The Degree Theorem in higher rank

Christopher Connell and Benson Farb ^{∗}

Final revision, to appear in J. Diff. Geom.

1 Introduction

The problem of relating volume to degree for maps between Riemannian manifolds is a fundamental one. Gromov’s Volume Comparison Theorem [Gr] gives such a relation for maps into negatively curved manifolds. In this paper we extend Gromov’s theorem to locally symmetric manifolds of nonpositive curvature. We derive this as a consequence of the following result, which we believe to be of independent interest.

Theorem 1.1 (Universal Jacobian bound). Let M be a closed, locally symmetric n-manifold with nonpositive sectional curvatures. Assume that M has no local direct factors locally isometric to R, H^{2}, or SL_{3}(R)/ SO_{3}(R). Then for any closed Riemannian manifold N and any continuous map f : N → M, there exists a constant C > 0, depending only on n and the smallest Ricci curvatures of N and M, so that f is homotopic to a C^{1} map F : N → M satisfying

| Jac F | ≤ C

Remark. By scaling the metrics it is easy to see that the dependence of the constant C on the smallest curvatures cannot be removed. Actually, we determine the constant explicitly in terms of the symmetric space and the volume entropy of N (see §2.1).

Theorem 1.1 together with a simple degree argument (see §6) gives the following generalization of Gromov’s Volume Comparison Theorem.

∗Both authors are supported in part by the NSF. They would like to thank Burt Totaro for useful comments.

Theorem 1.2 (The Degree Theorem). Let M be as in Theorem 1.1. Then for any closed Riemannian manifold N and any continuous map f : N → M,

deg(f) ≤ C \frac{Vol(N)}{Vol(M)}

where C > 0 depends only on n and the smallest Ricci curvatures of N and M.

Remarks.

1. As tori have self-maps of arbitrary degree, it is easy to see that the theorem would be false without the “no R factors” hypothesis. We believe that the “no H^{2} or SL_{3}(R)/ SO_{3}(R) local factors” hypothesis is unnecessary; we show in Example 4.6 below and in §6 of [CF2], however, that the issue is rather delicate.

2. As with Theorem 1.1, the dependence of the constant C on the smallest curvatures cannot be removed; this dependence is determined explicitly in §2.1.

3. In §6.2 we extend Theorem 1.2 to the case where N and M have finite volume (with “bounded geometry”) but are not necessarily compact, and where f is a coarse Lipshitz map.

When dim(M) = 2 the conclusion of the theorem follows easily from the Gauss-Bonnet Theorem. More generally, when M is any closed manifold with positive Gromov norm, Gromov has shown (see [Gr], p.8 and the Main Inequality on p.12) that a degree theorem as in Theorem 1.2 holds for M. Positivity of the Gromov norm for closed, locally symmetric M as in Theorem 1.2 is still an open question. However, this positivity was proved by Savage [Sa] for closed M locally isometric to SL_{n}(R)/ SO_{n}(R).

When rank(M) = 1, Besson-Courtois-Gallot [BCG1] proved the stronger entropy rigidity theorem, giving the exact best constant C. Entropy rigidity is still open in higher rank ^{1}; this would correspond to the above theorem with the constant C in the inequality being C = (\frac{h(g)}{h(g_{0})})^{n}, where h(g) and h(g_{0}) are the volume entropies of N and M (see [BCG1]), with equality being obtained iff N is a Riemannian cover of M after an appropriate rescaling.

1Entropy rigidity has recently been proved [BCG3, CF1] for manifolds locally modelled on products of rank one symmetric spaces with no H^{2} factors.

The Besson-Courtois-Gallot technique is a central ingredient here; indeed the main idea in our proof of Theorem 1.1 is to establish a higher rank version of the “canonical map” of [BCG1], and to give an a priori bound on its Jacobian. Obtaining this bound is the hardest part of the present paper (see §4 and §5). Our estimates in §4 and §5 can be viewed as a first step towards proving higher rank entropy rigidity.

The Minvol invariant. One of the basic invariants associated to a smooth manifold M is its minimal volume:

Minvol(M) := inf_{g}{Vol(M, g) : |K(g)| ≤ 1}

where g ranges over all smooth metrics on M and K(g) denotes the sectional curvature of g. The basic questions about Minvol(M) are: for which M is Minvol(M) > 0? When is Minvol(M) realized by some metric g?

When a nonpositively curved manifold M has a local direct factor locally isometric to R, it is easy to see that Minvol(M) = 0. By taking f to be the identity map (while varying the metric g on M), Theorem 1.2 immediately gives:

Corollary 1.3 (Positivity of Minvol). Let M be any finite volume, locally symmetric n-manifold (n ≥ 2) of nonpositive curvature. If M has no local direct factors locally isometric to R, H^{2}, or SL_{3}(R)/ SO_{3}(R), then Minvol(M) > 0.

For compact M, Corollary 1.3 was proved (without the H^{2} and SL(3, R) restriction) in [Gr2] (see also [Sa] for the case manifolds locally modelled on the symmetric space for SL(n, R)). When M admits a (real) hyperbolic metric, Besson-Courtois-Gallot [BCG1] proved that Minvol(M) is uniquely realized by the hyperbolic metric. It seems possible that this might hold in general.

Self maps and the co-Hopf property. As deg(f^{n}) = deg(f)^{n}, an immediate corollary of Theorem 1.2 is the following.

Corollary 1.4 (Self maps). Let M be a finite-volume locally symmetric manifold as in Theorem 1.1. Then M admits no self-maps of degree > 1. In particular, π_{1}(M) is co-Hopfian: every injective endomorphism of π_{1}(M) is surjective.

Note that Corollary 1.4 may also be deduced from Margulis’s Superrigidity theorem (for higher rank M). The co-Hopf property for lattices was first proved by Prasad [Pr].

More generally, if N and M are as in Theorem 1.2 and f : N → M and g : M → N are two maps of nonzero degree then |deg(f)| = 1 = |deg(g)| since f ∘ g is a self map of M.

Outline of the proof of Theorems 1.1 and 1.2. As noted above, a simple degree argument shows that it is enough to prove Theorem 1.1. Given f : N → M as in the hypothesis of the theorems, we use the method of [BCG1, BCG2] to construct a “canonical” map F : N → M which is homotopic to f (hence degF = degf) and has universally bounded Jacobian.

Step 1 (Constructing the map): First consider the case when the metric on N is nonpositively curved. 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 of the universal covers Y, X.

Morally what we do, following the method of [BCG2], is 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 §3). The inclusion Y → M(∂Y), denoted x ↦ μ_{x}, is given by the construction of the Patterson-Sullivan measures {μ_{x}}_{x∈X} corresponding to π_{1}(N) < Isom(Y) (see §2). 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 construction outline is that the metric on Y may not be nonpositively curved. So we must find an alternative to using the “visual boundary” of Y. This is done by constructing a certain family of smooth measures μ_{s} on Y itself, pushing them forward via f̃, and convolving with Patterson-Sullivan measure on X. Maps F̃_{s} are then defined by taking the barycenters of these measures; it is actually these maps which are considered instead of F. This idea was first introduced in [BCG1].

Two new features of F appear in higher rank. First, the non-strictness of convexity of the Busemann function (see §3) must be overcome to define F. Second, and more importantly, a theorem of Albuquerque shows that the support of each of the measures μ_{x} is codimension rank(X)−1 subset of ∂X called the Furstenberg boundary of X (see §2). This fact and its implications are crucial for later steps.

Step 2 (The Jacobian estimate): The heart of the paper ( §4 and §5) is obtaining a universal bound on F, independent of f. For this, we first realize the Jacobian of F as the ratio of determinants of two matrix integrals. We then show that whenever there are small eigenvalues in the denominator there are a sufficient number of small eigenvalues in the numerator with which to cancel them. The key is to find these eigenvalues independently of the integrating measure (which depends on μ_{s}), therefore reducing the problem to a problem about semisimple Lie groups.

Step 3 (Finishing the proof): Once a universal bound on | Jac(F)| is found, a simple degree argument finishes the proof. In the case when M and N are not compact, the main difficulty is proving that F_{s} is proper. This is quite technical, and requires extending some of the ideas of [BCS] to the higher rank setting.

2 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].

2.1 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.

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. 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. 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.

Let Γ be a lattice in Isom(X), so that h(g_{0}) < ∞ where (M, g_{0}) is Γ\X with the locally symmetric metric.

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

Theorem 2.1 (Existence Theorem, [Al]). There exists a family {ν_{x}} of probability measures on ∂X satisfying 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_{0})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.

