Minimal entropy rigidity for lattices in products of rank one symmetric spaces

Christopher Connell ^{∗}and Benson Farb ^{†}

May 17, 2001

1 Introduction

The volume entropy h(g) of a closed Riemannian n-manifold (M, g) is defined as

h(g) = lim_{R→∞} \frac{1}{R} log(Vol(B(x, R)))

where B(x, R) is the ball of radius R around a fixed point x in the universal cover X. (For noncompact M, see Section 6.2.) The number h(g) is independent of the choice of x, and equals the topological entropy of the geodesic flow on (M, g) when the curvature K(g) satisfies K(g) ≤ 0 (see [Ma]). Note that while the volume Vol(M, g) is not invariant under scaling the metric g, the normalized entropy

ent(g) = h(g)^{n} Vol(M, g)

is scale invariant.

Besson-Courtois-Gallot [BCG1] showed that, if n ≥ 3 and M admits a negatively curved, locally symmetric metric g_{0}, then ent(g) is minimized uniquely by g_{0} in the space of all Riemannian metrics on M. This striking result, called minimal entropy rigidity, has a great number of corollaries, including solutions to long-standing problems on geodesic flows, asymptotic harmonicity, Gromov’s minvol invariant, and a new proof of Mostow Rigidity in the rank one case (see [BCG2]).

Extending minimal entropy rigidity to all nonpositively curved, locally symmetric manifolds M has been a well-known open problem (see, e.g., [BCG2], Open Question 5). The case of closed manifolds locally (but not necessarily globally) isometric to products of negatively curved locally symmetric spaces of dimension at least 3 was announced in [BCG2] and later in [BCG3].

∗Supported in part by an NSF postdoctoral fellowship.

†Supported in part by NSF grant DMS 9704640 and by a Sloan foundation fellowship.

In this paper we prove minimal entropy rigidity in this case as well as in the more general setting of complete, finite volume manifolds. Although we haven’t seen Besson-Courtois-Gallot’s proof of this result, it is likely that our proof (in the compact case) overlaps with theirs. In particular, we apply the powerful method introduced in [BCG1, BCG2], with a few new twists (see below).

Statement of result. While the quantity ent(g) is invariant under scaling the metric on M, it is not invariant under scalings of the individual factors of M. Hence the iso-entropic inequality as in [BCG1, BCG2] does not hold as stated; one needs to find the locally symmetric metric g_{min} on M which minimizes h(g_{0})^{n} for a fixed volume Vol(M, g_{0}) among all locally symmetric metrics g_{0}. This minimization problem is easily solved by Lagrange multipliers; as we show in §2, such a g_{min} does indeed exist and is unique up to homothety.

We say that a Riemanninan manifold X has bounded geometry if for any ϵ > 0 there exists C so that every ϵ-ball in X has volume at least C. This holds for example when X has injectivity radius bounded below and sectional curvature bounded above.

Theorem A (Minimal Entropy Rigidity). Let M be an n-manifold which admits a complete, finite-volume Riemannian metric which is locally isometric to a product of negatively curved (rank 1) symmetric spaces of dimension at least 3. Let g be any other complete, finite-volume, Riemannian metric on M. If M is not compact then assume that M̃ has bounded geometry. Then

ent(g) ≥ ent(g_{min})

with equality iff g is homothetic to the locally symmetric metric g_{min}.

Remarks.

1. Theorem A is in fact true in more generality: if (N, g) is any finite volume Riemannian n-manifold of bounded geometry, and if f : N → M is any proper, coarsely Lipschitz map, then

ent(N, g) ≥ |degf| ent(M, g_{min})

with equality iff f is a homothetic Riemannian covering. It is this more general result that we prove.

2. P. Verovic [Ve] has shown that Theorem A no longer holds if g is allowed to be a Finsler metric. This behavior is different than in the rank one case.

3. By similar arguments to those in [BCG1, BCG2], Theorem A implies strong (Mostow) rigidity for the corresponding locally symmetric manifolds.

Outline of the proof of Theorem A.

In this outline we assume M and N are compact. We will outline the proof of Remark 1 after Theorem A; this implies Theorem A by taking f : N → M to be the identity. For a moment consider the case when the metric on N is nonpositively curved.

Endow M with the unique locally symmetric metric g_{min} minimizing ent(g). Denote by Y (resp. X) the universal cover of N (resp. M). Let M(∂Y), M(∂X) denote the spaces of atomless probability measures on the visual boundaries ∂Y, ∂X of the universal covers Y, X.

Morally what we do is, following the method of [BCG2], to define a map

F̃ : Y → M(∂Y) →^{ϕ_{∗}} M(∂X) →^{bar} X

where ϕ_{∗} = ∂f̃_{∗} is the pushforward of measures and bar is the “barycenter of a measure” (see §4). The inclusion Y → M(∂Y), denoted x ↦ ν_{x}, is given by the construction of the Patterson-Sullivan measures {ν_{x}}_{x∈X} corresponding to π_{1}(N) < Isom(Y) (see §3). An essential feature of these constructions is that they are all canonical, so that all of the maps are equivariant. Hence F̃ descends to a map F : N → M.

One problem with this outline is that the metric on Y is typically not nonpositively curved. In fact, if Y is a finite volume nonpositively curved manifold diffeomorphic to an irreducible higher rank locally symmetric space then Y is locally symmetric (see Chapter 9 of [Eb]). So we must find an alternative to using the “visual boundary” of Y. This is done by constructing smooth measures (parameterized by s > h(g)) on Y itself, pushing them forward via f̃, and convolving with Patterson-Sullivan measure on X. The maps F̃_{s} are then defined by taking the barycenters of these measures. In the equality case, the maps F_{s} limit (as s tends to h(g) from above) to a Lipschitz map F which turns out to be the desired locally isometric covering. This idea was first introduced in [BCG1].

As in [BCG1, BCG2], the main step in the proof is bounding the Jacobian | Jac F_{s}|. A simple degree computation finishes the proof of the compact case. The extension to the noncompact, finite volume case is perhaps the most technical part of the proof, and requires extending some of the ideas of [BCS] to the higher rank setting to show that F_{s} is proper.

New features. As noted above, our proof is an application of the method of [BCG2]. The new features which occur in the present case are:

• In nonpositive curvature, Busemann functions are convex but not strictly convex; hence a global argument is needed to prove that the barycenter map is well-defined.

• As shown by Albuquerque [Al], each Patterson-Sullivan measure ν_{x} on a symmetric space X is supported on a subset ∂_{F} X ⊂ ∂X identified with the Furstenberg boundary. We make essential use of this fact in several places; in particular, the boundary ∂_{F} X decomposes naturally as a product when Isom(X) does. The visual boundary ∂X does not. The product decomposition of ∂_{F} X allows one to reduce the key estimate on | Jac F_{s}| to the rank one case and an algebraic lemma of [BCG2]. Note that in the negatively curved case of [BCG1, BCG2], ∂_{F} X = ∂X.

