Abstract
We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns out possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.
We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space.
For reader’s convenience, in one appendix of this paper we provide a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. In the other appendix, we provide a comparison of different variations of Gromov’s pancake method.
The author Arseniy Akopyan was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117).
The author Roman Karasev was supported by the Federal professorship program grant 1.456.2016/1.4 and the Russian Foundation for Basic Research grants 18-01-00036 and 19-01-00169.
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References
A. Akopyan, R. Karasev, A tight estimate for the waist of the ball. Bull. Lond. Math. Soc. 49(4), 690–693 (2017). arXiv:1608.06279. http://arxiv.org/abs/1608.06279
A. Akopyan, R. Karasev, Waist of balls in hyperbolic and spherical spaces. Int. Math. Res. Not. (2018). arXiv:1702.07513. https://arxiv.org/abs/1702.07513
A. Akopyan, A. Hubard, R. Karasev, Lower and upper bounds for the waists of different spaces. Topol. Methods Nonlinear Anal. 53(2), 457–490 (2019)
M. Berger, Quelques problèmes de géométrie Riemannienne ou deux variations sur les espaces symétriques compacts de rang un. Enseignement Math. 16, 73–96 (1970)
Y. Brenier, Polar factorization and monotone rearrangement of vector-values functions. Commun. Pure Appl. Math. XLIV(10), 375–417 (1991)
L.A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities. Commun. Math. Phys. 214(3), 547–563 (2000)
G. de Rham, On the Area of Complex Manifolds. Notes for the Seminar on Several Complex Variables (Institute for Advanced Study, Princeton, 1957–1958)
H. Federer, Some theorems on integral currents. Trans. Am. Math. Soc. 117, 43–67 (1965)
M. Gromov, Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)
R. Harvey, H.B. Lawson Jr., Calibrated geometries. Acta Math. 148(1), 47–157 (1982)
W.-Y. Hsiang, H.B. Lawson Jr., Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1–38 (1971)
R. Karasev, A. Volovikov, Waist of the sphere for maps to manifolds. Topol. Appl. 160(13), 1592–1602 (2013). arXiv:1102.0647. http://arxiv.org/abs/1102.0647
B. Klartag, Convex geometry and waist inequalities. Geom. Funct. Anal. 27(1), 130–164 (2017). arXiv:1608.04121. http://arxiv.org/abs/1608.04121
B. Klartag, Eldan’s stochastic localization and tubular neighborhoods of complex-analytic sets (2017). arXiv:1702.02315. http://arxiv.org/abs/1702.02315
A.V. Kolesnikov, Mass transportation and contractions (2011). arXiv:1103.1479. https://arxiv.org/abs/1103.1479
Y. Memarian, On Gromov’s waist of the sphere theorem. J. Topol. Anal. 03(01), 7–36 (2011). arXiv:0911.3972. http://arxiv.org/abs/0911.3972
N. Palić, Grassmannians, measure partitions, and waists of spheres, PhD thesis, Freie Universität Berlin (2018). https://refubium.fu-berlin.de/handle/fub188/23043
W. Wirtinger, Eine determinantenidentität und ihre anwendung auf analytische gebilde und Hermitesche massbestimmung. Monatsh. Math. Phys. 44, 343–365 (1936)
Acknowledgements
The authors thank Alexey Balitskiy, Michael Blank, Alexander Esterov, Sergei Ivanov, Bo’az Klartag, Jan Maas, and the unknown referee for useful discussions, suggestions, and questions.
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Akopyan, A., Karasev, R. (2020). Gromov’s Waist of Non-radial Gaussian Measures and Radial Non-Gaussian Measures. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2256. Springer, Cham. https://doi.org/10.1007/978-3-030-36020-7_1
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DOI: https://doi.org/10.1007/978-3-030-36020-7_1
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