The third property implies no two measures are the same as measures. 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.

2.2 Symmetric spaces of noncompact type

Before we present Albuquerque’s theorem we will need some necessary background about higher rank symmetric spaces.

By definition, the symmetric space X is G/K where G is a semisimple Lie group and K a maximal compact subgroup. 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 orthogonal to 𝔨 with respect to the killing form B(·, ·) on 𝔤. Therefore, 𝔭 is also identified with the tangent space T_{p}X.

Let 𝔞 be, once and for all, a fixed maximal abelian subalgebra of 𝔤. It follows from the Cartan decomposition that 𝔞 ⊂ 𝔭. The set exp(𝔞) · p will be a maximal flat (totally geodesically embedded Euclidean space of maximal dimension) in X. 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).

Let 𝔞^{∗} be the dual to 𝔞, then for each α ∈ 𝔞^{∗} define

𝔤_{α} = {Y ∈ 𝔤| ad_{A} Y = α(A)Y for all A ∈ 𝔞}.

We call α a root if 𝔤_{α} ≠ 0. Therefore the roots form a finite set Λ.

If θ_{p} is the Cartan involution associated to the point p, which is Id on 𝔨 and − Id on 𝔭, then we may define a positive definite inner product ϕ_{p} on 𝔤 by ϕ_{p}(Y, Z) = −B(θ_{p}Y, Z). With respect to ϕ_{p}, the folowing root space decomposition

𝔤 = 𝔤_{0} + ∑_{α∈Λ} 𝔤_{α}

is orthogonal.

The following is proposition can be found in 2.7.3 of [Eb].

Proposition 2.2. Some properties of the roots and root space decomposition are:

1. [𝔤_{α}, 𝔤_{β}] ⊂ 𝔤_{α+β} if α + β ∈ Λ or is 0 otherwise.

2. If α ∈ Λ then −α ∈ Λ and θ_{p} : 𝔤_{α} → 𝔤_{−α} is an isomorphism.

3. If α is not an integer multiple of some other λ ∈ Λ then the only possible multiples of α in Λ are ±α and ±2α.

4. We have 𝔤_{0} = (𝔤_{0} ∩ 𝔨) + 𝔞.

5. If α, β ∈ Λ then β − 2\frac{⟨α,β⟩}{⟨α,α⟩}α ∈ Λ where ⟨·, ·⟩ is the dual inner product

to ϕ_{p} on 𝔞^{∗}. Furthermore, 2\frac{⟨α,β⟩}{⟨α,α⟩} is always an integer and if α and β are not collinear then it is ±1.

We call a subset Δ ⊂ Λ a base for Λ if

1. the elements of Δ form a basis (over R) for 𝔞^{∗},

2. and every root in Λ can be written as a linear combination of of elements in Δ with coefficients being either all nonnegative integers or all nonpositive integers.

If we choose an regular element A ∈ 𝔞 then define the set of positive roots corresponding to A,

Λ_{A}^{+} = {α ∈ Λ|α(A) ≥ 0}

The subset Δ_{A}^{+} ⊂ Λ_{A}^{+} consisting of elements which cannot be written as a sum of two elements in Λ_{A}^{+} is a base for Λ. Sometimes Δ_{A}^{+} is called a fundamental system of positive roots.

For A ∈ 𝔞 the associated (open) Weyl chamber W (A) is the connected component of the set of regular vectors in 𝔞 which contains A. We also call the set exp W (A) ⊂ exp(𝔞), as well as exp(W (A)) · p ⊂ X, a Weyl chamber which we again denote by W (A) using the context to determine where exactly it lies.

The union of all the singular geodesics in the flat exp(𝔞) · p passing through p is a finite set of hyperplanes forming the boundaries of the Weyl chambers. This provides another description of the Weyl chamber W (A) as

W (A) = {Y ∈ 𝔞|α(Y) > 0 for all α ∈ Δ_{A}^{+}}.

For each subset I ⊂ Δ_{A}^{+} the set W_{I}(A) = ∩_{α∈I}(ker α ∩ \overline{W (A)} is called the Weyl chamber face corresponding to the set I, and we designate W_{∅}(A) = W (A). The subgroup of K which stabilizes the face W_{I}(A) we denote by K_{I}.

2.3 The Furstenberg Boundary

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 in Isom(X).

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].

For a fixed regular vector A ∈ 𝔞 and associated set of positive roots Λ_{A}^{+} the barycenter b of the Weyl chamber W (A) is defined to be

b = ∑_{α∈Λ_{A}^{+}} m_{α}H_{α}

where m_{α} = dim 𝔤_{α} is the multiplicty of α and H_{α} is the dual vector (with respect to ϕ_{p}) of α. 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 we have \overline{Γ · x} = ∂X.

2.4 Albuquerque’s Theorem

Theorem 7.4 and Proposition 7.5 of [Al] combine to give the following theorem, which will play a crucial role in our proof of Theorem 1.2.

Theorem 2.3 (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 rank(X) − 1 in ∂X.

2.5 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 2.4 (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 2.3 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.

3 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.

In other words, 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(x, θ) = B(f̃(p), x, θ) be the Busemann function on X with respect to the basepoint f̃(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(x, θ)dσ_{y}^{s}(θ)

By Theorem 2.4, the support of ν_{z}, hence of σ_{y}^{s}, is all of ∂_{F} X, which in turn equals the G-orbit G · b^{+}(∞). Hence

B_{s,y}(x) = ∫_{∂_{F} X} B(x, θ)dσ_{y}^{s}(θ) = ∫_{G·b^{+}(∞)} B(x, θ)dσ_{y}^{s}(θ).

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

Proposition 3.1 (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. 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(γ(t), ξ) is strictly convex in t, and hence on an open positive μ_{y}-measure set around ξ. We know it is convex by the comment preceding the statement of the proposition.

If B(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 (i.e. γ is not contained in the boundary of a Weyl chamber). In particular, γ is contained in a unique 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(γ(t), ξ) is strictly convex in t.

For fixed z ∈ X, by the last property listed in Theorem 2.1, we see that

∫_{∂_{F} X} B(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(p_{1}, x, θ) = B(p_{2}, x, θ) + B(p_{1}, p_{2}, θ)

we see that B_{s,y} only changes by an additive constant when we change the basepoint of B. Also, B_{s,y} only changes by a multiplicative constant when we change the basepoint in the definition of μ_{y}. Since neither change affects the critical point of B_{y,s}, we see that F̃_{s} is independent of choice of basepoints.

The equivariance of f̃ and of {μ_{y}} implies that F̃_{s} is also equivariant. Hence F̃_{s} descends to a map F_{s} : N → M. It is easy to see that F_{s} is homotopic to f.

Proposition 3.2. 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 for any s > h(g).

Proof. From its definitions, F̃_{s}(y) is continuous in s and y. If (s_{i}, y_{i}) is a sequence converging to (s_{0}, y) for s_{0} ≥ s and y ∈ M then from the definition it is easy to verify that F̃_{s_{i}}(y_{i}) converges to F̃_{s_{0}}(y). If on the other hand s_{i} → ∞, then observe that lim_{i→∞} σ_{y_{i}}^{s_{i}} = ν_{ϕ(y)}. If follows that lim_{i→∞} F̃_{s_{i}}(y_{i}) = ϕ(y). This implies the proposition.

⋄

As in [BCG1], we will see that F_{s} is C^{1}, and will estimate its Jacobian.

4 The Jacobian estimate

Let X be expressed as a product of its irreducible factors X = X_{1} ×…×X_{k}, and let g_{i} denote the restricted symmetric metric on each factor X_{i}. As above, h(g_{i}) denotes the volume entropy of (X_{i}, g_{i}). The main estimate of this paper is the following.

Theorem 4.1 (The Jacobian Estimate). For all s > h(g) and all y ∈ N we have

| Jac F_{s}(y)| ≤ C (\frac{s}{h(g_{1})h(g_{2}) ⋯ h(g_{k})})^{n}

for some constant C, depending only on dim M.

Dependence of constants. Up to scaling of the metric, there are only a finite number of irreducible symmetric spaces of noncompact type in a given dimension. Therefore it is sufficient to show that C depends only on the individual symmetric spaces (X_{i}, g_{i}). Furthermore, when we apply Theorem 4.1, we will take the limit as s → h(g) so that the quantity C (\frac{h(g)}{h(g_{1})h(g_{2})···h(g_{k})})^{n} is the constant appearing in Theorem 1.2. It is evident then that the right hand side of inequality of Theorem 1.2 is scale invariant with respect to the metrics g and g_{i}.