• In the noncompact, finite volume case, direct geometric estimates on N and M and asymptotic analysis of the maps F_{s} are used to show that the maps F_{s} are proper.

2 The best locally symmetric metric

In this section we find the locally symmetric metric g_{min} which minimizes ent(g_{0}) over all locally symmetric metrics g_{0}. Recall that the entropy h(g_{0}) of a negatively curved, locally symmetric n-manifold (M, g_{0}) with maximum sectional curvature −1 equals n + d − 2, where d = 1, 2, 4, or 8 according to whether (M, g_{0}) is a real, complex, quaternionic, or Cayley hyperbolic space.

Now suppose that (M, g_{0}) is a complete, finite volume Riemannian manifold which is locally isometric to a product

(X_{1}, g_{1}) × ⋯ × (X_{k}, g_{k})

of negatively curved symmetric spaces. After possibly scaling each factor, we may assume that (X_{i}, g_{i}) has maximum sectional curvature −1. In that case, X_{i} has entropy h_{i} = n_{i} + d_{i} − 2 where n_{i} is the dimension of X_{i} and d_{i} = 1, 2, 4, or 8 is the dimension of the division algebra which classifies the symmetric space X_{i}. The space of locally symmetric metrics on M is then the set of metrics

g_{β} = β_{1}^{2}g_{1} × ⋯ × β_{k}^{2}g_{k}

where each β_{i} is a positive real number. Note that the entropy of β_{i}^{2}g_{i} is β_{i}^{−1}(n_{i} + d_{i} − 2).

We now wish to minimize ent(g_{β}) over all locally symmetric g_{β} on M. This is the same as minimizing h(g_{β}) while keeping volume fixed, i.e. while keeping ∏_{i=1}^{k} β_{i}^{n_{i}} = 1.

It is not hard to see that the exponential growth rate of the volume of balls in the universal cover is given by h_{0} = |l| where l = ∑_{α>0} m_{α}α^{∗} where α^{∗} is the dual of the positive root α with multiplicity m_{α} (see [Kn] for a sharp asympotic for the growth of balls). In the case of products of rank

one spaces with entropies β_{i}^{−1}h_{i}, this becomes h(g_{β}) = \sqrt{∑_{i=1}^{k} β_{i}^{−2}h_{i}^{2}}, since the roots are all orthogonal with multiplicity h_{i}.

Now, with scalings as above, we wish to minimize

h_{β} = \sqrt{∑_{i=1}^{k} β_{i}^{−2}h_{i}^{2}} = \sqrt{∑_{i=1}^{k} β_{i}^{−2}(n_{i} + d_{i} − 2)^{2}}

subject to ∏_{i=1}^{k} β_{i}^{n_{i}} = 1. An easy computation using Lagrange multipliers now gives that the locally symmetric metric g_{min} which minimizes ent(g) is

g_{min} = α_{1}^{2}g_{1} × ⋯ × α_{k}^{2}g_{k}

where

α_{i} = \frac{h_{i}}{\sqrt{n_{i}}} ∏_{i=1}^{k} (\frac{\sqrt{n_{i}}}{h_{i}} )^{\frac{n_{i}}{n}}

In this case

h(g_{min}) = \sqrt{n} ∏_{i=1}^{k} (\frac{h_{i}}{\sqrt{n_{i}}} )^{n_{i}/n}

3 Patterson-Sullivan measures on symmetric spaces

In this section we briefly recall Albuquerque’s theory [Al] of Patterson-Sullivan measures in higher rank symmetric spaces. For background on nonpositively curved manifolds, symmetric spaces, visual boundaries, Busemann functions, etc., we refer the reader to [BGS] and [Eb].

3.1 Basic properties

Let X be a Riemannian symmetric space of noncompact type. Denote by ∂X the visual boundary of X; that is, the set of equivalence classes of geodesic rays in X, endowed with the cone topology. Hence X ∪ ∂X is a compactification of X which is homeomorphic to a closed ball. Let Γ be a lattice in Isom(X), so that h(g_{0}) < ∞ where (M, g_{0}) is Γ\X with the induced metric.

Generalizing the construction of Patterson-Sullivan, Albuquerque constructs in [Al] a family of Patterson-Sullivan measures on ∂X. This is a family of measures {ν_{x}}_{x∈X} on ∂X which provide a particularly natural embedding of X into the space of measures on ∂X.

Proposition 3.1. The family {ν_{x}} satisfies the following properties:

1. Each ν_{x} has no atoms.

2. The family of measures {ν_{x}} is Γ-equivariant:

γ_{∗}ν_{x} = ν_{γx} for all γ ∈ Γ

3. For all x, y ∈ X, the measure ν_{y} is absolutely continuous with respect to ν_{x}. In fact the Radon-Nikodym derivative is given explicitly by:

\frac{dν_{x}}{dν_{y}} (ξ) = e^{h(g)B(x,y,ξ)}  (1)

where B(x, y, ξ) is the Busemann function on X. For points x, y ∈ X and ξ ∈ ∂X, the function B : X × X × ∂X → R is defined by

B(x, y, ξ) = lim_{t→∞} d_{X}(y, γ_{ξ}(t)) − t

where γ_{ξ} is the unique geodesic ray with γ(0) = x and γ(∞) = ξ.

Since Isom(X) acts transitively on X, the first property above implies that the ν_{x} are all probability measures. The second property implies no two measures are the same. Thus the assignment x ↦ ν_{x} defines an injective map

ν : X → M(∂X)

where M(∂X) is the space of probability measures on X. Such a mapping satisfying the above properties is called an h(g_{0})-conformal density.

3.2 The Furstenberg boundary and Albuquerque’s Theorem

The Furstenberg boundary of a symmetric space X of noncompact type is abstractly defined to be G/P where P is a minimal parabolic subgroup of the connected component G of the identity of Isom(X).

Fix once and for all a basepoint p ∈ X. This choice uniquely determines a Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭 of the Lie algebra of G where 𝔨 is the Lie algebra of the isotropy subgroup K = Stab_{G}(p) of p in G and 𝔭 is identified with the tangent space T_{p}X.

Let 𝔞 be a fixed maximal abelian subspace of 𝔭. The rank of X, denoted rk(X), is the dimension of 𝔞. If A = exp(𝔞) then A · p will be a maximal flat (totally geodesically embedded Euclidean space of maximal dimension). Recall, a vector v ∈ T X is called a regular vector if it is tangent to a unique maximal flat. Otherwise it is a singular vector. A geodesic is called regular (resp. singular) if one (and hence all) of its tangent vectors are regular (singular). A point ξ ∈ ∂X is regular (singular) if any (and hence all) of the geodesics in the corresponding equivalence class are regular (singular).

A Weyl chamber 𝔞^{+} is a choice of connected component of the set of regular vectors in 𝔞. There corresponds to 𝔞^{+} a choice of positive roots. Similarly if A^{+} = exp(𝔞^{+}) then A^{+} · p is called a Weyl chamber of the flat A·p. The union of all the singular geodesics in the flat A·p passing through p is a finite set of hyperplanes forming the boundaries of the Weyl chambers.

