An inequality for length and volume in the complex projective plane
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An inequality for length and volume in the complex projective plane Mikhail G. Katz1 Received: 5 April 2020 / Accepted: 7 September 2020 © Springer Nature B.V. 2020
Abstract We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer–Mrowka proof of the Thom conjecture. Keywords Minimal surface · Regularity · Systole · Closed geodesics · Croke–Rotman inequality · Gromov’s stable systolic inequality for complex projective space · Kronheimer–Mrowka theorem Mathematics Subject Classification (2010) Primary 53C23
1 Introduction The 1-systole sys1 of a Riemannian manifold M is the least length of a noncontractible loop in M. In the 1950s, Carl Loewner proved an inequality relating the systole and the area of an arbitrary metric on the 2-dimensional torus; see [14,17,34]. Gromov obtained a variety of inequalities relating the 1-systole and the volume of M. We will now present those that will be used in this paper. Thus, we have the inequality sys21 (S) ≤
4 area(S) 3
(1.1)
from Ref. [13, p. 49, Corollary 5.2.B], valid for every closed aspherical surface S. There are also bounds that improve as the genus g grows. Thus, Gromov proved that a surface Sg of genus g satisfies 64 sys21 (Sg ) ≤ √ area(Sg ) 4 g + 27
B 1
(1.2)
Mikhail G. Katz [email protected] Department of Mathematics, Bar Ilan University, Ramat Gan 5290002, Israel
123
Geometriae Dedicata
(cf. [18, p. 1216], [23]). An asymptotically better result is the following: sys21 (Sg ) area
≤
1 log2 g (1 + o(1)) when g → ∞ π g
(1.3)
from Ref. [18, p. 1211, Theorem 2.2], improving the multiplicative constant in Gromov’s 2 similar upper bound. Apart from the multiplicative constant, the asymptotic behavior logg g is the correct one due to the existence of arithmetic hyperbolic surfaces satisfying lower bounds of this type; see e.g., Buser–Sarnak [3], Katz et al. [16,20,21]. The literature contains a number of results on the higher systoles, as well. Recall that the 2-systole sys2 of M can be defined as the least area of a homologically nontrivial surface in M, or more generally 2-cycle with integer coefficients: sys2 (M) = min area(S) : [S] ∈ H2 (M; Z) \ {0} . The 2-systole of M is typically not controlled by the volume of M, a phenomenon referred to as systolic freedom. For example, the complex projective plane CP2 admits metrics of arbitrarily small volume such that every homologically nontrivial surface in CP2 has at least unit area; see Katz–Suciu ([22], Theorem 1.1, p. 113). Gromov has also defined a modified invariant of M called the stable 2-systole, stsys2 (M). It is a result of Federer [9] that the limits below exist and therefore can be used to define stsys2 as follows: stsys2 (M) = min lim n1 sys2 (nα) : α ∈ H2 (M; Z) \ {torsion} (1.4) n→∞
where sys2 (x) denotes the infimum of areas of surfaces representing the class x ∈ H2 (M; Z). Gromov’s stable systolic in
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