We claim that the quantities h(g) and h(g_{i}) can be bounded by Ricci curvatures. The Bishop volume comparison Theorem ([BC]) states that if the Ricci curvatures of (Y, g) are all greater than (n − 1)κ for some κ ≤ 0 then for any y ∈ Y and r > 0,

Vol B(y, r) ≤ V_{κ}(r)

where V_{κ}(r) is the volume of the ball of radius r in the space form of constant curvature κ. In particular this implies that

h(g) ≤ lim_{r→∞} \frac{log V_{κ}(r)}{r} = (n − 1)\sqrt{−κ}

Similarly, in the course of the proof of Theorem 4.1 we will see explicitly that

h(g_{i}) = Tr \sqrt{−R_{i}(b^{+}, ·, b^{+}, ·)}

where R_{i} is the curvature tensor on (X_{i}, g_{i}). In particular

h(g_{i}) ≥ min{1, − Ricci(b^{+}, b^{+})}

Therefore the constant C in Theorem 1.2 depends only on the Ricci curvatures of N and M.

We will prove Theorem 4.1 in several steps.

4.1 Finding the Jacobian

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

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

Hence as 2-forms

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

−s ∫_{Y} ∫_{∂_{F} X} dB_{(F_{s}(y),θ)}(·) ⟨∇_{y}d(y, z), ·⟩ dν_{ϕ(z)}(θ)dμ_{y}^{s}(z)

The distance function d(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 the chain rule,

Jac F_{s} = s^{n} \begin{matrix}det & (∫_{Y} ∫_{∂_{F} X} dB_{(F_{s}(y),θ)}(·) ⟨∇_{y}d(y, z), ·⟩ dν_{ϕ(z)}(θ)dμ_{y}^{s}(z)) \\ det & (∫_{∂_{F} X} DdB_{(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_{(F_{s}(y),θ)}^{2}dσ_{y}^{s}(θ))^{1/2} det (∫_{Y} ⟨∇_{y}d(y, z), ·⟩^{2} dμ_{y}^{s}(z))^{1/2} ; det (∫_{∂_{F} X} DdB_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ))

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

det (∫_{Y} ⟨∇_{y}d(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_{(F_{s}(y),θ)}^{2}dσ_{y}^{s}(θ)) & 1/2 \\ det & (∫_{∂_{F} X} DdB_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ)) \end{matrix}  (2)

4.2 Reduction to Irreducible Case

In this subsection we make, following [CF1], a reduction to the case when X = M̃ is irreducible.

If X = X_{1} ×…×X_{k} is the irreducible expression for X as a product, the group G = Isom(X) can also be written as a product G = G_{1} × G_{2} ⋯ × G_{k}, where each G_{i} ≠ SL(2, R), SL(3, R) is a simple Lie group. Theorem 2.3 implies that for all y ∈ Y, the measure σ_{y}^{s} is supported on the G-orbit

G · b^{+}(∞) = {(G_{1} × G_{2} ⋯ × G_{k}) · b^{+}(∞)}

Hence

∂_{F} X = G · b^{+}(∞) = ∂_{F} X_{1} × · × ∂_{F} X_{k}

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

∂_{F} X = ∂X_{1} × ⋯ × ∂X_{k}

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(x, y, θ_{i}) = B_{i}(x, y, θ_{i}). Since the factors X_{i} are orthogonal in X with respect to the metric g_{0}, the Busemann function of (X, g_{0}) with basepoint p ∈ X at a point θ = (θ_{1}, …, θ_{k}) ∈ ∂_{F} X is given by

B(x, θ) = \frac{1}{\sqrt{k}} ∑_{i=1}^{k} B_{i}(x_{i}, θ_{i}).

The Schur estimate for the determinant of symmetric semidefinite block matrices states,

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

Applying the dual form of this estimate to our symmetric tensors we have

det (∫_{∂_{F} X} (∑_{i=1}^{k} d(B_{i})_{(π_{i}F_{s}(y),π_{i}θ)})^{2} dσ_{y}^{s}(θ)) ≤

∏_{i=1}^{k} det (∫_{∂_{F} X_{i}} (d(B_{i})_{(π_{i}F_{s}(y),θ_{i})})^{2} d(π_{i})_{∗}σ_{y}^{s}(θ_{i})) ,

where π_{i} : X → X_{i} and π_{i} : ∂_{F} X → ∂_{F} X_{i} are the canonical projections.

Since DdB_{(F_{s}(y),θ)} = \frac{1}{\sqrt{k}} ∑_{i=1}^{k} DdB_{i}_{(π_{i}F_{s}(y),π_{i}θ)}, the denominator already splits as,

det (∫_{∂_{F} X} DdB_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ)) = ∏_{i=1}^{k} det (∫_{∂_{F} X_{i}} (Dd(B_{i})_{(π_{i}F_{s}(y),θ_{i})}) d(π_{i})_{∗}σ_{y}^{s}(θ_{i}))

Putting these together we obtain,

| Jac F_{s}(y)| ≤ (\frac{s}{\sqrt{n}})^{n} ∏_{i=1}^{k} \begin{matrix}det & (∫_{∂_{F} X_{i}} (d(B_{i})_{(π_{i}F_{s}(y),θ_{i})})^{2} d(π_{i})_{∗}σ_{y}^{s}(θ_{i})) & 1/2 \\ det & (∫_{∂_{F} X_{i}} (Dd(B_{i})_{(π_{i}F_{s}(y),θ_{i})}) d(π_{i})_{∗}σ_{y}^{s}(θ_{i})) \end{matrix}

Therefore we only need to bound each term in the product seperately. It suffices then to prove that for an irreducible symmetric space (X, g_{0}) ≠ H^{2}, SL(3, R)/ SO(3, R), and for any measure μ on ∂_{F} X, that

\begin{matrix}det & (∫_{∂_{F} X} dB_{(F_{s}(y),θ)}^{2}dμ(θ)) & 1/2 \\ det & (∫_{∂_{F} X} DdB_{(F_{s}(y),θ)}(·, ·)dμ(θ)) \end{matrix} ≤ \frac{C}{h(g_{0})}

We will continue to write σ_{y}^{s} instead of μ or (π_{i})_{∗}σ_{y}^{s}. The only property we use of σ_{y}^{s} from this point on is that it is fully supported on ∂_{F} X. Since supp((π_{i})_{∗}σ_{y}^{s}) = π_{i}(supp(σ_{y}^{s})) = ∂_{F} X_{i} there is no harm by this imprecision.

4.3 Simplifying the Jacobian

As stated above we need only now consider irreducible (X, g_{0}). For each point x ∈ X, we let F_{x} denote the canonical flat passing through x, i.e. F_{x} = exp(𝔞) · x. We denote the tangent space to F_{x} simply as F with the base point suppressed since it is naturally isomorphic to the Lie algebra exp(𝔞).

We wish to bound the quantity

\begin{matrix}det & (∫_{∂_{F} X} dB_{(F_{s}(y),θ)}^{2}dσ_{y}^{s}(θ)) & 1/2 \\ det & (∫_{∂_{F} X} DdB_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ)) \end{matrix}

Let F denote the tangent space to the flat F_{F_{s}(y)}. Choose an orthonormal basis {e_{i}} for the tangent space T_{F_{s}(y)}X such that e_{1}, …, e_{rank(X)} is a basis for F with e_{1}(∞) = b^{+}(∞). We may write the term

∫_{∂_{F} X} DdB_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ)  (3)

in matrix form as

∫_{∂_{F} X} O_{θ} (\begin{matrix}0 & 0 \\ 0 & D_{λ} \end{matrix}) O_{θ}^{∗} dσ_{y}^{s}(θ)

where O_{θ} is the orthogonal matrix in the e_{i} basis corresponding to the derivative of the unique isometry in K = Stab_{G}(F_{s}(y)) which sends e_{1} to v_{(F_{s}(y),θ)} (the vector in the tangent space of the point F_{s}(y) in the direction θ ∈ ∂_{F} X). In the above expression, the upper left zero matrix sub-block has dimensions rank(X) × rank(X), and D_{λ} has the form

