Transactions of the Moscow Mathematical Society},
 volume={74}, year={2013}, pages={229--244},
 doi={10.1090/S0077-1554-2014-00221-1},
 eprint={1307.1029}, archivePrefix={arXiv}, primaryClass={math.GT},
 url={https://arxiv.org/pdf/1307.1029v1},
 note={Definitions 1.1--1.2 cited from arXiv:1307.1029v1}
}

@article{bolotovDranishnikov2010,
 author={Bolotov, Dmitry and Dranishnikov, Alexander},
 title={On {Gromov}'s scalar curvature conjecture},
 journal={Proceedings of the American Mathematical Society},
 volume={138}, number={4}, year={2010}, pages={1517--1524},
 eprint={0901.4503}, archivePrefix={arXiv}, primaryClass={math.GT},
 url={https://arxiv.org/pdf/0901.4503v1},
 note={Theorem 4.3 cited from arXiv:0901.4503v1}
}

@misc{simonGMT,
  author = {Simon, Leon},
  title = {Introduction to Geometric Measure Theory},
  year = {2014},
  howpublished = {Author-hosted lecture notes},
  note = {Tsinghua lecture notes, draft updated 3 December 2014},
  url = {https://web.stanford.edu/class/math285/ts-gmt.pdf}
}

@article{gromov2019Mean,
  author = {Gromov, Misha},
  title = {Mean Curvature in the Light of Scalar Curvature},
  journal = {Annales de l'Institut Fourier},
  volume = {69},
  number = {7},
  pages = {3169--3194},
  year = {2019},
  doi = {10.5802/aif.3347},
  url = {https://www.numdam.org/articles/10.5802/aif.3347/}
}

@misc{simonPDE,
  author = {Simon, Leon},
  title = {Lectures on {PDE}},
  year = {2015},
  howpublished = {Author-hosted lecture notes},
  note = {Updated 5 March 2015},
  url = {https://math.stanford.edu/~lms/lecs-on-pde.pdf}
}

@article{nash1956,
  author = {Nash, John},
  title = {The Imbedding Problem for {Riemannian} Manifolds},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {63},
  number = {1},
  pages = {20--63},
  year = {1956},
  doi = {10.2307/1969989},
  url = {https://mccuan.math.gatech.edu/courses/6455/nash55.pdf}
}

@article{chodoshLi,
  author = {Chodosh, Otis and Li, Chao},
  title = {Generalized soap bubbles and the topology of manifolds with positive scalar curvature},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {199},
  number = {2},
  pages = {707--740},
  year = {2024},
  doi = {10.4007/annals.2024.199.2.3},
  url = {https://arxiv.org/abs/2008.11888v4},
  note = {Numbering follows arXiv:2008.11888v4, 20 September 2023}
}

@article{schoenYau1979Structure,
  author = {Schoen, Richard and Yau, Shing-Tung},
  title = {On the structure of manifolds with positive scalar curvature},
  journal = {Manuscripta Mathematica},
  volume = {28},
  pages = {159--183},
  year = {1979},
  doi = {10.1007/BF01647970}
}

@article{gromovLawson1983,
 author={Gromov, Mikhael and Lawson, Jr., H. Blaine},
 title={Positive scalar curvature and the {Dirac} operator on complete {Riemannian} manifolds},
 journal={Publications Math{\'e}matiques de l'IH{\'E}S},
 volume={58}, year={1983}, pages={83--196},
 doi={10.1007/BF02953774}, url={https://numdam.org/articles/10.1007/BF02953774/}
}

@article{gromov1983Filling,
  author = {Gromov, M.},
  title = {Filling {Riemannian} manifolds},
  journal = {Journal of Differential Geometry},
  volume = {18},
  number = {1},
  pages = {1--147},
  year = {1983},
  url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/fillingRiemannianManifolds.pdf}
}

@article{gromov2018Metric,
  author = {Gromov, Misha},
  title = {Metric Inequalities with Scalar Curvature},
  journal = {Geometric and Functional Analysis},
  volume = {28},
  number = {3},
  pages = {645--726},
  year = {2018},
  doi = {10.1007/s00039-018-0453-z},
  url = {https://www.ihes.fr/~gromov/wp-content/uploads/2019/02/Inequalities-scalar-Apr-6-2018Oct25.pdf},
  note = {Section 2 cited from the author version dated 28 October 2018}
}

@misc{gromov2020Aspherical,
  author = {Gromov, Misha},
  title = {No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds},
  year = {2020},
  eprint = {2009.05332},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  url = {https://arxiv.org/abs/2009.05332v1},
  note = {Locators follow arXiv:2009.05332v1}
}

