THE LOEWNER-PU INEQUALITY, ISOSYSTOLIC ESTIMATES, AND QUASI-CONFORMAL GEOMETRY

RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA

                             HOWARD@MATH.SC.EDU

Other than the last section the following is based on a lecture I gave in a class on integral geometry at the Royal Institute of Technology in Stockholm in the Fall of 1993.

1. Introduction

Let M^{2} be a compact two dimensional manifold and let g be any smooth Riemannian metric on M^{2}. Let Area(g) be the area of this metric and sys_{1}(g) the length of the shortest non-contractible closed geodesic in (M^{2}, g). In 1949 Loewner [3, Note on page 71] proved that when M^{2} is the two dimensional torus the remarkable inequality

sys_{1}(g)^{2} ≤ \frac{2}{\sqrt{3}} Area(g)

holds. The proof is based on the fact any metric on a torus is globally conformal to a flat metric and that by a clever but elementary averaging argument the inequality for general metrics on can be reduced to verifying the inequality for flat metrics. This inequality is sharp and equality holds for the flat torus based on the hexagonal lattice.

A few years latter Pu [3] showed Loewner’s method can be extended to any compact Riemannian manifold with a transitive group of isometries to show that for certain “isosytolic” inequalities (defined below) that in a given conformal class the invariant metric will give the extremal inequality (see Theorem 2.1 below). As every smooth metric on the real projective space RP^{2} is conformal to a metric with constant curvature +1 this implies for any metric g on RP^{2} there is the sharp inequality

sys_{1}(g)^{2} ≤ \frac{π}{2} Area(g).

In this note we make the elementary observation that the method of Loewner and Pu works not only for metrics conformal to a an invariant metric, but also for metrics λ-quasi-conformal to an invariant metric. The resulting inequality relates the constant λ to the to isosytolic inequalities. Thus the constants in the isosystolic inequalities can be used to estimate the quasi-conformal distance of a metric from an invariant metric. As an application we use recent results of Babenko and Katz [1] to show there are metrics on the three dimensional torus that have arbitrarily large conformal distance from all flat metrics.

Date: October 1997.

2. The Quasi-Conformal Version of the Loewner-Pu Inequality.

Let M be a smooth compact manifold and g_{0} and g two Riemannian metrics on M. Then for a constant λ ≥ 1. the metric g is λ quasiconformal to g_{0} iff there is a diffeomorphism φ : M → M and a positive function u : M → R so that

u^{2}g_{0} ≤ φ^{∗}g ≤ λ^{2}u^{2}g_{0}. (2.1)

When λ = 1 the g_{0} and g are conformal.

Let H_{k}(M) be the k dimensional homology of M with integer coefficients. Then any homology class α ∈ H_{k}(M) can be represented by a chain S = ∑_{i} a_{i}s_{i} so that each singular simplex s_{i} : Δ_{k} → M is a C^{1} map (here Δ_{k} is the standard k dimensional simplex). For a smooth metric g on M can then define the volume of such a chain S by

Vol_{k}^{g}(S) = ∑_{i} |a_{i}| Vol_{k}^{g}(s_{i}),

where the volume of s_{i} is the volume the singular Riemannian space (Δ_{k}, s^{∗}g). (Thus this volume depends on the Riemannian metric on M.) The volume of a homology class α ∈ H_{k}(M) is then

Vol_{k}^{g}(α) := inf_{S∈α} Vol_{k}^{g}(S)

where the infimum is over all C^{1} chains representing α. Also for any smooth metric g on M define, following Berger [2], the k-systole of g by

sys_{k}(g) := inf_{0≠α∈H_{k}(M)} Vol_{k}^{g}(α).

The following is generalization of a result of Pu [3, Theorem 2 p. 62] form conformally related to quasi-conformally related metrics. The proof here only involves trivial changes form that given in [3].

Theorem 2.1. Let M = G/K, dim M = n, be a homogeneous space with G a compact Lie group and K a closed subgroup. Let g_{0} be a Riemannian metric on M that is invariant under the action of G and let g be any Riemannian metric on M that is λ quasi-conformal to g_{0}. Then

\frac{sys_{k}(g)}{Vol_{n}(g)^{\frac{k}{n}}} ≤ λ^{k} \frac{sys_{k}(g_{0})}{Vol_{n}(g_{0})^{\frac{k}{n}}}

for 1 ≤ k ≤ n − 1.

Corollary 2.2. With the same hypothesis

\frac{Vol_{n}(g)}{sys_{k}(g) sys_{n−k}(g)} ≥ \frac{Vol_{n}(g_{0})}{λ^{n} sys_{k}(g_{0}) sys_{n−k}(g_{0})}.

Proof. If g is λ quasi-conformal to g_{0} then for some diffeomorphism φ : M → M the inequalities (2.1) hold. By replacing g by φ^{∗}g we can assume

u^{2}g_{0} ≤ g ≤ λ^{2}u^{2}g_{0}.

We denote by dV_{n}^{g} the volume element of the metric g and the volume element induced on k dimensional submanifolds by g will be denoted by dV_{k}^{g} with similar notation for volumes induced by g_{0}. The inequalities between g and g_{0} imply

u^{k} dV_{k}^{g_{0}} ≤ dV_{k}^{g} ≤ λ^{k}u^{k} dV_{k}^{g_{0}}.

