We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.
@article{bwmeta1.element.doi-10_2478_s11533-013-0387-5,
author = {Misha Gromov},
title = {Plateau-Stein manifolds},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {923-951},
zbl = {1293.31006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0387-5}
}
Misha Gromov. Plateau-Stein manifolds. Open Mathematics, Tome 12 (2014) pp. 923-951. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0387-5/
[1] Cheeger J., Naber A., Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math., 2013, 191(2), 321–339 http://dx.doi.org/10.1007/s00222-012-0394-3 | Zbl 1268.53053
[2] Gromov M., Sign and geometric meaning of curvature, Rend. Sem. Mat. Fis. Milano, 1991, 61, 9–123 http://dx.doi.org/10.1007/BF02925201 | Zbl 0820.53035
[3] Gromov M., Hilbert volume in metric spaces. Part 1, Cent. Eur. J. Math., 2012, 10(2), 371–400 http://dx.doi.org/10.2478/s11533-011-0143-7
[4] Gromov M., Lawson H.B. Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math., 1980, 111(3), 423–434 http://dx.doi.org/10.2307/1971103 | Zbl 0463.53025
[5] Lawson H.B. Jr., Michelsohn M.-L., Embedding and surrounding with positive mean curvature, Invent. Math., 1984, 77(3), 399–419 http://dx.doi.org/10.1007/BF01388830 | Zbl 0555.53027
[6] Lawson H.B. Jr., Michelsohn M.-L., Approximation by positive mean curvature immersions: frizzing, Invent. Math., 1984, 77(3), 421–426 http://dx.doi.org/10.1007/BF01388831 | Zbl 0555.53028
[7] Lott J., Villani C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 2009, 169(3), 903–991 http://dx.doi.org/10.4007/annals.2009.169.903 | Zbl 1178.53038
[8] Micallef M.J., Moore J.D., Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math., 1988, 127(1), 199–227 http://dx.doi.org/10.2307/1971420 | Zbl 0661.53027
[9] Ollivier Y., Ricci curvature of Markov chains on metric spaces, J. Funct. Anal., 2009, 256(3), 810–864 http://dx.doi.org/10.1016/j.jfa.2008.11.001 | Zbl 1181.53015
[10] Sha J.-P., Handlebodies and p-convexity, J. Differential Geom., 1987, 25(3), 353–361 | Zbl 0661.53028
[11] Sormani C., Wenger S., The intrinsic flat distance between Riemannian manifolds and other integral current spaces, J. Differential Geom., 2011, 87(1), 117–199 | Zbl 1229.53053
[12] Wenger S., Isoperimetric inequalities of Euclidean type in metric spaces, Geom. Funct. Anal., 2005, 15(2), 534–554 http://dx.doi.org/10.1007/s00039-005-0515-x | Zbl 1084.53037
[13] White B., A local regularity theorem for mean curvature flow, Ann. of Math., 2005, 161(3), 1487–1519 http://dx.doi.org/10.4007/annals.2005.161.1487 | Zbl 1091.53045