@article{fischerColbrieSchoen1980,
  author = {Fischer-Colbrie, Doris and Schoen, Richard},
  title = {The structure of complete stable minimal surfaces in
           3-manifolds of non-negative scalar curvature},
  journal = {Communications on Pure and Applied Mathematics},
  volume = {33},
  number = {2},
  pages = {199--211},
  year = {1980},
  doi = {10.1002/cpa.3160330206},
  url = {https://math.jhu.edu/~js/Math748/fischer-colbrie-schoen.pdf}
}

@article{miao2002Corners,
  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},
  doi = {10.4310/ATMP.2002.v6.n6.a4},
  eprint = {math-ph/0212025},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/math-ph/0212025v3},
  note = {Locators follow arXiv:math-ph/0212025v3}
}

@misc{lohkamp2022SplittingsII,
  author = {Lohkamp, Joachim},
  title = {Scalar Curvature Splittings {II}: Removal of Singularities},
  year = {2022},
  eprint = {2203.15531},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  url = {https://arxiv.org/abs/2203.15531v2},
  note = {Version 2, 24 April 2022}
}

@article{massari1975,
  author = {Massari, Umberto},
  title = {Frontiere orientate di curvatura media assegnata in {$L^p$}},
  journal = {Rendiconti del Seminario Matematico della Universit{\`a} di Padova},
  volume = {53},
  pages = {37--52},
  year = {1975},
  url = {https://www.numdam.org/item/RSMUP_1975__53__37_0/}
}

@article{massariPepe1975,
  author = {Massari, Umberto and Pepe, L.},
  title = {Successioni convergenti di ipersuperfici di curvatura media assegnata},
  journal = {Rendiconti del Seminario Matematico della Universit{\`a} di Padova},
  volume = {53},
  pages = {53--68},
  year = {1975},
  url = {https://www.numdam.org/item/RSMUP_1975__53__53_0/}
}

@misc{biZhu2026,
  author = {Bi, Yuchen and Zhu, Jintian},
  title = {Positive Scalar Curvature Obstructions via Singular Dimension Descent},
  year = {2026},
  eprint = {2606.20528},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  url = {https://arxiv.org/abs/2606.20528v3},
  note = {Version 3, 11 September 2026}
}

@article{bolotov2003,
 author={Bolotov, Dmitry V.},
 title={Macroscopic dimension of 3-manifolds},
 journal={Mathematical Physics, Analysis and Geometry},
 volume={6}, number={3}, year={2003}, pages={291--299},
 doi={10.1023/A:1024994930786}
}

@article{bolotov2009,
 author={Bolotov, Dmitry},
 title={About the macroscopic dimension of certain {PSC}-manifolds},
 journal={Algebraic \& Geometric Topology},
 volume={9}, number={1}, year={2009}, pages={21--27},
 doi={10.2140/agt.2009.9.21}
}

@article{bolotovDranishnikov2016,
 author={Bolotov, Dmitry and Dranishnikov, Alexander},
 title={On {Gromov}'s conjecture for totally non-spin manifolds},
 journal={Journal of Topology and Analysis},
 volume={8}, number={4}, year={2016}, pages={571--587},
 doi={10.1142/S1793525316500230},
 eprint={1402.4510}, archivePrefix={arXiv}, primaryClass={math.GT},
 url={https://arxiv.org/pdf/1402.4510v6},
 note={Theorem 3.4 cited from arXiv:1402.4510v6}
}

@misc{daherDranishnikov2019,
 author={Daher, Michelle and Dranishnikov, Alexander},
 title={On macroscopic dimension of non-spin 4-manifolds},
 year={2019}, eprint={1904.00846}, archivePrefix={arXiv}, primaryClass={math.GT},
 note={Version 1. Published in Journal of Topology and Analysis; doi:10.1142/S1793525321500084},
 url={https://arxiv.org/abs/1904.00846v1}
}

@article{liokumovichWang2023,
 author={Liokumovich, Yevgeny and Wang, Zhichao},
 title={On the waist and width inequality in complete 3-manifolds with positive scalar curvature},
 journal={Cambridge Journal of Mathematics},
 volume={14}, number={2}, year={2026}, pages={349--375},
 doi={10.4310/CJM.260514230515},
 eprint={2308.04044}, archivePrefix={arXiv}, primaryClass={math.DG},
 note={Theorem 1.1 cited from arXiv version 1 (2023)},
 url={https://arxiv.org/abs/2308.04044v1}
}