D_{λ} = (\begin{matrix}λ_{1} & 0 & ⋯ & 0 \\ 0 & λ_{2} & 0 & ⋮ \\ ⋮ & 0 & ... & 0 \\ 0 & ⋯ & 0 & λ_{n−rank(X)} \end{matrix})

where {λ_{1}, …, λ_{n−rank(X)}} is the set of nonzero eigenvalues of DdB_{(F_{s}(y),θ)}. Since DdB_{(x,θ)} is G equivariant, its eigenvalues do not depend on x but only on which K-orbit in ∂X the point θ lies in. In particular, DdB_{(x,θ)} is flow invariant and hence the Ricatti equation shows that it is simply related to the curvature tensor by

DdB_{(x,θ)} = \sqrt{−R(v_{(x,θ)}, ·, v_{(x,θ)}, ·)}

On the other hand in a symmetric space R(v, ·, v, ·) = −(ad_{v})^{2}|_{𝔭}. Therefore the eigenvalues of DdB_{(F_{s}(y),θ)} are those of DdB_{(p,b^{+}(∞))} which in turn

are those of \sqrt{ad_{b^{+}}^{2}}|_{𝔭}. (Note that while ad_{b^{+}} does not preserve 𝔭, (ad_{b^{+}})^{2}|_{𝔭} is a symmetric endomorphism of 𝔭.) Recall, b^{+} = b/∥b∥ where b = ∑_{β∈Λ_{A}^{+}} m_{β}H_{β} for any choice of A ∈ 𝔞 (the choice of A only determines the Weyl chamber containing b). Setting

𝔭_{α} = 𝔭 ∩ (𝔤_{α} ⊕ 𝔤_{−α}),

we have 𝔭_{α} = {X − θ_{p}X : X ∈ 𝔤_{α}}.

By definition of 𝔤_{α}, for each α ∈ Λ_{A}^{+} we may write

(ad_{b^{+}})^{2}|_{𝔤_{α}} = α(b^{+})^{2} Id = (\frac{1}{∥b∥} ∑_{β∈Λ_{A}^{+}} α(m_{β}H_{β}))^{2} Id

The same expression clearly holds for (ad_{b^{+}})^{2}|_{𝔤_{−α}}. Therefore, for any α ∈ Λ, \sqrt{(ad_{b^{+}})^{2}|_{𝔭_{α}}} = |α(b^{+})|. For 𝔭_{0} = 𝔞 the same formula holds with α = 0. In particular, the ratio of the largest eigenvalue (denoted by λ_{max}) among the λ_{i}’s in D_{λ} to the smallest nonzero eigenvalue (denoted by λ_{min}) only depends on X.

Furthermore, since α(b^{+}) > 0 for all α ∈ Λ_{A}^{+} and dim 𝔭_{α} = m_{α}, we have

Tr \sqrt{ad_{b^{+}}^{2} |_{𝔭}} = ∑_{α∈Λ_{A}^{+}} m_{α}α(b^{+}) = \frac{1}{∥b∥} ∑_{α,β∈Λ_{A}^{+}} m_{α}m_{β}α(H_{β}) =

\frac{1}{∥b∥} ⟨ ∑_{β∈Λ_{A}^{+}} m_{β}H_{β}, ∑_{α∈Λ_{A}^{+}} m_{α}H_{α}⟩ = \frac{∥b∥^{2}}{∥b∥} = h(g_{0})

where the last equality follows from Theorem 2.3. As a result, there is a constant c only depending on X such that

\frac{h(g_{0})}{c} ≤ λ_{i} ≤ c h(g_{0})  (4)

for i = 1, …, (n − rank(X)). We now use the following.

Lemma 4.2. The determinant of a sum of n × n positive semidefinite matrices is a nondecreasing homogeneous polynomial of degree n in the eigenvalues of each summand. Furthermore, if the sum is positive definite, then the determinant is strictly increasing in the eigenvalues of the summands.

Proof. Let M be the sum of positive semidefinite matrices. Then there exist fixed orthogonal matrices O_{l} and real numbers λ_{l,j} such that M may be written as

M = ∑_{l} O_{l} (\begin{matrix}λ_{l,1} & 0 & ⋯ & 0 \\ 0 & λ_{l,2} & 0 & ⋮ \\ ⋮ & 0 & ... & 0 \\ 0 & ⋯ & 0 & λ_{l,n} \end{matrix}) O_{l}^{∗}

Then we have the differentiation formula (see, e.g. Prop. 2.8 of [Ch]):

\frac{d}{dλ_{l,j}} det M = Tr (\frac{d}{dλ_{l,j}} M) M^{adj}

where M^{adj} is the adjunct matrix of M. Now,

\frac{d}{dλ_{l,j}} M = O_{l}E_{(j,j)}O_{l}^{∗}

where E_{(j,j)} is the elementary matrix with 1 in the (j, j) position and zeros elsewhere. Therefore, by cyclically permuting O_{l} in the trace above we find that \frac{d}{dλ_{l,j}} det M is the (j, j) the entry of O_{l}^{∗}M^{adj}O_{l} which is nonnegative since M is positive semidefinite. Lastly, if M is positive definite then O_{l}^{∗}M^{adj}O_{l} is also, which means that \frac{d}{dλ_{l,j}} det M is positive. The lemma follows. ⋄

Applying Lemma 4.2 to the Riemann sums for the integral (3) above, using the bound in Equation (4), and taking limits, gives

det ∫_{∂_{F} X}DdB_{(F_{s}(y),θ)}(·, ·)dσ_{y}^{s}(θ) ≥

(\frac{h(g_{0})}{c})^{n} det ∫_{∂_{F} X} O_{θ} (\begin{matrix}0 & 0 \\ 0 & I_{n−rank(X)} \end{matrix}) O_{θ}^{∗} dσ_{y}^{s}(θ)

where I_{n−rank(X)} is the identity matrix of dimension n − rank(X).

Next we observe that, relative to the orthonormal basis {e_{1}, …, e_{rank(X)}} for T_{F_{s}(y)}X, the expression

∫_{∂_{F} X} dB_{(F_{s}(y),θ)}^{2}dσ_{y}^{s}(θ)

may be written in the form

Q_{1} = ∫_{∂_{F} X} O_{θ} (\begin{matrix}1 & 0_{(n−1)×1} \\ 0_{1×(n−1)} & 0_{(n−1)×(n−1)} \end{matrix}) O_{θ}^{∗} dσ_{y}^{s}(θ)

where O_{θ} is the same matrix as above. Let

Q_{2} = ∫_{∂_{F} X} O_{θ} (\begin{matrix}0 & 0 \\ 0 & I_{n−rank(X)} \end{matrix}) O_{θ}^{∗} dσ_{y}^{s}(θ)

We have just shown that, to prove Theorem 4.1, it suffices to prove that

\frac{det Q_{1}}{(det Q_{2})^{2}} ≤ C  (5)

for some constant C. The rest of this section will be devoted to proving this.

4.4 Eigenvalue matching

Here is the general idea of our proof of Theorem 4.1, which we have reduced to showing (5) above. Since the numerator is bounded above, we consider when the matrix Q_{2} in the denominator has any eigenvalues smaller than a certain constant depending only on the dimension of X. When this occurs, Theorem 4.4 below will show that each such eigenvalue is matched by at least two smaller (up to a universal constant) eigenvalues of the matrix Q_{1} in the numerator.

Let {v_{i}} be an orthonormal eigenbasis for the symmetric matrix Q_{2}, and recall that {e_{i}} is a basis for the tangent space F to the fixed, chosen flat. Note that the i-th eigenvalue of the matrix Q_{2} may be written as

L_{i} = v_{i}^{∗}Q_{2}v_{i} = ∫_{∂_{F} X} ∑_{j=rank(X)+1}^{n} ⟨O_{θ}.e_{j}, v_{i}⟩^{2} dσ_{y}^{s}(θ)

We first argue that no L_{i} equals zero. Since s > h(g) we have that the measures μ_{y}^{s} is a finite measure in the Lebesgue class (dg). Since the ν_{x} for x ∈ X are positive on any open set (with respect to the cone topology) of ∂_{F} X, it follows that σ_{y}^{s} is as well. In particular, {O_{θ}|θ ∈ supp(σ_{y}^{s}) = ∂_{F} X} is isomorphic to the group K and therefore there is no nonzero subspace V ⊂ T_{F_{s}(y)}X such that O_{θ}V ⊂ F for all θ ∈ ∂_{F} X. Hence none of the eigenvalues L_{i} are 0.

Let ϵ = 1/(rank(X) + 1). Note that ϵ is a constant depending only on n, as there are only finitely many symmetric spaces of a given rank and given dimension. Suppose k of the eigenvalues are strictly less than ϵ. Since each L_{i} ≤ 1, and since

∑_{i} L_{i} = Tr Q_{2} = n − rank(X)

it follows easily that k ≤ rank(X). By rearranging the order we may assume that L_{i} < ϵ for i = 1, ⋯, k.

