AN INEQUALITY BETWEEN ENERGY AND INTERSECTION.^{1}

by C. Croke & A. Fathi

University of Pennsylvania University of Florida & CNRS

Although the identity map of a closed Riemannian manifold is always harmonic, it is not necessarily energy minimizing in its homotopy class. In fact, if π_{1} and π_{2} are zero then there are maps homotopic to the identity with arbitrarily small energy—see [Pl] and [Wh] or [EL]. The identity map is known to minimize for a surface, a Kähler manifold, a manifold of nonpositive curvature, or the standard RP^{n}—see [Sm] and [Cr] or [EL]. The reader is referred to [EL] for a survey of harmonic maps.

In this note we prove:

Theorem. Let (M, g) be a closed Riemannian manifold without conjugate points, then the identity Id : M → M is energy minimizing in its homotopy class. Moreover, any energy minimizing map in that homotopy class is an isometry.

Corollary. On a closed Riemannian manifold without conjugate points of dimension ≥ 2, every conformal self diffeomorphism homotopic to the identity is an isometry.

To prove the theorem, we introduce a quantity i_{f}(g, g′) for any map f : M → N between two Riemannian manifolds (M, g) and (N, g′) with M closed. We call this quantity intersection because it is a generalization of the Thurston intersection theory for curves and measured foliations on surfaces—see also the work of Bonahon [Bo1] and [Bo2]. The quantity i_{f}(g, g′) depends only on the homotopy class of f. We prove that there exists an inequality between the energy of the map f and the intersection, namely:

E(f) ≥ \frac{n}{2α_{n−1}^{2} Vol_{g}(M)}i_{f}(g, g′)^{2},

where n is the dimension of M and α_{n−1} is the volume of the unit sphere in the Euclidean space R^{n}. In the case where M has no conjugate points we see that i_{Id}(g, g) = α_{n−1} Vol(M). The theorem is an easy consequence of these general facts.

We now introduce some notation and terminology. Let (M, g) be a Riemannian manifold, we will denote by S(M) its unit tangent bundle and by p_{M} : S(M) → M

Typeset by AMS-TEX

the canonical projection. When M is complete, the geodesic flow obtained from g on S(M) is denoted by g_{t} : S(M) → S(M), t ∈ R. We will denote by Vol_{g} the volume measure obtained from g on M and by Liou_{g} the Liouville measure obtained from g on the unit tangent bundle S(M). We will say that a path α in M is homotopy minimizing if it realizes the minimum length in the homotopy class of paths with the same endpoints, we will call an infinite path homotopy minimizing if each segment is homotopy minimizing.

1. Intersection.

Definition 1.1. (Intersection). Let (M, g) an (N, g′) be Riemannian manifolds with M closed. If f : M → N is a continuous map, for v ∈ S(M) define φ_{t}(v) as the infinimum of length of all paths in N homotopic with fixed endpoints to s ↦ f(p_{M}g_{s}v), 0 ≤ s ≤ t. We define the intersection of g, g′ under f as:

i_{f}(g, g′) = inf_{t∈R} \frac{1}{t} ∫_{S(M)} φ_{t}(v) d Liou_{g}(v).

Lemma 1.2. The maps φ_{t} form a subadditive cocycle for the geodesic flow g_{t}, this means:

∀v ∈ S(M), ∀t, t′ ∈ R, φ_{t+t′}(v) ≤ φ_{t′}(g_{t}(v)) + φ_{t}(v).

In particular, the family ∫_{S(M)} φ_{t}(v) d Liou_{g}(v) is subadditive and:

i_{f}(g, g′) = lim_{t→∞} \frac{1}{t} ∫_{S(M)} φ_{t}(v) d Liou_{g}(v)

Proof. The subadditivity comes from the fact that the Riemannian distance obtained from g′ on the universal cover of N satisfies the triangle inequality! The second statement follows from the first and the fact that the Liouville measure is invariant under the geodesic flow.

Lemma 1.3. The intersection i_{f}(g, g′) depends only on the homotopy class of f.