In our current special case of products, rk(X) = k (recall k is the number of factors) since a product of geodesics, one from each factor, produces a maximal flat. Also, the Weyl chambers are simply the 2^{rk(X)} orthants of 𝔞 ≃ R^{rk(X)} and b^{+} is the unit vector corresponding to the barycenter of the extremal unit vectors on the interior boundary of a fixed Weyl chamber.

The Furstenberg boundary can be identified with the orbit of G acting on any regular point v(∞) ∈ ∂X, the endpoint of a geodesic tangent to a regular vector v. of a Weyl chamber in a fixed flat 𝔞. This follows from the fact that the action of any such P on ∂X fixes some regular point.

Because of this, for symmetric spaces of higher rank, behaviour on the visual boundary can often be aptly described by its restriction to the Furstenberg boundary. Here we will use only some very basic properties of this boundary. For more details on semisimple Lie groups and the Furstenberg boundary, see [Zi].

Let b be the sum of the dual vectors of the positive roots corresponding to 𝔞^{+}. The vector b^{+} is called the algebraic centroid of 𝔞^{+}. Set b^{+} = b/∥b∥.

Define the set ∂_{F} X ⊂ ∂X to be ∂_{F} X = G · b^{+}(∞). Henceforth we will refer to the Furstenberg boundary as this specific realization. We point out that for any lattice Γ in Isom(X), the induced action on the boundary is transitive only on ∂_{F} X. That is, \overline{Γ · b^{+}(∞)} = G · b^{+}(∞) even though for any interior point x ∈ X, \overline{Γ · x} = ∂X.

Theorem 7.4 and Proposition 7.5 of [Al] combine to give the following categorization which will play a crucial role in our proof of Theorem A.

Theorem 3.2 (Description of ν_{x}). Let (X, g_{0}) be a symmetric space of noncompact type, and let Γ be a lattice in Isom(X). Then

1. h(g_{0}) = ∥b∥,

2. b^{+}(∞) is a regular point, and hence ∂_{F} X is a regular set,

3. For any x ∈ X, the support supp(ν_{x}) of ν_{x} is equal to ∂_{F} X, and

4. ν_{x} is the unique probability measure invariant under the action on ∂_{F} X of the compact isotropy group Stab_{G}(x) at x. In particular, ν_{p} is the unique K-invariant probability measure on ∂_{F} X.

Note that when X is has rank one, ∂_{F} X = ∂X. In general ∂_{F} X has codimension rk(X) − 1 in ∂X.

3.3 Limits of Patterson-Sullivan measures

We now describe the asymptotic behaviour of the ν_{x} as x tends to a point in ∂X.

For any point ξ of the visual boundary, let S_{θ} be the set of points ξ ∈ ∂_{F} X such that there is a Weyl chamber W whose closure ∂\overline{W} in ∂X contains both θ and ξ. Let K_{θ} be the subgroup of K which stabilizes S_{θ}. K_{θ} acts transitively on S_{θ} (see the proof below).

Theorem 3.3 (Support of ν_{x}). Given any sequence {x_{i}} tending to θ ∈ ∂X in the cone topology, the measures ν_{x_{i}} converge in M(∂_{F} X) to the unique K_{θ}-invariant probability measure ν_{θ} supported on S_{θ}.

Proof. Let x_{i} = g_{i} · p, for an appropriate sequence g_{i} ∈ G. Recall that ν_{x_{i}} = (g_{i})_{∗}ν_{p}. Then combining part (4) of Theorem 3.2 with Proposition 9.43 of [GJT] have that some subsequence of the ν_{x_{i}} converges to a K_{θ}- invariant measure ν_{θ} supported on S_{θ}.

Note that in [GJT], the notation I refers to a subset of a fundamental set of roots corresponding to the face of a Weyl chamber containing θ in its boundary. If g_{i} · p = k_{i}a_{i} · p converges then both k = lim k_{i} and a^{I} = lim_{i}a_{i}^{I} exist (note the definition of a^{I} in [GJT]). Again in the notation of [GJT], K_{θ} is the conjugate subgroup (ka^{I})K^{I}(ka^{I})^{−1} in K. Moreover, S_{θ} is the orbit ka^{I}K^{I} · b^{+}(∞).

By Corollary 9.46 and Proposition 9.45 of [GJT], any other convergent subsequence of the ν_{x_{i}} produces the same measure in the limit, and therefore the sequence ν_{x_{i}} itself converges to ν_{θ} uniquely. ⋄

In the case when θ is a regular point, the above theorem implies that S_{θ} is a single point and the limit measure ν_{θ} is simply the Dirac probability measure at that point point in ∂_{F} X.

4 The barycenter of a measure

In this section we describe the natural map which is an essential ingredient in the method of Besson-Courtois-Gallot.

Let ϕ denote the lift to universal covers of f with basepoint p ∈ Y (resp. f(p) ∈ X), i.e. ϕ = f̃ : Y → X. We will also denote the metric and Riemannian volume form on universal cover Y by g and dg respectively. Then for each s > h(g) and y ∈ Y consider the probability measure μ_{y}^{s} on Y in the Lebesgue class with density given by

\frac{dμ_{y}^{s}}{dg} (z) = \frac{e^{−sd(y,z)}}{∫_{Y} e^{−sd(y,z)}dg} .

The μ_{y}^{s} are well defined by the choice of s.

Consider the push-forward ϕ_{∗}μ_{y}^{s}, which is a measure on X. Define σ_{y}^{s} to be the convolution of ϕ_{∗}μ_{y}^{s} with the Patterson-Sullivan measure ν_{z} for the symmetric metric.

More precisely, for U ⊂ ∂X a Borel set, define

σ_{y}^{s}(U) = ∫_{X} ν_{z}(U)d(ϕ_{∗}μ_{y}^{s})(z)

Since ∥ν_{z}∥ = 1, we have

∥σ_{y}^{s}∥ = ∥μ_{y}^{s}∥ = 1.

Let B_{0}(x, θ) = B_{0}(ϕ(p), x, θ) be the Busemann function on X with respect to the basepoint ϕ(p) (which we will also denote by p). For s > h(g) and x ∈ X, y ∈ Y define a function

B_{s,y}(x) = ∫_{∂X} B_{0}(x, θ)dσ_{y}^{s}(θ)

By Theorem 3.3, the support of ν_{z}, hence of σ_{y}^{s}, is all of ∂_{F} X, which in turn equals the G-orbit G · b^{+}(∞), where G = Isom(X_{0}). Hence

B_{s,y}(x) = ∫_{∂_{F} X} B_{0}(x, θ)dσ_{y}^{s}(θ) = ∫_{G·ξ} B_{0}(x, θ)dσ_{y}^{s}(θ)

Since X is nonpositively curved, the Busemann function B_{0} is convex on X. Hence B_{s,y} is convex on X, being a convex integral of convex functions. While B_{0} is strictly convex only when X is negatively curved, we have the following.

Proposition 4.1 (Strict convexity of B). For each fixed y and s, the function x ↦ B_{y,s}(x) is strictly convex, and has a unique critical point in X which is its minimum.

