The first and second widths of the real projective space

Batista, Márcio; de Lima, Anderson

doi:10.1090/proc/16388

The first and second widths of the real projective space
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by Márcio Batista and Anderson de Lima

Proc. Amer. Math. Soc. 151 (2023), 3985-3997

DOI: https://doi.org/10.1090/proc/16388

Published electronically: April 13, 2023

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Abstract:

In this paper, we deal with the first and second widths of the real projective space $\mathbb {RP}^{n}$, for $n$ ranging from $4$ to $7$, and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces.

References

Nicolau Sarquis Aiex, The width of ellipsoids, Comm. Anal. Geom. 27 (2019), no. 2, 251–285. MR 4003008, DOI 10.4310/CAG.2019.v27.n2.a1
Márcio Batista and Anderson Lima, Min-max widths of the real projective 3-space, Trans. Amer. Math. Soc. 375 (2022), no. 7, 5239–5258. MR 4439504, DOI 10.1090/tran/8682
Thomas Beck, Spencer Becker-Kahn, and Boris Hanin, Nodal sets of smooth functions with finite vanishing order and $p$-sweepouts, Calc. Var. Partial Differential Equations 57 (2018), no. 5, Paper No. 140, 13. MR 3848453, DOI 10.1007/s00526-018-1406-y
Sidney Donato, The first $p$-widths of the unit disk, J. Geom. Anal. 32 (2022), no. 6, Paper No. 177, 38. MR 4404859, DOI 10.1007/s12220-022-00913-3
Manfredo do Carmo, Manuel Ritoré, and Antonio Ros, Compact minimal hypersurfaces with index one in the real projective space, Comment. Math. Helv. 75 (2000), no. 2, 247–254. MR 1774704, DOI 10.1007/PL00000373
M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178–215. MR 1978494, DOI 10.1007/s000390300004
Larry Guth, Minimax problems related to cup powers and Steenrod squares, Geom. Funct. Anal. 18 (2009), no. 6, 1917–1987. MR 2491695, DOI 10.1007/s00039-009-0710-2
Daniel Ketover, Fernando C. Marques, and André Neves, The catenoid estimate and its geometric applications, J. Differential Geom. 115 (2020), no. 1, 1–26. MR 4081930, DOI 10.4310/jdg/1586224840
Yevgeny Liokumovich, Fernando C. Marques, and André Neves, Weyl law for the volume spectrum, Ann. of Math. (2) 187 (2018), no. 3, 933–961. MR 3779961, DOI 10.4007/annals.2018.187.3.7
Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782. MR 3152944, DOI 10.4007/annals.2014.179.2.6
Fernando C. Marques and André Neves, Morse index and multiplicity of min-max minimal hypersurfaces, Camb. J. Math. 4 (2016), no. 4, 463–511. MR 3572636, DOI 10.4310/CJM.2016.v4.n4.a2
Fernando C. Marques and André Neves, Topology of the space of cycles and existence of minimal varieties, Surveys in differential geometry 2016. Advances in geometry and mathematical physics, Surv. Differ. Geom., vol. 21, Int. Press, Somerville, MA, 2016, pp. 165–177. MR 3525097
Fernando C. Marques and André Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math. 209 (2017), no. 2, 577–616. MR 3674223, DOI 10.1007/s00222-017-0716-6
Laurent Mazet and Harold Rosenberg, Minimal hypersurfaces of least area, J. Differential Geom. 106 (2017), no. 2, 283–315. MR 3662993, DOI 10.4310/jdg/1497405627
C. Nurser, Low min-max widths of the round three-sphere, Ph. D. Thesis, Department of Mathematics of Imperial College, October (2016). DOI 10.25560/42503
Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR 626027, DOI 10.1515/9781400856459
A. Ramirez-Luna, Compact minimal hypersurfaces with index one and the width of real projective space, arXiv:1902.08221v2, 2019.
A. Ramirez-Luna, Orientability of min-max hypersurfaces in manifolds of positive Ricci curvature, arXiv:1907.12519, 2019.
Luis A. Santaló, Integral geometry and geometric probability, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. With a foreword by Mark Kac. MR 2162874, DOI 10.1017/CBO9780511617331
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
Xin Zhou, Min-max hypersurface in manifold of positive Ricci curvature, J. Differential Geom. 105 (2017), no. 2, 291–343. MR 3606731, DOI 10.4310/jdg/1486522816