Let H be an inner product space over R, and denote by SO(H) the special orthogonal group of H. Scale the bi-invariant metric on SO(H) so that SO(H) has diameter π/2. Define the angle between two subspaces V, W ⊂ H as

∠(V, W) := inf{d_{SO(H)}(I, P) : P ∈ SO(H) with P V ⊂ W or P W ⊂ V }

Let π_{V} (W) represent the orthogonal projection of W onto V. Then it is routine to verify the following properties of the angle:

1. ∠(V, W) ≤ \frac{π}{2}

2. ∠(V, W) = ∠(W^{⊥}, V^{⊥})

3. ∠(V, W) = ∠(W, V)

4. If V ⊆ U and dim U ≤ dim W then ∠(V, W) ≤ ∠(U, W), or if V ⊆ U and dim V ≥ dim W then ∠(V, W) ≥ ∠(U, W)

5. If ∠V, W = 0 then V ⊆ W or W ⊆ V

6. If U ⊆ W then ∠(π_{V} (U), U) ≤ ∠(π_{V} (U), W) ≤ ∠(V, W) For a 1-dimensional subspace V spanned by a vector v, our definition of

angle agrees with the usual definition:

7. V = span\{v\} ⇒ cos(∠(V, W)) = \frac{⟨v,π_{W} (v)⟩}{|v|·|π_{v}(W)|} Finally, ∠ satisfies the following form of the triangle inequality.

Lemma 4.3 (Triangle inequality for ∠). Let U, V, W be subspaces of a fixed inner product space H. Suppose that dim U = dim W ≤ dim V. Then

∠(V, W) ≤ ∠(U, V) + ∠(U, W)

Proof. By definition of ∠ there exist P_{1}, P_{2}, P_{3} ∈ SO(H) with • P_{1}W ⊆ V and ∠(V, W) = d_{SO(H)}(I, P_{1}).

• P_{2}U ⊆ V and ∠(U, V) = d_{SO(H)}(I, P_{2}).

• P_{3}U = W and ∠(U, W) = d_{SO(H)}(I, P_{3}). Now P_{2}P_{3}^{−1}W ⊆ V so that

d(I, P_{1}) ≤ d(I, P_{2}P_{3}^{−1})
= d(P_{2}, P_{3})
≤ d(I, P_{2}) + d(I, P_{3})

and we are done.

⋄ One of the main ingredients in the proof of Theorem 4.1 is the following.

Theorem 4.4 (Eigenvalue Matching Theorem). For any k-frame given by orthonormal vectors v_{1}, …, v_{k} of T_{x}X with k ≤ rank(X) there is an orthonormal 2k-frame given by vectors v′_{1}, v′′_{1} …, v′_{k}, v′′_{k}, each perpendicular to span{v_{1}, …, v_{k}}, such that for i = 1, …, k and all h ∈ K, there is a constant C, depending only on dim X, such that

∠(hv′_{i}, F^{⊥}) ≤ C∠(hv_{i}, F)

and

∠(hv′′_{i}, F^{⊥}) ≤ C∠(hv_{i}, F)

where hv represents the linear (derivative) action of K on v ∈ T_{x}X.

We will prove Theorem 4.4 in Section 5; its proof is independent of the rest of the paper.

4.5 Proof of the Jacobian Estimate

Assuming Theorem 4.4 for the moment, we now complete the proof of Theorem 4.1.

Proof of Theorem 4.1. From equation (2) and the reduction in §4.3 we see that it is sufficient to show that

\frac{det Q_{1}}{(det Q_{2})^{2}} ≤ C

for some constant C depending only on n.

As before let L_{1}, …, L_{k} be the k ≤ rank(X) eigenvalues of Q_{2} which are strictly less than ϵ = 1/(rank(X) + 1). If no such eigenvalues exist, then there is a lower bound on Q_{2} depending only on rank(X). As there is an upper bound on Q_{1}, we are done (see the discussion on dependency of constants above). So we assume k ≥ 1.

Let v_{1}, …, v_{k} be an orthonormal set of associated eigenvectors. Recall that {e_{i}} denotes the chosen orthonormal basis for the T_{F_{s}(y)}X such that e_{1}, …, e_{rank(X)} spans the tangent space F to the fixed maximal flat.

For any vector v ∈ T_{F_{s}(y)}X let

r(v) = ∑_{j=rank(X)+1}^{n} ⟨e_{j}, v⟩^{2}

so that

L_{i} = ∫_{∂_{F} X} r(O_{θ}^{∗}v_{i}) dσ_{y}^{s}(θ).

Since e_{1}, …, e_{rank(X)} form an orthonormal basis for F, for any unit vector v we have

cos(∠(v, F)) = ⟨v, π_{F}(v)⟩ /|π_{F}(v)|
= ⟨v, ∑ ⟨v, e_{j}⟩ e_{j}⟩ /(∑ ⟨v, e_{j}⟩^{2})^{1/2}
= (∑ ⟨v, e_{j}⟩^{2})^{1/2}

so that

cos(∠(v, F))^{2} = ∑_{j=1}^{rank(X)} ⟨v, e_{j}⟩^{2}

Hence

r(v) = 1 − ∑_{j=1}^{rank(X)} cos^{2}(∠v, e_{j})
= 1 − cos^{2}(∠v, F)
= sin^{2}(∠v, F)

Similarly

⟨v, e_{1}⟩^{2} ≤ ∑_{j=1}^{rank(X)} ⟨v, e_{j}⟩^{2} = sin^{2}(∠v, F^{⊥})

For each i = 1, …, k, let v′_{i} and v′′_{i} be the pair of vectors corresponding to v_{i} produced by the Eigenvalue Matching Theorem (Theorem 4.4). That theorem together with the concavity of sin^{2} θ for 0 ≤ θ ≤ π/2 gives, for all θ ∈ ∂_{F} X and for each w_{i} = v′_{i} or v′′_{i}, that

sin^{2}(∠O_{θ}^{∗}w_{i}, F^{⊥}) ≤ sin^{2}(C∠O_{θ}^{∗}v_{i}, F) ≤ C^{2} sin^{2}(∠O_{θ}^{∗}v_{i}, F)

where C > 1 is the constant in the Eigenvalue Matching Theorem.

Furthermore, Q_{1} is the integral (against a probability measure) of matrices with all eigenvalues less than 1 so no eigenvalue of Q_{1} is greater than one. Hence we may estimate

det Q_{1} ≤ ∏_{i=1}^{k}(v′_{i}.Q_{1}.v′_{i})(v′′_{i}.Q_{1}.v′′_{i})

= ∏_{i=1}^{k} (∫_{∂_{F} X} ⟨e_{1}, O_{θ}^{∗}.v′_{i}⟩^{2} dσ_{y}^{s}(θ)) (∫_{∂_{F} X} ⟨e_{1}, O_{θ}^{∗}.v′′_{i}⟩^{2} dσ_{y}^{s}(θ))

≤ ∏_{i=1}^{k} (∫_{∂_{F} X} sin^{2}(∠O_{θ}^{∗}.v′_{i}, F^{⊥}) dσ_{y}^{s}(θ)) (∫_{∂_{F} X} sin^{2}(∠O_{θ}^{∗}.v′′_{i}, F^{⊥}) dσ_{y}^{s}(θ))

≤ ∏_{i=1}^{k} (∫_{∂_{F} X} C^{2} sin^{2}(∠O_{θ}^{∗}.v_{i}, F) dσ_{y}^{s}(θ)) (∫_{∂_{F} X} C^{2} sin^{2}(∠O_{θ}^{∗}.v_{i}, F) dσ_{y}^{s}(θ))

= C^{2k} ∏_{i=1}^{k} L_{i}^{2}

= C^{2k} det Q_{2}^{2} ∏_{i=k+1}^{n} L_{i}^{−2}

≤ C^{2k} det Q_{2}^{2}(rank(X) + 1)^{2(n−k)}

The last inequality follows from the definition of k, whereby L_{i} ≥ \frac{1}{rank(X)+1} for each i > k.

