Minimax Problems Related to Cup Powers and Steenrod Squares

By Larry Guth. 2007

109 citations recorded by the source corpus.

Abstract.If F is a family of mod 2 k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3c85c76481afc78eb29f7fee1a501af2a133d9c2

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In Applied Mathematics

In Computer Graphics and Computer-Aided Design

In Geometry and Topology

Citations in the corpus

Morse index of multiplicity one min-max minimal hypersurfaces (2015, background): “[18] L. Guth, Minimax problems related to cup powers and Steenrod squares, Geom.”
Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles (2009, methodology): “See [27], [29] for the proof and the related discussion.”
The Allen–Cahn equation on closed manifolds (2016, methodology): “The idea behind Guth’s bend-and-cancel argument is to deform a sweepout using a map that projects the complement of a small neighborhood of c(k) onto the (n − 1)-skeleton K (k)n−1.”
Weyl law for the volume spectrum (2016, background): “In the case of higher codimension, the cohomology ring of the space of relative cycles is richer (see [9]) and so another question would be to understand the asymptotic limit for the widths associated with Steenrod powers.”
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature (2013, methodology): “Finally, to apply Guth’s bend–and–cancel argument, we need a Lipschitz map homotopic to the identity that maps the complement of a small neighborhood of c(k) in M into the n-skeleton G(K (k)n).”
Geometry of the faithfulness assumption in causal inference (2012, background): “For more details on Crofton’s formula for real algebraic hypersurfaces see, for example, [2] or [4], pages 45 and 46.”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “One way would be to construct an explicit p-sweepout in Pp as in [26, Theorem 5.1] using the bend-and-cancel argument of Guth [15].”
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]).”
Equidistribution of minimal hypersurfaces for generic metrics (2017, background): “It follows from the Gromov-Guth bound ([4], [6], see Theorem 5.1 of [12]) that there exists C = C(g̃) such that ωp(M, g̃) ≤ Cp 1 (n+1) for every p ∈ N.”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “ surfaces. A well known conjecture of Yau posits that any closed 3-manifold admits inﬁnitely many immersed minimal surfaces [Yau82]. By considering the p-widths introduced by Gromov [Gro03] (see also [Gut09]), Marques–Neves proved [MN17] that a closed Riemannian manifold (Mn,g) (for 3 ≤ n≤ 7) with p”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “APPLICATION TO VOLUME SPECTRUM In this part, we will show how to apply the result in Section 4 to study volume spectrum introduced by Gromov, Guth, and Marques-Neves.”