Proof. For the strict convexity, it suffices to show that given a geodesic segment γ(t) between two points γ(0), γ(1) ∈ X, there exists some ξ ∈ ∂_{F} X such that function B_{0}(γ(t), ξ) is strictly convex in t, and hence on an open positive σ_{y}^{s}-measure set around ξ. We know it is convex by the comment preceding the statement of the proposition.

If B_{0}(γ(t), ξ) is constant on some geodesic subsegment of γ for some ξ, then γ must lie in some flat F such that the geodesic between ξ ∈ ∂F and γ (which meets γ at a right angle) also lies in F. On the other hand, ξ ∈ ∂_{F} X is in the direction of the algebraic centroid in a Weyl chamber, and γ is perpendicular to this direction. By the properties of the roots, γ is a regular geodesic (not contained in the boundary of a Weyl chamber). In particular, γ is contained in exactly one flat F. Furthermore, ∂_{F} X ∩ ∂F is a finite set (an orbit of the Weyl group). As a result, for almost every ξ ∈ ∂_{F} X B_{0}(γ(t), ξ) is strictly convex in t.

For fixed z ∈ X, by the last property listed in Proposition 3.1, we see that

∫_{∂_{F} X} B_{0}(x, θ)dν_{z}(θ)

tends to ∞ as x tends to any boundary point ξ ∈ ∂X. Then for fixed y and s > h(g), B_{y,s}(x) increases to ∞ as x tends to any boundary point ξ ∈ ∂X. Hence it has a local minimum in X, which by strict convexity must be unique. ⋄

We call the unique critical point of B_{s,y} the barycenter of the measure σ_{y}^{s}, and define a map F̃_{s} : Y → X by

F̃_{s}(y) = the unique critical point of B_{s,y}

Since for any two points p_{1}, p_{2} ∈ X

B_{0}(p_{1}, x, θ) = B_{0}(p_{2}, x, θ) + B_{0}(p_{1}, p_{2}, θ),

B_{s,y} only changes by an additive constant when we change the basepoint of B_{0}. This change does not affect the location of critical point of B_{y,s}. As a result, F̃_{s} is independent of choice of basepoints.

The equivariance of ϕ and of {μ_{y}} implies that F̃_{s} is also equivariant. Hence F̃_{s} descends to a map F_{s} : N → M. As in [BCG1], we will see that F_{s} is C^{1}, and will estimate its Jacobian.

5 The Jacobian estimate

The goal of this section is to prove a sharp estimate on the magnitude of the Jacobian of F_{s} : N → M.

We obtain the differential of F_{s} by implicit differentiation:

0 = D_{x=F_{s}(y)}B_{s,y}(x) = ∫_{∂_{F} X} dB_{0}_{(F_{s}(y),θ)}(·)dσ_{y}^{s}(θ).

Hence, as two forms,

0 = D_{y}D_{x=F_{s}(y)}B_{s,y}(x) = ∫_{∂_{F} X} DdB_{0}_{(F_{s}(y),θ)}(D_{y}F_{s}(·), ·)dσ_{y}^{s}(θ) (2)
−s ∫_{Y} ∫_{∂_{F} X} dB_{0}_{(F_{s}(y),θ)}(·) ⟨∇_{y}d_{Y} (y, z), ·⟩ dν_{ϕ(z)}(θ)dμ_{y}^{s}(z)  (3)

The distance function d_{Y} (y, z) is Lipschitz and C^{1} off of the cut locus which has Lebesgue measure 0. It follows from the Implicit Function Theorem (see [BCG2]) that F_{s} is C^{1} for s > h(g). By chain rule,

Jac F_{s} = s^{n} \begin{matrix}det & (∫_{Y} ∫_{∂_{F} X} dB_{0}_{(F_{s}(y),θ)}(·) ⟨∇_{y}d(y, z), ·⟩ dν_{ϕ(z)}(θ)dμ_{y}^{s}(z)) \\ det & (∫_{∂_{F} X} DdB_{0}_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ)) \end{matrix}.

Applying Hölder’s inequality to the numerator gives:

| Jac F_{s}| ≤ s^{n} det (∫_{∂_{F} X} dB_{0}_{(F_{s}(y),θ)}^{2}dσ_{y}^{s}(θ))^{1/2} det (∫_{Y} ⟨∇_{y}d_{Y} (y, z), ·⟩^{2} dμ_{y}^{s}(z))^{1/2} ; det (∫_{∂_{F} X} DdB_{0}_{(F_{s}(y),θ)}(D_{y}F_{s}(·), ·)dσ_{y}^{s}(θ)) .

Using that Tr ⟨∇_{y}d_{Y} (y, z), ·⟩^{2} = |∇_{y}d_{Y} (y, z)|^{2} = 1, except possibly on a measure 0 set, we may estimate

det (∫_{Y} ⟨∇_{y}d_{Y} (y, z), ·⟩^{2} dμ_{y}^{s}(z))^{1/2} ≤ (\frac{1}{\sqrt{n}})^{n} .

Therefore

| Jac F_{s}| ≤ (\frac{s}{\sqrt{n}})^{n} \begin{matrix}det & (∫_{∂_{F} X} dB_{0}_{(F_{s}(y),θ)}^{2}dσ_{y}^{s}(θ)) & 1/2 \\ det & (∫_{∂_{F} X} DdB_{0}_{(F_{s}(y),θ)}(D_{y}F_{s}(·), ·)dσ_{y}^{s}(θ)) \end{matrix}. (4)

Theorem 5.1 (The Jacobian Estimate). For all s > h(g) and all y ∈ N we have:

| Jac F_{s}(y)| ≤ (\frac{s}{h(g_{min})})^{n}

with equality at any y ∈ N if and only if D_{y}F_{s} is a homothety.

As discussed in the introduction, the main idea is to use the fact that the measures σ_{y}^{s} on ∂X are supported on ∂_{F} X, which decomposes as a product of Furtsenberg boundaries of the rank one factors of X. This can then be used to reduce the estimate on | Jac F_{s}(y)| to the rank one case and an algebraic lemma of [BCG1].

Proof. Item (1) is clear; we prove item (2).

By the hypothesis on X, the group G = Isom(X) can be written as a product G = G_{1} × G_{2} ⋯ × G_{rk(X)}, where each G_{i} ≠ SL(2, R) is a simple rank one Lie group. Theorem 3.3 states that there exists ξ ∈ ∂X so that for all y ∈ Y, the measure σ_{y}^{s} is supported on some G-orbit

G · ξ = {(G_{1} × G_{2} ⋯ × G_{rk(X)}) · ξ}

Hence

∂_{F} X = G · ξ = ∂_{F} X_{1} × · × ∂_{F} X_{rk(X)}

Since each X_{i} has rank one, ∂_{F} X_{i} = ∂X_{i} so that

∂_{F} X = ∂X_{1} × ⋯ × ∂X_{rk(X)}

Let B_{i} denote the Busemann function for the rank one symmetric space X_{i} with metric g_{i}. Then for θ_{i} ∈ ∂X_{i} ⊂ ∂X and x, y ∈ X_{i} we have B_{0}(x, y, θ_{i}) = α_{i}B_{i}(x, y, θ_{i}). Since the factors X_{i} are orthogonal in X with respect to the metric g_{min}, the Busemann function of (X, g_{min}) with basepoint p ∈ X at a point θ = (θ_{1}, …, θ_{rk(X)}) ∈ ∂_{F} X is given by

B_{0}(x, θ) = ∑_{i=1}^{rk(X)} \frac{α_{i}}{\sqrt{rk(X)}}B_{i}(x_{i}, θ_{i}).