Thus

∫_{M} u(z)^{n} dV_{k}^{g_{0}}(z) ≤ ∫_{M} 1 dV_{n}^{g}(z) = Vol_{n}^{g}(M).

and for and k dimensional submanifold S

Vol_{k}^{g}(S) ≤ λ^{k} ∫_{S} u(x)^{k} dV_{k}^{g_{0}}(x).

We use the normalized measure Haar measure dξ on G so that ∫_{G} 1 dξ = 1. Then for any point z ∈ M using the invariance of the metric g_{0} we have for any function f : M → R and z ∈ M

∫_{G} f(ξz) dξ = Vol_{n}^{g_{0}}(M)^{−1} ∫_{M} f(y) dV_{n}^{g_{0}}(y).

We now assume S is a k dimensional submanifold of M so that Vol_{k}^{g_{0}}(S) = sys_{k}(g_{0}). (Here we are cheating a bit. By a basic result of Federer and Fleming there is an integral currant S with Vol_{k}^{g_{0}}(S) = sys_{k}(g_{0}) and by a regularity theorem due to Almgren this current is regular almost everywhere. The use of these results can be avoided in the following argument by working with a minimizing sequences of singular chains, at the expense of making the proof slightly more complicated notationally.) Note for any ξ ∈ G we have sys_{k}(g) ≤ Vol_{k}^{g}(ξS). Therefore, using the inequalities above and Hölder’s inequality,

sys_{k}(g) ≤ ∫_{G} Vol_{k}^{g}(ξS) dξ
≤ ∫_{G} λ^{k} ∫_{S} u(ξx)^{k} dV_{k}^{g_{0}}(x) dξ
= λ^{k} ∫_{S} ∫_{G} u(ξx)^{k} dξ dV_{k}^{g_{0}}(x)

= λ^{k} sys_{k}(g_{0}) ∫_{G} u(ξx_{0})^{k} dξ (This is independent of choice of x_{0})

≤ λ^{k} sys_{k}(g_{0}) (∫_{G} u(ξx_{0})^{n} dξ)^{\frac{k}{n}}

= λ^{k} sys_{k}(g_{0}) Vol_{n}^{g_{0}}(M)^{− \frac{k}{n}} (∫_{M} u(z)^{n} dV_{n}^{g_{0}}(z))^{\frac{k}{n}}
≤ λ^{k} sys_{k}(g_{0}) Vol_{n}^{g_{0}}(M)^{− \frac{k}{n}} (Vol_{n}^{g}(M))^{\frac{k}{n}}

which is equivalent to the required inequality. The proof of the corollary is straightforward.

3. Application to Quasi-Conformal Geometry.

Let g_{0}, g be two smooth Riemannian metrics on the compact Riemannian manifold M^{n}. Then define the quasi-conformal distance between g_{0} and g by

Conf Dist(g, g_{0}) := inf{λ : g is λ quasi-conformal to g_{0}}.

A little work then shows

Conf Dist(g, g_{0}) = Conf Dist(g_{0}, g)

and if g_{1} is anther smooth metric then

Conf Dist(g_{1}, g_{0}0) ≤ Conf Dist(g_{1}, g) Conf Dist(g, g_{0}).

Thus the function d(g, g_{0}) := ln Conf Dist(g, g_{0}) makes the space of all conformal classes of metrics on M^{n} into a metric space. Corollary 2.2 implies

(3.1) Conf Dist(g, g_{0})^{n} ≥ \frac{Vol_{n}(g_{0}) sys_{k}(g) sys_{n−k}(g)}{Vol_{n}(g) sys_{k}(g_{0}) sys_{n−k}(g_{0})}.

Lemma 3.1. There is a constant c(n) so that for any flat metric g_{0} on the n dimensional torus T^{n} the inequality

\frac{Vol_{n}(g_{0})}{sys_{k}(g_{0}) sys_{n−k}(g_{0})} ≥ c(n)

holds for 1 ≤ k ≤ n − 1.

Proof. A (non-trivial) exercise.

Combining this with (3.1) implies that for any smooth metric g and any flat metric on T^{n} that

(3.2) Conf Dist(g, g_{0})^{n} ≥ c(n)\frac{sys_{k}(g) sys_{n−k}(g)}{Vol_{n}(g)}.

Theorem 3.2. For each n ≥ 3 and D_{0} > 0 there is smooth metric g on the n dimensional torus T^{n} so that for all flat metrics g_{0} on T^{n} Conf Dist(g, g_{0}) ≥ D_{0}.

Proof. By a recent result of Babenko and Katz [1] there is a sequence of smooth metrics g_{ℓ} on T^{n} so that

lim_{ℓ→∞} \frac{sys_{1}(g_{ℓ}) sys_{n−1}(g_{ℓ})}{Vol_{n}(g_{ℓ})} = ∞.

This together with (3.2) completes the proof.

References

1. Ivan Babenko and Mikhail Katz, Systolic freedom of orientable manifolds, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 6, 787–809.

2. Marcel Berger, Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4) 5 (1972), 1–44.

3. P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71.