@incollection{liokumovichMaximo2023,
  author = {Liokumovich, Yevgeny and Maximo, Davi},
  title = {Waist inequality for 3-manifolds with positive scalar curvature},
  booktitle = {Perspectives in Scalar Curvature},
  volume = {2},
  publisher = {World Scientific},
  pages = {799--831},
  year = {2023},
  eprint = {2012.12478},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  url = {https://arxiv.org/abs/2012.12478v4},
  note = {Theorem 1.1 cited from arXiv:2012.12478v4}
}

@article{deGiorgi1954,
  author = {De Giorgi, Ennio},
  title = {Su una teoria generale della misura {$({r}-1)$}-dimensionale in uno spazio ad {$r$} dimensioni},
  journal = {Annali di Matematica Pura ed Applicata},
  series = {4},
  volume = {36},
  pages = {191--213},
  year = {1954},
  doi = {10.1007/BF02412838},
  url = {https://doi.org/10.1007/BF02412838}
}

@article{deGiorgi1955,
  author = {De Giorgi, Ennio},
  title = {Nuovi teoremi relativi alle misure {$({r}-1)$}-dimensionali in uno spazio ad {$r$} dimensioni},
  journal = {Ricerche di Matematica},
  volume = {4},
  pages = {95--113},
  year = {1955}
}

@article{federerFleming1960,
  author = {Federer, Herbert and Fleming, Wendell H.},
  title = {Normal and Integral Currents},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {72},
  number = {3},
  pages = {458--520},
  year = {1960},
  doi = {10.2307/1970227},
  url = {https://doi.org/10.2307/1970227}
}

@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://doi.org/10.2307/1970868}
}

@article{simons1968,
  author = {Simons, James},
  title = {Minimal varieties in {Riemannian} manifolds},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {88},
  number = {1},
  pages = {62--105},
  year = {1968},
  doi = {10.2307/1970556},
  url = {https://doi.org/10.2307/1970556}
}

@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://doi.org/10.1090/S0002-9904-1970-12542-3}
}

@article{michaelSimon1973,
  author = {Michael, J. H. and Simon, L. M.},
  title = {Sobolev and mean-value inequalities on generalized submanifolds of {$R^n$}},
  journal = {Communications on Pure and Applied Mathematics},
  volume = {26},
  number = {3},
  pages = {361--379},
  year = {1973},
  doi = {10.1002/cpa.3160260305},
  url = {https://doi.org/10.1002/cpa.3160260305}
}

@misc{antonelli2026Volume,
  author        = {Antonelli, Gioacchino},
  title         = {Universal Volume Growth Bounds from Positive Intermediate Curvature},
  year          = {2026},
  eprint        = {2608.14507},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  version       = {2},
  note          = {arXiv:2608.14507v2, August 19, 2026},
  doi           = {10.48550/arXiv.2608.14507},
  url           = {https://arxiv.org/abs/2608.14507v2},
  urldate       = {2026-09-23}
}

@misc{kumarSen2026Width,
  author        = {Kumar, Aditya and Sen, Balarka},
  title         = {{Urysohn} Width and Macroscopic Scalar Curvature},
  year          = {2026},
  eprint        = {2601.14669},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  version       = {2},
  note          = {arXiv:2601.14669v2, February 1, 2026},
  doi           = {10.48550/arXiv.2601.14669},
  url           = {https://arxiv.org/abs/2601.14669v2},
  urldate       = {2026-09-23}
}

@article{nabutovskyRotmanSabourau2021,
  author = {Nabutovsky, Alexander and Rotman, Regina and Sabourau, St{\'e}phane},
  title = {Sweepouts of closed {Riemannian} manifolds},
  journal = {Geometric and Functional Analysis},
  volume = {31},
  pages = {721--766},
  year = {2021},
  doi = {10.1007/s00039-021-00575-3},
  eprint = {2007.14954},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  url = {https://arxiv.org/abs/2007.14954v1},
  note = {Definitions 3.1 and 6.1 and equation (6.1) cited from arXiv:2007.14954v1}
}

@misc{gromov2017Dozen,
  author = {Gromov, Misha},
  title = {A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature},
  year = {2017},
  howpublished = {Author-hosted preprint},
  note = {19 September 2017},
  url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/dozen_problems_Sept_2017.pdf}
}

@misc{openai2026RationalInessentiality,
  author = {{OpenAI}},
  title = {{Positive scalar curvature forces rational inessentiality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026}},
  year = {2026}
}

@book{agmon1982,
  author = {Agmon, Shmuel},
  title = {Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of {N}-Body {Schr{\"o}dinger} Operators},
  series = {Mathematical Notes},
  volume = {29},
  publisher = {Princeton University Press},
  year = {1982},
  url = {https://api.pageplace.de/preview/DT0400.9781400853076_A23704097/preview-9781400853076_A23704097.pdf}
}