Since ∇_{x}^{g_{min}}B_{i} = \frac{1}{α_{i}^{2}} ∇_{x}^{g_{i}}B_{i}, we may verify that |∇_{x}^{g_{min}}B_{0}(x, θ)|_{g_{min}}^{2} = 1. Similarly,

∇_{x}^{g_{min}}B_{0} = ∑_{i=1}^{rk(X)} \frac{1}{α_{i}\sqrt{rk(X)}}∇_{x}^{g_{i}}B_{i}

Differentiating again with respect to a fixed orthonormal basis of T_{x}X with respect to g_{min} we obtain,

Hess_{x}^{g_{min}}B_{0} = ⊕_{i=1}^{rk(X)} \frac{1}{α_{i}\sqrt{rk(X)}}Hess_{x}^{g_{i}}B_{i}

Since σ_{y}^{s} is supported on G_{1} × ⋯ × G_{rk(X)} · ξ, we can use product coordinates on X to write the right hand side of (4) as,

(\frac{s}{\sqrt{n}})^{n} det (∫_{∂_{F} X} (∑_{i=1}^{rk(X)} ∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}) (∑_{i=1}^{rk(X)} ∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})^{∗} dσ_{y}^{s}(θ))^{\frac{1}{2}} ; det (∫_{∂_{F} X} ⊕_{i=1}^{rk(X)}Hess_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}dσ_{y}^{s}(θ))

where the superscript ∗ means transpose and we have cancelled the factor

(\frac{1}{α_{i}\sqrt{rk(X)}})^{n} from the numerator and denominator.

Recall the following estimate for positive semi-definite block matrices,

det (\begin{matrix}A & B \\ B^{∗} & C \end{matrix}) ≤ det(A) det(C)

Then since the matrix

∫_{∂_{F} X} ( ∑_{i=1}^{rk(X)} ∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}) ( ∑_{i=1}^{rk(X)} ∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})^{∗} dσ_{y}^{s}(θ)

is positive semi-definite, by iteratively using the above estimate (on subblocks) we obtain

det (∫_{∂_{F} X} ( ∑_{i=1}^{rk(X)} ∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}) ( ∑_{i=1}^{rk(X)} ∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})^{∗} dσ_{y}^{s}(θ)) ≤

∏_{i=1}^{rk(X)} det (∫_{∂_{F} X}(∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})(∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})^{∗}dσ_{y}^{s}(θ))

Also, we have

det (∫_{∂_{F} X} ⊕_{i=1}^{rk(X)}Hess_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}dσ_{y}^{s}(θ)) = ∏_{i=1}^{rk(X)} det (∫_{∂_{F} X} Hess_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}dσ_{y}^{s}(θ))

Hence we have

| Jac F_{s}(y)| ≤ (\frac{s}{\sqrt{n}})^{n} ∏_{i=1}^{rk(X)} det (∫_{∂_{F} X} (∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}) (∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})^{∗} dσ_{y}^{s}(θ))^{1/2} ; det (∫_{∂_{F} X} Hess_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}dσ_{y}^{s}(θ))

Since B_{i} is the Busemann function of rank one symmetric space X_{i}, it follows that, setting the tensor H_{i} = (∇_{(F_{s}(y),θ)}^{g_{i}}B_{i})^{2}, we have

Hess_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i} = I −H_{i} − ∑_{k=1}^{d_{i}−1} J_{k}H_{i}J_{k}

where the J_{k} are the matrices representing the underlying complex structure of (the division algebra corresponding to) the symmetric space. Lemma 5.5 of [BCG2] says that for any n × n matrix H with Tr H = 1, the following holds:

\frac{det H^{1/2}}{det (I −H − ∑_{k=1}^{d_{i}−1} J_{k}HJ_{k})} ≤ (\frac{\sqrt{n}}{n + d − 2})^{n}

with equality if and only if H = \frac{1}{n} I. Applying this estimate to each term of the above product now gives

det (∫_{∂_{F} X} (∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}) (∇_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i})^{∗} dσ_{y}^{s}(θ))^{1/2} ; det (∫_{∂_{F} X} Hess_{(F_{s}(y),θ_{i})}^{g_{i}}B_{i}dσ_{y}^{s}(θ)) ≤ (\frac{\sqrt{n_{i}}}{n_{i} + d_{i} − 2})^{n_{i}}

with equality if and only if H_{i} = \frac{1}{n_{i}} I. Hence

| Jac F_{s}(y)| ≤ (\frac{s}{\sqrt{n}})^{n} ∏_{i=1}^{rk(X)} (\frac{\sqrt{n_{i}}}{n_{i} + d_{i} − 2})^{n_{i}} = \frac{s^{n}}{h(g_{min})^{n}} .

If equality is attained then we have equality for each term. Let Q_{i} ⊂ T_{y}Y be the subspace which is mapped by D_{y}F_{s} onto T_{F_{s}(y)}X_{i}. So for any v_{i} ∈ T_{F_{s}(y)}X_{i} and any u_{i} ∈ Q_{i} we have

|⟨D_{y}F_{s}(u_{i}), v_{i}⟩_{g_{i}} | ≤ \frac{\sqrt{n}\sqrt{n_{i}}s∥v_{i}∥_{g_{i}}}{n_{i} + d_{i} − 2} (∫_{∂_{F} X} dB_{y,ϕ^{−1}(θ)}(u_{i})^{2}dσ_{y}^{s}(θ))^{1/2}

Recall that α_{i} = \frac{h_{i}}{\sqrt{n_{i}}} ∏_{i=1}^{rk(X)} (\frac{\sqrt{n_{i}}}{h_{i}} )^{\frac{n_{i}}{n}} . So multiplying each side by α_{i}^{2} we write the above with respect to the metric g_{min}:

|⟨D_{y}F_{s}(u_{i}), v_{i}⟩_{g_{min}} | ≤ \frac{α_{i}\sqrt{n}\sqrt{n_{i}}s∥v_{i}∥_{g_{min}}}{h_{i}} (∫_{∂_{F} X} dB_{y,ϕ^{−1}(θ)}(u_{i})^{2}dσ_{y}^{s}(θ))^{1/2}

= \sqrt{n}∥v_{i}∥_{g_{min}} \frac{s}{h(g_{min})} (∫_{∂_{F} X} dB_{y,ϕ^{−1}(θ)}(u_{i})^{2}dσ_{y}^{s}(θ))^{1/2} .