The constant C in Theorem 4.1 may be taken to be the product (over factors X_{j} of X with dimension n_{j}),

\frac{1}{\sqrt{n}^{n}} ∏_{j} C_{j}^{rank(X)}c_{j}^{n_{j}}(rank(X_{j}) + 1)^{(n_{j})}

where C_{j} ≥ 1 is the constant C from Theorem 4.4, c_{j} is the constant c in Equation (4) and k_{j} is the constant k above. This combined constant depends only on n = dim X. ⋄

4.6 A cautionary example

In the general method of [BCG1] as well as here, one is solving a minimization problem without regard to the measure. However, at least in the SL_{3}(R)/ SO_{3}(R) case, to get a bound on the Jacobian of F_{s} one must use further properties of the measure, as indicated by the example we now give.

If for a single flat F_{x} and a sequence of y_{i} ∈ F_{x}, the measures σ_{y_{i}}^{s} tend to the sum of Dirac measures \frac{1}{2}δ_{b^{+}(∞)} + \frac{1}{2}δ_{wb^{+}(∞)} where w is in the Weyl group for F_{x}, then we claim that Jac F_{s}(y_{i}) → ∞. First note that the sum

dB_{(F_{s}(y_{i}),b^{+}(∞))}^{2} + dB_{(F_{s}(y_{i}),wb^{+}(∞))}^{2}

has only a 3-dimensional kernel, while

DdB_{(F_{s}(y_{i}),b^{+}(∞))} + DdB_{(F_{s}(y_{i}),wb^{+}(∞))}

has a 2-dimensional kernel. Furthermore

Q_{1} = ∫_{∂_{F} X} dB_{(F_{s}(y_{i}),θ)}^{2}dσ_{y_{i}}^{s} and Q_{2} = ∫_{∂_{F} X} DdB_{(F_{s}(y_{i}),θ)}dσ_{y_{i}}^{s}

degenerate in the same way, so that det(Q_{1})/ det(Q_{2})^{2} is unbounded. This can be easily verified explicitly in the case of a sum of five Dirac measures for which both integrals are nonsingular degenerating to the sum of the two Dirac measures given above.

A similar problem occurs when there are H^{2} factors. These and other examples are worked out in full detail in Section 6 of [CF2].

5 Proof of the Eigenvalue Matching Theorem

In order to prove Theorem 4.4 we will need a series of lemmas.

5.1 Dimension inequalities

For any x ∈ X and any subspace V ⊆ T_{x}X, denote by K_{V} the elements of K which stabilize V (i.e. leave V invariant). For V ⊂ F, if Fix_{K}(V) is the subgroup of K which fixes V pointwise then K_{V} = U · Fix_{K}(V) where U is the subgroup stabilizing V of the (discrete) Weyl group which stabilizes F (see [Eb]).

The following lemma is a basic algebraic ingredient in the proof of Theorem 4.4.

Lemma 5.1 (Dimension inequality, I). With the above notations,

dim (span{K_{V} · F}^{⊥}) ≥ 2 dim(V).

Proof. First we show that K_{V} · F is itself a subspace hence equal to its span.

Recognize that K_{V} · F is the union of all tangent spaces to flats which contain V. Pick a basis v_{1}, …, v_{l} of V note that K_{V} · F = ∩_{i=1}^{l}F(v_{i}) where F(v_{i}) is the union of all the tangent spaces to flats containing v_{i} using the notation of [Eb]. Proposition 2.11.4 of [Eb] states that F(v_{i}) = R^{r} × X_{i} for some symmetric space of noncompact type and r ≤ rank(X). In particular it is a manifold and the tangent space to it corresponds to K_{v_{i}} · F, which is a vector space. Then K_{V} · F is a vector space.

Let K_{F} be the stabilizer of F in K. Then K_{F} ⊂ W · K_{V} where W denotes the Weyl group (a finite group). Hence dim K_{F} = dim(K_{F} ∩ K_{V}). Hence

dim K_{V} · F = dim K_{V} + dim F − dim K_{F}

Since X = K · F we obtain

dim M = dim K + dim F − dim K_{F}.

Putting this together we obtain,

(dim span{K_{V} · F })^{⊥} = dim M − dim K_{V} · F = dim K − dim K_{V} .

But Lemma 5.2 below gives that this final term is ≥ 2 dim V, as desired. ⋄

The following lemma was used in the proof of Lemma 5.1. Recall that, at this point, we are assuming that the symmetric space X is irreducible and has rank(X) ≥ 2.

Lemma 5.2 (Dimension inequality, II). Assume that X ≠ SL_{3}(R)/ SO_{3}(R). Then for any subspace V ⊂ F, we have

dim K ≥ 2 dim V + dim K_{V} .

This lemma is the only place where X ≠ SL_{3}(R)/ SO_{3}(R) is used.

Proof. For a root α ∈ Λ in F, define 𝔨_{α} = (Id +θ_{p})𝔤_{α}, where θ_{p} is the Cartan involution at p = F_{s}(y). Then by Proposition 2.14.2 of [Eb] we have that 𝔨_{α} = 𝔤_{α} ⊕ 𝔤_{−α} ∩ 𝔨, 𝔨_{α} = 𝔨_{−α}, and dim 𝔨_{α} ≥ 1.

Note that from the definition of 𝔤_{α} it follows immediately that

𝔨_{α} = {Y ∈ 𝔨|[X, Y] = 0 for all X ∈ ker α}.

Note that in G the normalizer mod centralizer is finite for any flat subspace. Therefore for any V ⊂ F we may write the Lie algebra 𝔨_{V} of K_{V} as,

𝔨_{V} = {Y ∈ 𝔨|[X, Y] = 0 for all X ∈ V }

It then follows from the previous statements that,

𝔨_{V} = 𝔨_{0} + ∑_{V ⊂ker α^{α∈Λ}} 𝔨_{α}

Consequently, we may assume that V in the statement of the lemma is maximally singular: V may be written as the intersection of the kernels of the greatest number of roots among all subspaces of dimension dim V. Otherwise dim K_{V} = dim 𝔨_{V} is strictly smaller than it would be if V were maximally singular.

Recall that we have the invariant inner product ϕ_{p} on 𝔞 and hence on F. Let Λ denote the collection of roots. For α ∈ Λ, let H_{α} ∈ F denote the dual root vector (with respect to ϕ_{p}) corresponding to α. For any subset V ⊂ F we define the function

card_{R}(V) := \frac{1}{2} card\{α ∈ Λ|H_{α} ∈ V \}

Since root vectors lying in a subspace always come in opposing pairs, card_{R} is a positive integer.

Let α be any root. Note that if a subspace V ⊂ ker α, then H_{α} lies in V^{⊥}. Therefore the statement of the lemma reduces to showing that

dim 𝔨_{0} + ∑_{α∈Λ} dim 𝔨_{α} ≥ 2 dim V + dim 𝔨_{0} + ∑_{V ⊂ker α^{α∈Λ}} dim 𝔨_{α}

or more simply,

∑_{H_{α}∈𝔉\V^{⊥}} dim 𝔨_{α} ≥ 2 dim V

Swapping V^{⊥} for V and vice versa, and using dim 𝔨_{α} ≥ 1 for each α, it is sufficient to prove that

card_{R}(F \ V) ≥ 2(rank(X) − dim V)  (6)

Since we are assuming that G is simple, we could check this condition by using a classification of root vectors in the simple algebras such as in [Va].

However, because this would be tedious we will instead give a synthetic proof.

For each i = 0, …, rank(X), we say that W_{i} ⊂ F is a maximally rooted subspace of dimension i if

card_{R}(W_{i}) = max{card_{R}(V) : V ⊂ F with dim V = i}

In other words, W_{i} is maximally rooted if W_{i}^{⊥} is maximally singular. We claim that if 0 = W_{0}, W_{1}, …, W_{rank(X)} = F are any maximally rooted subspaces of F with dim W_{i} = i, then for 0 < i ≤ rank(X),

card_{R}(W_{i}) ≥ i + card_{R}(W_{i−1})  (7)

This is true for i = 1 since W_{1} is one dimensional it contains a root vector pair and the trivial subspace W_{0} contains none. By induction, assume the claim holds for all maximally rooted subspace W_{i} of dimension i < j. In particular, for such a space W_{j−1} and for any subspace Z ⊂ W_{j−1} of codimension one, card_{R}(Z) ≤ card_{R}(W_{j−2}) so

card_{R}(W_{j−1} \ Z) = card_{R}(W_{j−1}) − card_{R}(Z) ≥ j − 1

We claim that there exists a root vector H_{α} which is not in W_{j−1} or its perpendicular W_{j−1}^{⊥} (with respect to ϕ_{p}). If not, then every root vector either lies in W_{j−1} or W_{j−1}^{⊥} which implies the root system is reducible (e.g. Corollary 27.5 of [Hu2]), and hence G is reducible, contrary to assumption.