Proof. Let f_{u}, u ∈ [0, 1] be a homotopy with f_{0} = f. Since M is compact, we can find K > 0 such that for every m ∈ M the path f_{u}(m), u ∈ [0, 1] is homotopic with endpoints fixed to a path in N whose g′-length is ≤ K. It follows that if we define the cocycle ψ_{t} using f_{1} instead of f we obtain ∀v ∈ S(M), ψ_{t}(v) ≤ φ_{t}(v) + 2K. Since we divide by t and let t tend to ∞ to obtain the intersection it is easy to finish the proof.

Lemma 1.4. If (M, g) is a Riemannian compact manifold without conjugate points of dimension n and Id : M → M is the identity, then:

i_{Id}(g, g) = α_{n−1} Vol_{g}(M),

where α_{n−1} is the volume of the sphere of radius 1 in the Euclidean space R^{n}.

Proof. In that case φ_{t} = t.

Comments 1.5. a) As we said in the introduction the intersection is a generalization of Thurston’s theory of intersection for curves and measured foliations on surfaces. Previous appearances of the intersection of metrics occur in Thurston formulation of the Weil-Peterson metric on Teichmüller space (see Wolpert’s work [Wo3]), in Bonahon’s work on Teichmüller space [Bo] and in Katok’s work on entropy of geodesic flows [Ka]. In [Fa] there will be a more detailed study of the intersection of metrics and its relation to the dynamics of geodesic flows. In all the papers mentioned above the map f is the identity.

b) By the subadditive ergodic theorem the limit lim_{t→∞} \frac{φ_{t}(v)}{t} exists for Liou_{g} almost every v ∈ S(M) and the integral of this limit with respect to Liou_{g} is the intersection.

2. The inequality.

Let (M, g)and (N, g′) be Riemannian manifolds. If f : M → N is a C^{1} map, then its energy is:

(1)  E(f) = \frac{1}{2} ∫_{M} ∥Df_{m}∥^{2} d Vol_{g}(m) = \frac{n}{2α_{n−1}} ∫_{S(M)} ∥Df(v)∥^{2} d Liou_{g}(v).

The last equality can be found in [Cr], with n = dim(M).

Theorem 2.1. Let (M, g) and (N, g′) be Riemannian manifolds with M closed and n-dimensional. If f : M → N is a C^{1} map, we have:

E(f) ≥ \frac{n}{2α_{n−1}^{2} Vol_{g}(M)}i_{f}(g, g′)^{2}.

Moreover, the inequality above is strict unless f is a Riemannian homothety, i.e. f^{∗}g′ = cg with c ≥ 0 and the image under f of every geodesic in M is homotopy minimizing in N, (in particular (M, g) has no conjugate points, if f is not constant).

Proof. By Cauchy-Schwarz inequality, we have:

(2) α_{n−1} Vol_{g}(M) ∫_{S(M)} ∥Df(v)∥^{2} d Liou_{g}(v) ≥ (∫_{S(M)} ∥Df(v)∥ d Liou_{g}(v))^{2}.

By the invariance of the Liouville measure under the geodesic flow we have:

(3)  ∫_{S(M)} ∥Df(v)∥ d Liou_{g}(v) = \frac{1}{t} ∫_{S(M)} ∫_{0}^{t} ∥Df(g_{s}v)∥ ds d Liou_{g}(v).

From the definition of φ_{t}, we obtain:

(4)  ∫_{0}^{t} ∥Df(g_{s}v)∥ ds ≥ φ_{t}(v).

From the definition of the intersection we have:

(5)  \frac{1}{t} ∫_{S(M)} φ_{t}(v) d Liou_{g}(v) ≥ i_{f}(g, g′).

The inequality follows easily from (1) to (5).

If we have equality, we must have equality in (2), (4) and (5). But equality in (2) forces ∥Df(v)∥ to be independent of v ∈ S(M), this shows that there exists c ≥ 0 with f^{∗}g′ = cg. Now equality in (4) implies φ_{t}(v) = ct from which the rest follows

Corollary 2.2. If (M, g) is a closed Riemannian manifold without conjugate points then the identity is energy minimizing in its homotopy class. Moreover, any other energy minimizing map in that homotopy class must be an isometry.