Hence for all u ∈ T_{y}Y we obtain

∥D_{y}F_{s}(u)∥_{g_{min}} ≤ \sqrt{n}\frac{s}{h(g_{min})} (∫_{∂_{F} X} dB_{y,ϕ^{−1}(θ)}(u)^{2}dσ_{y}^{s}(θ))^{1/2} .

Now we follow Section 5 of [BCG2]. It follows that for any orthonormal basis {v_{i}} of T_{F_{s}(y)}X we have

Tr (D_{y}F_{s}^{∗}) ∘ (D_{y}F_{s}) = ∑_{i=1}^{n} ⟨D_{y}F_{s}(v_{i}), D_{y}F_{s}(v_{i})⟩_{g_{min}} ≤ n (\frac{s}{h(g_{min})})^{2} .

Lastly it follows that

(\frac{s}{h(g_{min})})^{2} = | Jac F_{s}|^{2} = det(D_{y}F_{s})^{∗} ∘ (D_{y}F_{s})

≤ (\frac{Tr (D_{y}F_{s})^{∗} ∘ (D_{y}F_{s})}{n}) ≤ (\frac{s}{h(g_{min})})^{2} .

The equality implies that

(D_{y}F_{s})^{∗} ∘ (D_{y}F_{s}) = (\frac{s}{h(g_{min})})^{2} I .

Hence D_{y}F_{s} is a homothety of ratio \frac{s}{h(g_{min})} as desired. ⋄

Remark. The equality case in item (3) will not be used in what follows, but will provide insight into what happens as we take limits s → h(g).

6 Finishing the proof of Theorem A

We now prove the statement given in Remark 1 after Theorem A, from which the theorem immediately follows. We first establish that F_{s} is homotopic to f:

Proposition 6.1. For any s > h(g), the map Ψ_{s}: [0, 1] × N → M defined by

Ψ_{s}(t, y) = F_{s+ \frac{t}{1−t}} (y)

is a homotopy between Ψ_{s}(0, ·) = F_{s} and Ψ_{s}(1, ·) = f.

Proof. From its definitions, F̃_{s}(y) is continuous in s and y. Observe that for fixed y, lim_{s→∞} σ_{y}^{s} = ν_{ϕ(y)}. It follows that lim_{s→∞} F̃_{s}(y) = ϕ(y). This implies the proposition. ⋄

6.1 The compact case

Suppose M and N are compact. Since for s > h(g), F_{s} is a C^{1} map, we may simply compute the following using elementary integration theory:

|deg(f)| Vol(M) = |deg(f)| ∫_{M} dg_{min} = |∫_{N} f^{∗}dg_{min}|
≤ ∫_{N} |F_{s}^{∗}dg_{min}| = ∫_{N} |Jac F_{s}dg|
= ∫_{N} |Jac F_{s}| dg ≤ (\frac{s}{h(g_{min})})^{n} Vol(N) (5)

Letting s → h(g) gives the inequality in Theorem A. In the case when equality is achieved, after scaling the metric g by the constant \frac{h(g)}{h(g_{min})}, we have h(g) = h(g_{min}) and Vol(N) = |deg(f)| Vol(M).

We remark that in the notation used in [BCG1] for the sequence of Lemmas 7.2-7.9 and Proposition 8.2, the measure Φ^{2}(y, θ)dθ is simply σ_{y}^{s} in the rank one case (recall the Poisson kernel is p_{0}(y, θ) = e^{−h(g)B_{0}(y,θ)}).

In higher rank this equivalence still holds so long as we replace dθ by the Patterson-Sullivan measure ν_{p} at the basepoint p. The proofs of Lemmas 7.2-7.9 and Proposition 8.2 in [BCG1] then hold in our more general context once we replace the visual boundary by the Furstenberg boundary and dθ by the measure ν_{p} as the analysis is the same. These show that F_{s} converges to a nonexpanding Lipschitz map F as s → h(g). Applying Theorem C.1 from Appendix C of [BCG1] we obtain Theorem A.

6.2 The noncompact case

We now consider the case when M has finite volume but is not compact. In this setting, it is not known whether the limit in the definition of h(g) always exists. For this reason we define the quantity h(g) to be

h(g) = inf {s ≥ 0 | ∃C > 0 such that ∀x ∈ X, ∫_{X} e^{−sd(x,z)}dg(z) < C } .

In fact this agrees with the previous definition for h(g) when M is compact.

If N has infinite volume the main theorem is automatically satisfied so long as h(g) > 0. The main difficulty is that, in order for the proof given above to work, we need to know that F_{s} is proper (and thus surjective since deg(F_{s}) = deg(f) ≠ 0). For this, we will need to prove higher rank analogs of some lemmas used in [BCS] for the rank one case. For the basics of degree theory for proper maps between noncompact spaces, see [FG].

We will show that F_{s} is proper by essentially showing that the barycenter of σ_{x}^{s} lies nearby a convex set containing large mass for this measure. This convex set is in turn far away from ϕ(p) whenever x is far from p ∈ Y. We achieve this by first estimating the concentration of the mass of σ_{x} in certain cones which will be our convex sets. One difficulty that arises in the higher rank is that these cones must have a certain angle when restricted to a flat. Another difficulty is that the ends of M can have large angle at infinity. In fact our methods breakdown unless we control the asymptotic expansion of f down the ends (see Remarks 6.2).

First, we localize the barycenter of the measure σ_{x}^{s}. Let v_{(x,θ)} be the unit vector in S_{x}X pointing to θ ∈ ∂X.

Lemma 6.2. Let K ⊂ X and y ∈ Y be such that (ϕ_{∗}μ_{y}^{s})(K) > C for some constant 1 > C > \frac{1}{2}. Suppose that for all x ∈ X there exists v ∈ S_{x}X such that for all z ∈ K:

∫_{∂_{F} X} ⟨v_{(x,θ)}, v⟩ dν_{z}(θ) ≥ \frac{1}{C} − 1

Then

x ≠ F̃_{s}(y)

Proof. If F̃_{s}(y) = x then ∇_{x}B_{s,y}(x) = 0. However, ∇_{x}B_{s,y}(x) may be expressed as

∫_{X} ∫_{∂_{F} X} v_{(x,θ)}dν_{z}(θ)dϕ_{∗}μ_{y}^{s}(z)

where v_{(x,θ)} is the unit vector in S_{x}X pointing to θ ∈ ∂_{F} X. Then we have

∥D_{x}B_{s,y}∥ = ∥∫_{X} ∫_{∂_{F} X} v_{(x,θ)}dν_{z}(θ)dϕ_{∗}μ_{y}^{s}(z)∥

≥ ∥∫_{K} ∫_{∂_{F} X} v_{(x,θ)}dν_{z}(θ)dϕ_{∗}μ_{y}^{s}(z)∥ −

∥∫_{X−K} ∫_{∂_{F} X} v_{(x,θ)}dν_{z}(θ)dϕ_{∗}μ_{y}^{s}(z)∥

≥ ∫_{K} ∫_{∂_{F} X} ⟨v_{(x,θ)}, v⟩ dν_{z}(θ)dϕ_{∗}μ_{y}^{s}(z) − ϕ_{∗}μ_{y}^{s}(X − K)

≥ ϕ_{∗}μ_{y}^{s}(K) (\frac{1}{C} − 1) − 1 + ϕ_{∗}μ_{y}^{s}(K)

> C (\frac{1}{C} − 1) − 1 + C = 0

The strictness of the inequality finishes the proof. ⋄

For v ∈ SX and α > 0 consider the convex cone,

E_{(v,α)} = exp_{π(v)} {w ∈ T_{π(v)}X | ∠_{π(v)}(v(∞), w(∞)) ≤ α} ,

where π : T X → X is the tangent bundle projection.