Therefore, H_{α}^{⊥} ∩W_{j−1} is a codimension one subspace of W_{j−1} and by inductive hypothesis there are at least j − 1 distinct pairs of root vectors ±H_{α_{1}}, …, ±H_{α_{j−1}} in W_{j−1} \ (H_{α}^{⊥} ∩ W_{j−1}). For each of these we have ϕ_{p}(H_{α}, H_{α_{l}}) ≠ 0. By the standard calculus of roots (e.g. Proposition 2.9.3 of [Eb]) this implies that for each 1 ≤ l ≤ j−1 either ±(H_{α}+H_{α_{l}}) or ±(H_{α}− H_{α_{l}}) is a pair of root vectors lying in W_{j−1}⊕⟨H_{α}⟩ which does not lie in W_{j−1}. Including H_{α}, these form at least j pairs of root vectors which are contained in W_{j−1} ⊕ ⟨H_{α}⟩ \ W_{j−1}. Therefore card_{R}(W_{j−1} ⊕ ⟨H_{α}⟩) ≥ card_{R}(W_{j−1}) + j. Since by definition of W_{j}, card_{R}(W_{j}) ≥ card_{R}(W_{j−1} ⊕ ⟨H_{α}⟩), the claim follows.

Recursively applying Equation 7 shows that for 0 ≤ i < j ≤ rank(X),

card_{R}(W_{j}) − card_{R}(W_{i}) ≥ ∑_{k=i}^{j} k = \frac{j(j + 1)}{2} − \frac{i(i + 1)}{2}

Now to prove the inequality (6), as noted before we may assume V of dimension q is maximally rooted, since then V^{⊥} is maximally singular. Since F is a maximally rooted space, the above expression reads

card_{R}(F \ V) = card_{R}(F) − card_{R}(V) = \frac{rank(X)(rank(X) + 1)}{2} − \frac{q(q + 1)}{2}

This is readily seen to be greater that 2(rank(X)−q) unless rank(X) = 2 and q = 0 (V = F). However, every irreducible Lie algebras of rank two other than 𝔰𝔩(3, R) has at least four pairs of roots (see [Hu1], p.44, Figure 1), and hence the inequality (6) is satisfied in all of the required cases. ⋄

5.2 Angle inequalities

Lemma 5.3 (Angle inequality, I). For any subspace V ⊆ F there is a subspace V′ ⊂ V^{⊥} with dim V′ ≥ 2 dim V and a constant C depending only on the symmetric space X such that for all k ∈ K,

∠(kV′, F^{⊥}) ≤ C∠(kV, F)

where kV represents the linear (derivative) action of K on V ⊂ T_{x}X.

Proof. For any subspace V ⊂ F, let U_{1}, U_{2}, …, U_{l(V)} be the maximally singular subspaces of dimension dim V which have minimal angle with V. Define S_{V} = U_{1} ⊕…⊕U_{l(V)} ⊂ F. If G(r, F) denotes the Grassmann variety of subspaces in F with dimension r, then the set of V ∈ G(r, F) for which l(V) is constant has codimension l(V) − 1 in G(r, F).

For any subspace V ⊂ F we define a subspace V′ ⊂ F^{⊥} by

V′ = (span{K_{S_{V}} · F})^{⊥}

where K_{S_{V}} is the subgroup of K which stabilizes S_{V}. By Proposition 5.1, V′ has dimension at least 2 dim V since we always have K_{S_{V}} ⊂ K_{U} for some U ⊂ F with dim U = dim V.

If no such constant C as in the lemma exists then there is a sequence k_{i} ∈ K and V_{i} ⊂ F with dim V_{i} = r such that

\frac{∠(k_{i} V_{i}, F)}{∠(k_{i} V′_{i}, F^{⊥})} → 0.  (8)

Now since S_{V} and hence V′ varies upper semicontinuously in V (thinking of the map V → V′ as a self-map of G(r, F)), it follows from the continuity of the ∠ function that

\frac{∠(k V, F)}{∠(k V′, F^{⊥})}

is lower semicontinuous in V.

However since both K and G(r, F) are compact, for some subsequence of the k_{i}V_{i}, the k_{i} converge to k_{0} ∈ K and the V_{i} converge to a fixed subspace V_{0} ⊂ F. Furthermore, k_{0}V_{0} lies in F since ∠(k_{0}V_{0}, F) must be 0. It follows that k_{0} ∈ W · K_{V_{0}} where W is the Weyl group stabilizing F.

By construction, K_{V_{0}} ⊂ K_{V′_{0}} and for any w ∈ W,

∠(wV′_{0}, F^{⊥}) = ∠(V′_{0}, w^{−1}F^{⊥}) = ∠(V′_{0}, F^{⊥})

Therefore, we also have ∠(k_{0}V′_{0}, F^{⊥}) = 0. Continuity of ∠ along with the fact that W ⊂ K acts isometrically implies that it is sufficient to show that for any fixed subspace V ⊂ F the quantity

lim inf_{k→K_{V}} \frac{∠(kV, F)}{∠(kV′, F^{⊥})}

is bounded away from 0. Note that since this quantity is lower semicontinuous in V, and since G(r, F) is compact, it is unnecessary to show that the bound is independent of V.

First we handle the denominator. Using the bi-invariance of the metric on SO(n), the properties of the angle function, and the fact that for all k_{0} ∈ K_{S_{V}} we have k_{0} k k^{−1}F ⊂ K_{S_{V}} F, it follows that

d_{SO(n)}(k, K_{S_{V}}) = d_{SO(n)}(k^{−1}, K_{S_{V}}) = d_{SO(n)}(K_{S_{V}} · k, Id)

≥ inf{d_{SO(n)}(Id, P) : P ∈ SO(n) with P k^{−1}F ⊂ K_{S_{V}} F}

= ∠(span{K_{S_{V}} F}, k^{−1}F)
= ∠(kK_{S_{V}} F, F)
= ∠ ((kK_{S_{V}} F)^{⊥}, F^{⊥})
= ∠(kV′, F^{⊥})

So it remains to show that for any sequence k_{i} → k^{V} ∈ K_{V} in any fixed neighborhood U of K_{V}, that ∠(k_{i}V, F) ≥ Cd_{SO(n)}(k_{i}, K_{S_{V}}). Furthermore, since ∠(k_{i}V, F) = ∠(k_{i}(k_{i}^{V})^{−1}V, F) for any k_{i}^{V} ∈ K_{V}, we may assume that k_{i} → Id.

By Theorem 2.10.1 of [Va], in a sufficiently small neighborhood of Id we may uniquely write k_{i} as k_{i} = exp(k_{i}^{⊥}) exp(k_{i}^{S}) where k_{i}^{S} ∈ 𝔨_{S_{V_{0}}} and k_{i}^{⊥} ∈ 𝔨_{S_{V_{0}}}^{⊥}. Furthermore k_{i}^{S} → 0 and k_{i}^{⊥} → 0.

Bi-invariance of the metric on SO(n) implies that for |k_{i}^{⊥}| < \frac{π}{2} ,

d_{SO(n)}(k_{i}, K_{S_{V}}) = d_{SO(n)}(exp(k_{i}^{⊥}), K_{S_{V}}) = |k_{i}^{⊥}|

Now K_{V} is the only subgroup of K which both leaves V in F and also intersects all sufficiently small neighborhoods of the identity. Therefore, in order to show that ∠(k_{i}V, F) ≥ C|k_{i}^{⊥}|, we need only show that

d_{SO(n)}(k_{i}, K_{V})/|k_{i}^{⊥}| ↛ 0

Well, the Cambell-Baker-Hausdorff formula implies that

exp(k_{i}^{⊥}) exp(k_{i}^{S}) = exp (k_{i}^{⊥} + k_{i}^{S} + O(|k_{i}^{⊥}| · |k_{i}^{S}|))

Since the definition of S_{V} implies that 𝔨_{S_{V}} ⊃ 𝔨_{V} and k_{i}^{⊥} is perpendicular to 𝔨_{S_{V}}, we have

d_{SO(n)}(k_{i}, K_{V}) ≥ |k_{i}^{⊥}| + O(|k_{i}^{⊥}| · |k_{i}^{S}|)

Since we had |k_{i}^{S}| → 0 this finishes the lemma.