Proof. We have E(Id) = \frac{n}{2} Vol_{g}(M) and i_{Id}(g, g) = α_{n−1} Vol_{g}(M) by lemma 1.4.

Comment 2.3. To conclude we would like to add a small comment on the properness of the energy function on Teichmüller space as considered by Schoen-Yau [SY] and Wolf [Wo2]. This will make one more connection between the Thurston point of view of Teichmüller space, i.e. using intersection, and the harmonic point of view of Teichmüller space—see [EL, pages 424–427] for a survey and references. Let us first recall the setting. If M is a closed surface of genus p and g, h are metrics of constant curvature -1 on M, define E(g, h) as the infinimum of the energy of maps f : (M, g) → (M, h) homotopic to the identity. It is not difficult to see that E(g, h) depends only on the classes of g and h in T_{p} the Teichmüller space of M. For h fixed Schoen and Yau have shown that g ↦ E(g, h) is proper on T_{p}. For g fixed Wolf has

shown that h ↦ E(g, h) is proper. From theorem 2.1, we have: E(g, h) ≥ \frac{i(g, h)^{2}}{8π^{2}(p − 1)},

where i(g, h) = i_{Id}(g, h) is the usual intersection—for example as geodesic currents—of two elements in Teichmüller space. Using Thurston’s compactification of Teichmüller space, it is easy to show that the intersection function as a function of either one of its argument is proper on Teichmüller space. This implies that the energy has the same property. Of course, this should not come as a surprise, and we do not claim that this proof of the properness is easier than the original ones. Our only purpose is to point out one more connection between different aspects of Teichmüller space.

3. Applications.

Theorem 3.1. Let (M, g) be a closed Riemannian manifold without conjugate points. Any conformal self diffeomorphism f : (M, g) → (M, g) homotopic to the identity must be an isometry.

We will need the following well known lemma (we provide the easy proof):

Lemma 3.2. Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2. Any self conformal diffeomorphism f : (M, g) → (M, g) has energy less than or equal to

E(Id) = \frac{n}{2} Vol(M).

Proof. We call σ : M → R the conformal factor. We have ∥D_{x}f∥ = σ(x)∥Id∥. It follows by Hölder inequality that:

E(f) = \frac{n}{2} ∫_{M} σ^{2} d Vol_{g} ≤ \frac{n}{2}(∫_{M} d Vol_{g})^{\frac{n−2}{n}} (∫_{M} σ^{n}d Vol_{g})^{\frac{2}{n}} .

But Vol(M) = ∫_{M} d Vol_{g} = ∫_{M} σ^{n}d Vol_{g}.

Proof of theorem 3.1. By corollary 2.2, we have: E(f) ≥ E(Id). On the other hand E(f) ≤ E(Id), by lemma 3.2. Hence E(f) = E(Id). We conclude the result from the equality case in 2.2.

We give a different proof of the following result which is due to Andreas Knauf [7]:

Corollary 3.3. A closed Riemannian manifold without conjugate points which is conformal to a flat manifold is itself flat.

Proof. Since any flat Riemannian manifold is covered by a flat torus—see [12, theorem 3.3.1, page 105], we can assume that the closed manifold is R^{n}/Γ, where Γ is a discrete subgroup of R^{n} of rank n, and that the metric g we are considering is conformal to the flat metric obtained on R^{n}/Γ from the Euclidean metric on R^{n}. Since the translations on R^{n}/Γ are isometries for the flat metrics, they are conformal maps for g. But the translations are homotopic to the identity of R^{n}/Γ. It follows from 3.1 that they are also isometries for g. Since the translations are transitive on R^{n}/Γ, it is easy to conclude that g must be a (constant) multiple of the flat metric

Remark 3.4. a) Since, by the Uniformization theorem all metrics on the 2-torus are conformally flat, a corollary of 3.3 is Hopf’s theorem: metrics without conjugate points on the 2-torus are flat.

b) An immediate consequence of 4.1 is that two conformal metrics without conjugate points have the same group of isometries homotopic to the identity, moreover, these groups coincide with the group of self-conformal diffeomorphisms homotopic to the identity (for each of the two metrics).

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