Denote by ∂E_{(v,α)} ⊂ ∂X its boundary at infinity.

Lemma 6.3. There exists T_{0} > 0 and α_{0} > 0 such that for all t ≥ T_{0}, all x ∈ X, all v ∈ S_{x}X and all z ∈ E_{(g^{t}v,α_{0})},

∫_{∂_{F} X} ⟨v_{(x,θ)}, v⟩ dν_{z}(θ) ≥ \frac{\sqrt{2}}{3}.

Proof. Since the isometry group of the symmetric space X is transitive on X and for any isometry ψ, dψ(E_{(v,α)}) = E_{(dψ(v),α)}, it is sufficient to prove the lemma for a fixed x and all v ∈ S_{x}X.

For now choose α_{0} < π/4. Take a monotone sequence t_{i} → ∞, and any choice z_{i} ∈ E_{(g^{t_{i}}v,α)} for each t_{i}. It follows that some subsequence of the z_{i}, which we again denote by {z_{i}}, must tend to some point θ ∈ ∂E_{(v,α)}.

Let ν_{θ} be the weak limit of the measures ν_{z_{i}}. From Theorem 3.3, ν_{θ} is a probability measure supported on a set S_{θ} satisfying

∠_{x}(θ, ξ) ≤ \frac{π}{4} ∀ξ ∈ S_{θ}.

Therefore we have,

∫_{S_{θ}} ⟨v_{(x,ξ)}, v_{(x,θ)}⟩ dν_{θ}(ξ) ≥ \frac{\sqrt{2}}{2}  (6)

Now whenever θ ∈ ∂E_{(v,α)} then v = v_{(x,θ)} + ϵv′ for some unit vector v′ and ϵ ≤ sin(α). Using either case above we may write

∫_{∂_{F} X} ⟨v_{(x,ξ)}, v⟩ dν_{θ}(ξ) ≥ ∫_{∂_{F} X} ⟨v_{(x,ξ)}, v_{(x,θ)}⟩ dν_{θ}(ξ) − sin(α).

So choosing α small enough we can guarantee that

1. any two Weyl chambers intersecting E_{(g^{t}v,α)} for all t > 0 in the same flat must share a common face of dimension rk(M) − 1, and

2. for any θ ∈ ∂E_{(v,α)},

∫_{∂_{F} X} ⟨v_{(x,ξ)}, v⟩ dν_{θ}(ξ) ≥ \frac{\sqrt{2}}{2.5}.

Let

E_{(v(∞),α)} = ∩_{t>0}∂E_{(g^{t}v,α)}.

By the first property used in the choice of α above, for any two points θ_{1}, θ_{2} ∈ E_{(v(∞),α)}, either θ_{1} and θ_{2} are in the boundary of the same Weyl chamber, or else there is another point θ′ in the intersection of the boundaries at infinity of the closures of the respective Weyl chambers.

By maximality there is some θ_{0} ∈ E_{(v(∞),α)} intersecting the boundary at infinity of the closure of every Weyl chamber which intersects E_{(g^{t}v,α)} for all t > 0. Hence, for every θ ∈ E_{(v(∞),α)}, the support of the limit measure ν_{θ} satisfies S_{θ} ⊂ S_{θ_{0}}. (While θ_{0} is not necessarily unique, the support S_{θ_{0}} of the corresponding limit measure ν_{θ_{0}} is.)

As t increases, for any z ∈ E_{(g^{t}v,α)}, the measures ν_{z} uniformly become increasingly concentrated on S_{θ_{0}}. Then applying the estimate (6) to θ = θ_{0}, we may choose T_{0} sufficiently large so that for all z ∈ E_{(g^{t}v,α)} with t > T_{0},

∫_{∂_{F} X} ⟨v_{(x,ξ)}, v⟩ dν_{z}(ξ) ≥ \frac{\sqrt{2}}{3}.

⋄

Proposition 6.4. F_{s} is proper.

Proof. By way of contradiction, let y_{i} ∈ Y be an unbounded sequence such that {F̃_{s}(y_{i})} lies in a compact set K. We may pass to an unbounded subsequence of {y_{i}}, which we again denote as {y_{i}}, such that the sequence ϕ(y_{i}) converges within a fundamental domain for π_{1}(M) in X to a point θ_{0} ∈ ∂X. Since K is compact, the set

A = ⋂_{x∈K} E_{(g^{T_{0}}v_{(x,θ_{0})},α_{0})}

contains an open neighborhood of θ_{0} and d_{X}(A, K) ≥ T_{0}. Notice that A is itself a cone, being the intersection of cones on a nonempty subset of ∂X.

We now show that A contains the image ϕ(B(y_{i}, R_{i})) of increasingly large balls (R_{i} → ∞). However, we observe from the fact that A is a cone on an open neighborhood of θ_{0} in ∂X that A contains balls B(ϕ(y_{i}), r_{i}) with r_{i} → ∞. By assumption f, and hence ϕ, is coarsely Lipschitz:

d_{X}(ϕx, ϕy) ≤ Kd_{Y} (x, y) + C

for some constants C > 0 and K ≥ 1. Therefore ϕ^{−1}(B(ϕ(y_{i}), r_{i})) ⊃ B(y_{i}, R_{i}) where KR_{i} + C > r_{i}. In particular R_{i} → ∞.

Hence, there exists an unbounded sequence R_{i} such that B(y_{i}, R_{i}) ⊂ ϕ^{−1}(A). Furthermore, since Y has bounded geometry, we have that Vol(B(y_{i}, injrad)) is greater than some constant independent of y_{i} and hence ∫_{Y} e^{−sd(y_{i},z)}dg(z) > Q for some constant Q > 0. By choice of s there is a constant C_{s} depending only on s such that ∫_{Y} e^{−sd(y,z)}dg(z) < C_{s} for all y ∈ Y.

In polar coordinates we may write,

∫_{Y} e^{−sd(y,z)}dg(z) = ∫_{0}^{∞} e^{−st} Vol(S(y, t))dt

= ∫_{0}^{∞} e^{−st} \frac{d}{dt} Vol(B(y, t))dt

= − ∫_{0}^{∞} \frac{d}{dt} (e^{−st}) Vol(B(y, t))dt

= s ∫_{0}^{∞} e^{−st} Vol(B(y, t))dt.