⋄

Lemma 5.4 (Angle inequality, II). For any subspace V of T_{x}X with dim V ≤ rank(X), there is a subspace V′ ⊥ V with dim V′ ≥ 2 dim V, and a constant C depending only on n, such that

∠(kV′, F^{⊥}) ≤ C∠(kV, F) for all k ∈ K

Proof. The first step of the proof is to reduce to the case when V is a subspace of F, so that Lemma 5.3 may be applied.

We first observe that the lemma is true if and only if it is true with V replaced by k_{0}V for any fixed k_{0} ∈ K. Since K is compact we may therefore choose V among all kV, k ∈ K so that ∠(V, F) ≤ ∠(kV, F) for all k ∈ K.

With this assumption, consider the projection W = π_{F} (V) of V onto F. By Lemma 5.3, we obtain a subspace W′ such that

∠(kW′, F^{⊥}) ≤ C∠(kW, F)

for all k ∈ K. Then we let V′ be the projection of W′ onto V^{⊥}. By the properties of the angle function (see 4.4), it follows that

∠(kV′, F^{⊥}) ≤ ∠(kW′, F^{⊥}) + ∠(kV′, kW′) by Lemma 4.3
≤ C∠(kW′, F^{⊥}) + ∠(V′, W′)
≤ C∠(kW, F) + ∠(V′, W′)  since (W^{⊥})^{⊥} ⊇ W
≤ C∠(kW, F) + ∠V, F  for same reason
= C∠(kW, F) + ∠(V, F)  since W = π_{F} (V)

Thus it suffices to bound ∠(kW, F) by a constant times ∠(kV, F). But

∠(kW, F) ≤ ∠(kV, F) + ∠(kV, kW) by Lemma 4.3
= ∠(kV, F) + ∠(V, W)
= ∠(kV, F) + ∠(V, F)  as W = π_{F} (V)
≤ ∠(kV, F) + ∠(kV, F) by minimality
= 2∠(kV, F)

and we are done. ⋄

5.3 Finishing the proof of the Eigenvalue Matching Theorem

Armed with the lemmas of the previous two subsections, we now prove Theorem 4.4.

We begin by noting that the construction of V′ from V above respects subspace inclusion. I.e. if U ⊂ V then U′ ⊂ V′. This follows from the definition of V′ and the fact that for two singular subspaces U_{1} and U_{2} with U_{1} ⊂ U_{2}, we have K_{U_{1}} · (W ∩ K_{U_{2}}) ⊃ K_{U_{2}}, where W is the Weyl group.

Now we simply proceed by induction on the number of vectors k. For k = 1 we set V = v_{1} the statement of the proposition follows from Lemma 5.4. Order the vectors by increasing angle with F. Assume the proposition for k − 1 vectors, then set V_{k} = span{v_{1}, ⋯, v_{k}}. By Lemma 5.4 we have an orthogonal subspace of twice the dimension of V_{k}, namely V′_{k}, which we may write by the preceeding paragraph as V′_{k} = V′_{k−1} ⊕ W′ where W′ is two dimensional. The same lemma also guarantees that ∠W′, F^{⊥} ≤ C∠v_{k}, F, since ∠v_{k}, F = ∠V_{k}, F.

This completes the proof of Theorem 4.4.

6 Finishing the proof of the Degree Theorem

We will break the proof of Theorem 1.2 into the compact and noncompact cases.

6.1 The compact case

Suppose M and N are compact. Since for s > h(g), F_{s} is a C^{1} map, using Proposition 3.2 and elementary integration theory yields,

|deg(f)| Vol(M) = |deg(f)| ∫_{M} dg_{0} = |∫_{N} f^{∗}dg_{0}|
= |∫_{N} F_{s}^{∗}dg_{0}|
≤ ∫_{N} |Jac F_{s}| dg ≤ C (\frac{s}{h(g_{0})})^{n} Vol(N)  (9)

For the last inequality we have used the principal estimate from Theorem 4.1. Rearranging terms gives us the inequality in Theorem 1.2 since C depends only on the dimension and (\frac{s}{h(g_{0})})^{n} depends only on n and the smallest Ricci curvatures of M and N.

6.2 The noncompact case

We now consider the case when N (and/or 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 will define the quantity h(g) to be

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

In fact this agrees with the previous definition for h(g) when N is compact. In the case of the symmetric space (M, g_{0}) this definition of h(g_{0}) agrees with the previous definition for compact manifolds.

For the finite volume case, 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 need to assume that the geometry of N is bounded in the sense that its Ricci curvatures are bounded from above and that the injectivity radius of its universal cover Y is bounded from below. These are the specific assumptions implied in the third remark after the theorem.

We will show that F_{s} is proper by essentially showing that the barycenter of σ_{y}^{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_{y}^{s} 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_{y}^{s}. Let v_{(x,θ)} be the unit vector in S_{x}X pointing to θ ∈ ∂X.

Lemma 6.1. 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 6.2. 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 2.4, 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}  (10)

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 rank(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 (10) 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 6.3. 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 the Ricci curvature is assumed to be bounded from above and the injectivity radius from below, 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 6.1 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 6.1 in light of Lemma 6.2. ⋄

Remarks:

1. In the proof of the above proposition, we used that injrad is bounded from below and Ricci curvature is bounded from above only to show that the volume of balls of any fixed radius are bounded from below.

2. 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. However, 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 classification of higher rank locally symmetric ends see [Ha]). For products of such surfaces it is possible, by expanding a family of infinite cones, to construct a proper map f : M → M such that for a radial sequence y_{i} → ξ ∈ ∂X, ϕ maps the bulk of the mass of μ_{y_{i}}^{s} into a set (almost) symmetrically arranged about a 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 last proposition actually shows that d(F_{s_{i}}(x), f(x)) is bounded for s_{i} ≥ s > h(g). In particular, the homotopy in Proposition 3.2 is proper. The inequality in Theorem 1.2 now follows as in the compact case, with deg(f) and deg(F_{s}) suitably interpreted.

References

[Al] P. Albuquerque, Patterson-Sullivan theory in higher rank symmetric spaces, Geom. Funct. Anal. (GAFA), Vol. 9 (1999), 1-28.

[BCG1] G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, . Geom. Funct. Anal. 5 (1995), no. 5, 731–799.

[BCG2] , Minimal entropy and Mostow’s rigidity theorems, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 623–649.

[BCG3] , in preparation.

[BCS] J. Boland, C. Connell, and J. Souto, Volume rigidity for finite volume manifolds, 1999 preprint.

[BGS] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, Vol. 61, Birkaüser, 1985.

[BC] R. L. Bishop and R. Crittenden, Geometry of manifolds, Academic Press, New York, 1964.

[Ch] I. Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Math., Vol. 108, Cambridge University Press, 1993.

[CF1] C. Connell and B. Farb, Minimal entropy rigidity for lattices in products of rank one symmetric spaces, to appear in Commun. in Analysis and Geomtry.

[CF2] C. Connell and B. Farb, Some recent applications of the barycenter method in geometry, in “Topology and Geometry of Manifolds”, Proc. Symp. Pure Math., Vol.71, to appear.

[Eb] P. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Math. Univ. of Chicago Press, 1996.

[FG] I. Fonseca and W. Gangbo, Degree theory in analysis and applications, The Clarendon Press Oxford University Press, New York, 1995, Oxford Science Publications.

[GJT] Y. Guivarc’h, L. Ji, and J. C. Taylor, Compactifications of symmetric spaces, Progress in Mathematics, Vol. 156, Birkaüser, 1998.

[Gr] M. Gromov, Volume and bounded cohomology, Inst. Hautes tudes Sci. Publ. Math. No. 56, 5–99 (1983).

[Gr2] M. Gromov, Filling Riemannian manifolds, J. Diff. Geom 18 (1983), 1–147.

[Ha] T. Hattori, Asymptotic geometry of arithmetic quotients of symetric spaces, Math. Z. 222 (1996), no. 2, 247-277.

[Hu1] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer-Verlag, 1972.

[Hu2] J. Humphreys, Linear Algebraic Groups, Grad. Texts in Math. 21, Springer-Verlag, 1981.

[Kn] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. 7 (1997), no. 4, 755-782.

[Pr] G. Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups, Amer. J. Math. 98 (1976), no. 1, 241–261.

[Sa] R. Savage, The space of positive definite matrices and Gromov’s invariant, Trans. Amer. Math. Soc. 274 (1982), no. 1, 239–263.

[Va] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Springer-Verlag, New York, 1984.

[Zi] R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Math., Vol. 81, Birkhäuser, 1984.

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