Isoperimetry of Waists and Concentration of Maps

By Misha Gromov. 2003

343 citations recorded by the source corpus.

Let f : Sn → Rk be a continuous map where Sn is the unit n-sphere. Then there exists a point z ∈ Rk such that the spherical n-volumes of the e-neighbours of the level Yz = f−1(z) ⊂ Sn, denoted Yz + e ⊂ Sn, satisfy Vol(Yz + e) ≥ Vol(Sn−k + e) ( )Sn for all e > 0, where Sn−k ⊂ Sn denotes an equatorial (n− k)-sphere. This is proven in §5.9. Remarks. (a) If k = n, and card f−1(z) ≤ 2, z ∈ Rn, then ( )Sn applied to e = π/2 amounts to the Borsuk–Ulam theorem: some level f−1(z) of f : Sn → Rk equals a pair of opposite points. Not surprisingly, our argument in the general case depends on Z2-cohomological considerations. (b) If k = 1 one may take the Levy mean of f for z ∈ R, where the level f−1(z) ⊂ Sn divides the sphere into equal halves (i.e. where Vol(f−1(−∞, z]) and Vol(f−1[z,∞)) are both ≥ 12 Vol(Sn)). Then ( )SN follows from the spherical isoperimetric inequality (see 9.2.B). (c) The inequality ( )Sn for e → 0 shows that the Minkowski m-volume of f−1(z) for m = n − k is ≥ than that of the equatorial sphere Sm ⊂ Sn. Yet, it remains unclear if some level of f has the Hausdorff measure ≥ Volm(S). (If f is a generic smooth map, a level with Haumesm ≥ Volm(S) is delivered by the Almgren–Morse theory, see [Gr5].)

Subfields: Management Information Systems

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Citations in the corpus

Dwyer–Kan homotopy theory of enriched categories (2012, neutral): “]: x[1|234] + x[1|342] + x[1|243] + x[1|234] + x[12|34] + x[13|24] + x[14|23] + x[14|23] + x[13|24] + x[12|34] + x[123|4] + x[124|3] + x[134|2] + x[123|4] = 1, x[1|243] + x[1|432] + x[1|243] + x[1|234] + x[12|43] + x[13|24] + x[14|23] + x[14|23] + x[13|24] + x[12|43] + x[123|4] + x[124|3] + x[143|2]”
Stability properties and topology at infinity of $$f$$f-minimal hypersurfaces (2013, methodology): “Following Gromov [17], if we consider an isometrically immersed orientable hypersurface Σm in the weighted manifold M f , we can also define a generalization of the mean curvature vector field as H f = H + (∇ f )⊥.”
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature (2013, positive): “The following result was proven by Gromov (see [14, Section4.”
On the Multiplicity One Conjecture in min-max theory (2019, positive): “D G ] 5 F eb 2 01 the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one.”
Weyl law for the volume spectrum (2016, background): “For instance, Gromov and later Guth ([6] or Guth [9]) showed the existence of a constant C = C(M,g) for which Cvol(M) n n+1 p 1 n+1 ≤ ωp(B) ≤ Cvol(M) n n+1 p 1 n+1”
Density of minimal hypersurfaces for generic metrics. (2017, methodology): “The ﬁrst estimates for these numbers were proven by Gromov in the late 1980s [5] (see also Guth [7]).”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “In their proof, they recast the sublinear bounds for a certain sequence of min-max widths, proved by Gromov [12] and Guth [15], in the context of an extended Almgren-Pitts’ min-max theory and argue as follows.”
Equidistribution of minimal hypersurfaces for generic metrics (2017, methodology): “As in the Irie-Marques-Neves paper ([8]), the crucial tool in our proof is the Weyl law for the Volume Spectrum conjectured by Gromov ([5]) and recently proven by the first two authors jointly with Liokumovich in [9]: Weyl Law for the Volume Spectrum (Liokumovich, Marques, Neves, 2016): There exists”
The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture (2008, methodology): “For a hypersurface S ⊂ Rn, we define a directed volume function VS by the following formula.”
Minimax Problems Related to Cup Powers and Steenrod Squares (2007, background): “Recently, Gromov studied the problem of estimating V(a(k, n)) in Section 8 of [7].”
On the isoperimetric problem in Euclidean space with density (2006, background): “We define, as in [20] and [6, Chap.”
Convex equipartitions: the spicy chicken theorem (2013, methodology): “The so called Gromov–Borsuk– Ulam theorem of [11] and [19] is the case when p = 2 and μ is the O(d)-invariant probability measure on S.”
Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles (2009, background): “a manifold) Y 3 y and establish lower bounds on the topologies of Yy in terms of some geometric invariants (called ”waists” in [17]) of metric spaces Z receiving X via continuous maps α : X → Z.”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “[Gro03] M. Gromov.”