Using this we may estimate, using any δ < s − h(g),

μ_{y_{i}}^{s}(ϕ^{−1}(A)) > μ_{y_{i}}^{s}(B(y_{i}, R_{i}))

= 1 − \frac{∫_{R_{i}}^{∞} e^{−st} Vol(B(y_{i}, t))dt}{∫_{0}^{∞} e^{−st} Vol(B(y_{i}, t))dt}

≥ 1 − \frac{e^{−δR_{i}} ∫_{R_{i}}^{∞} e^{−(s−δ)t} Vol(B(y_{i}, t))dt}{∫_{0}^{∞} e^{−st} Vol(B(y_{i}, t))dt}

≥ 1 − e^{−δR_{i}} \frac{C_{s−δ}}{Q} .

Therefore for all sufficiently large i,

μ_{y_{i}}^{s}(ϕ^{−1}(A)) > \frac{3}{3 + \sqrt{2}}.

The constant \frac{3}{3+\sqrt{2}} is the constant C from Lemma 6.2 such that \frac{1}{C} −1 = \frac{\sqrt{2}}{3}.

Set v_{i} = g^{T_{0}+1}v_{(F̃_{s}(y_{i}),θ_{0})}. Recalling that A ⊂ E_{(v_{i},α_{0})} for all i, we have that for sufficiently large i,

ϕ_{∗}μ_{y_{i}}^{s}(E_{(v_{i},α_{0})}) > \frac{3}{3 + \sqrt{2}}

but d_{X}(F̃_{s}(y_{i}), E_{(v_{i},α_{0})}) > T_{0}, contradicting the conclusion of Lemma 6.2 in light of Lemma 6.3. ⋄

Remark. Ideas from coarse topology can be used to remove the coarse lipschitz assumption on f in the case that the ends of M have angle at infinity bounded away from π/2. Unfortunately, M may have ends containing pieces of flats with wide angle (consider the product of two rank one manifolds each with multiple cusps, or for a complete classification of higher rank ends see [Ha]). For such spaces it is possible to construct a proper map f : M → M such that for a radial sequence y_{i} → ∞, ϕ maps the bulk of the mass of μ_{y_{i}}^{s} into a set (almost) symmetrically arranged about the point p ∈ X thus keeping F̃_{s}(y_{i}) bounded. This explains the need for a condition on f akin to the coarse lipschitz hypothesis.

The inequality in Theorem A now follows as in the compact case, with deg(f) and deg(F_{s}) suitably interpreted.

Now we complete the proof of the rigidity when we have equality in Theorem A. First we assume that h(g) = h(g_{0}) by scaling the metric g by the constant \frac{h(g_{0})}{h(g)}. We note that the proofs of the lemmas in Section 7 of [BCG1] (done for the case f = Id) are identical so long as we restrict the uniformity of Lemmas 7.5 and 7.6 to be uniform only on compact subsets. These proofs go through with only minor modification in the case that f has (local) degree deg(f) ≠ 1; this is explained in Section 8.2 of [BCG1]. In this case we obtain the general versions of Lemma 7.6 and 7.7 of [BCG1],

Lemma 6.5. There is a subsequence s_{i} such that the maps F_{s_{i}} converges uniformly on compact sets to a continuous map F : N → M such that ||d_{y}F_{s_{i}}|| is uniformly bounded on compact subsets of N and converges to 1 almost everywhere.

The proof of Lemma 7.8 of [BCG1] then goes through without modification to obtain

Lemma 6.6. The map F is Lipschitz with Lipschitz constant less than or equal to one.

Before we proceed we must show:

Lemma 6.7. The map F is proper.

Proof. By the previous lemma, F is a contracting Lipschitz map. We note that the local notion of deg(F) given by

degF (x) = ∑_{y∈F^{−1}(x)} sign(Jac F (y))

is well defined for a.e. x ∈ N. Let P ⊂ M be the set of points which have unbounded preimage under F. The set P is clearly closed and of measure 0 since F is Lipschitz with Jacobian equal to 1 almost everywhere and N has finite volume. Clearly, F acts properly on M \ P.

Since F is homotopic to f, degF (x) = degf(x) for a.e. x in one (nonempty) connected component U of M \ P. We note that Lemma C.2 and C.4 in [BCG1] only require that the injectivity radius be bounded for any finite set of points. The proof of these two lemmas show that for every x ∈ U, card(F^{−1}(x) ≤ deg(f).

Now we will show that P is empty and therefore U = M and F is proper. If P is nonempty then there exists a finite length absolutely continuous

curve c : I → M such that c([0, 1)) ⊂ U and c(1) = x_{0} for some point x_{0} ∈ P. For example, c could be taken to be the shortest geodesic from some fixed point in U to P.

By Fubini’s theorem we may assume that c was chosen so that F restricted to the set F^{−1}(c) has derivative with norm 1 almost everywhere. If not, some absolutely continuous perturbation of c in the open set U with fixed endpoint x_{0} must satisfy this condition since D_{x}F has norm 1 almost everywhere.

Now observe that since c([0, 1)) ⊂ U, F^{−1}(c) is the (not necessarily disjoint) union of at most deg(f) curves c_{1}, …, c_{deg(f)} with F (c_{i}) = c for i = 1, …, deg(f). Since F is Lipschitz with Lipschitz constant 1 a.e. on F^{−1}(c), each c_{i} is absolutely continuous with length at most that of c (e.g. by the fundamental theorem of calculus). In particular, the 1 dimensional Hausdorff measure of F^{−1}(c) does not exceed deg(f) length(c) and so F^{−1}(c) is bounded. However, this contradicts that c(1) = x_{0} ∈ P. ⋄

To complete the equality case (rigidity) we must prove the following.

Proposition 6.8. Consider two n-dimensional complete oriented Riemannian manifolds of finite volume, N and M. Suppose F : N → M is a proper Lipschitz map satisfying d_{M}(F (x), F (y)) ≤ d_{N}(x, y) for all x, y ∈ M. Then if Vol(N) = |degf| Vol(M), the map F is a Riemannian covering homotopic to f.

We establish this following Appendix C of [BCG1] through two lemmas. If we set C(x) = card {F^{−1}x} The proof of Lemma C.2 in [BCG1]

establishes the following.

Lemma 6.9. For almost every y ∈ N, and a.e. x ∈ M, we have C(x) = deg(f) and D_{y}F is an isometry between T_{y}N and T_{F (y)}M.

Since F is proper, the preimage set {F^{−1}(x)} is compact and hence lies in a region with injectivity radius bounded from below. The rest of the proofs of Appendix C can then be followed verbatim to obtain Lemma 6.10. For every y ∈ N, C(y) = deg(f) and F is a local isometry.

This lemma, in particular implies that F is a Riemannian covering map. Taking limits in Proposition 6.1 establishes that F is homotopic to f and hence the Proposition follows. This also completes the proof of Theorem A.

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Christopher Connell: Dept. of Mathematics, University of Illinois at Chicago Chicago, IL 60680 E-mail: cconnell@math.uic.edu

Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu