ISOSCATTERING DEFORMATIONS FOR COMPLETE MANIFOLDS OF NEGATIVE CURVATURE

PETER PERRY AND DOROTHEE SCHUETH

Abstract. We construct continuous families of non-isometric metrics on simply connected manifolds of dimension n ≥ 9 which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.

Contents

1. Introduction 1 2. Scattering Theory 3 2.1. The Spectral Shift 3 2.2. Resolvent Resonances 5 3. Isoscattering via Effective Torus Actions 7 4. Applications 11 5. Non-Isometry 14 References 16

1. Introduction

Compact closed manifolds X of strictly negative sectional curvature are infinitesimally spectrally rigid in the sense that any continuous isospectral deformation of the metric takes place in a single isometry class. The purpose of this paper is to show that, by contrast, there are continuous families of non-isometric metrics on non-compact manifolds which have strictly negative sectional curvatures and the “same spectrum” in a sense which we will make precise.

Our examples are constructed from conformally compact, asymptotically hyperbolic manifolds; we give the precise definition in section 2 of what follows. Such manifolds have at most finitely many L^{2}-eigenvalues and nonempty continuous spectrum with no embedded eigenvalues.

There are two natural notions of “spectral” data carried by the continuous spectrum. The first is the spectral shift, a real-valued function on ℝ which is the analogue of the counting function for the eigenvalues of the Laplacian on a compact manifold.

Key words and phrases. Geometric scattering, isophasal manifold.

2000 Mathematics Subject Classification. 58J53, 58J50.

The first author was supported in part by NSF grant DMS-0100829 and DMS-0408419.

The second author was supported in part by the DFG Priority Programme 1154.

The spectral shift is defined with respect to a background metric g_{0} which renormalizes the problem (see section 2.1). A celebrated formula of Birman and Krein shows that the spectral shift is identical to the scattering phase. A second analogue of “spectral” data for the Laplacian on an asymptotically hyperbolic manifold is the set of eigenvalues and resolvent resonances, enumerated with multiplicity. Resolvent resonances are poles of Green’s function for the Laplacian, meromorphically continued from its initial domain of definition on a cut plane to covering surface (see section 2.2).

If (X, g_{0}) is a complete, non-compact Riemannian manifold, two complete metrics g_{1} and g_{2} on X are said to be isophasal if the spectral shift functions for the pairs (Δ_{1}, Δ_{0}) and (Δ_{2}, Δ_{0}) are equal almost everywhere, and isopolar if the eigenvalues and resolvent resonances of Δ_{1} and Δ_{2} coincide. If g_{1} and g_{2} are both isopolar and isophasal, we will say that these metrics are isoscattering. We will develop sufficient criteria for two metrics to be isoscattering; our main class of examples are isoscattering metrics on ℝ^{n}:

Theorem 1.1. Suppose that g_{0} is a complete metric on ℝ^{n≥9} which is invariant under the canonical O(n)-action, and let K be any O(n)-invariant compact subset of ℝ^{n}. There exist continuous families of complete, non-isometric metrics {g_{α}} on X so that the g_{α} are mutually isophasal. Moreover, the metrics g_{α} coincide with g_{0} on ℝ^{n}\K and may be chosen arbitrarily close to g_{0} in the C^{∞} topology on symmetric two-tensors. If g_{0} is asymptotically hyperbolic, then the metrics {g_{α}} are also isopolar.

Note that the g_{α} are claimed to be isophasal/isopolar to each other (not to g_{0}).

Remark 1.2. Letting m = n − 4, the parameter space for the continuous families of isophasal metrics on ℝ^{n} that we construct has dimension

d ≥ \frac{m(m − 1)}{2} − [\frac{m}{2}] ([\frac{m}{2}] + 2) > 1

if n = 9 or n ≥ 11. (If n = 10, the parameter space has dimension at least 1).

Remark 1.3. (i) The condition that g_{0} be asymptotically hyperbolic is only used to ensure that the resolvents of the operators Δ_{α} can be mermorphically continued, so that resolvent resonances are well-defined. Presumably, isopolar metrics can also be constructed for other classes of metrics in geometric scattering theory (such as those discussed in Remark 4.10) although we have not done so here.

(ii) Theorem 1.1 will follow from a specific application (see Remark 4.8(ii)) of some general isophasality/isopolarity principles (Theorems 3.2/3.3, successively specialized in Theorems 4.3 and 4.7), and from a nonisometry proof given for this specific class of metrics in section 5 (see Theorem 5.1).

From Theorem 1.1 and the fact that the sectional curvatures of a metric of negative sectional curvature remain negative under a sufficient small perturbation we immediately deduce our main result:

Corollary 1.4. Let g_{0} be an O(n)-invariant, asymptotically hyperbolic metric on ℝ^{n≥9} with negative sectional curvatures. There exist continuous multiparameter families g_{α} of non-isometric, mutually isoscattering metrics on ℝ^{n} which coincide with g_{0} outside a given O(n)-invariant compact set and have negative sectional curvatures.

Guillemin and Kazhdan [16] showed that negatively curved closed surfaces are infinitesimally spectrally rigid. Min-Oo [25] extended this result to n-dimensional compact manifolds with negative curvature operator; Croke and Sharafutdinov [7] extended it to n-dimensional compact manifolds of strictly negative sectional curvature. All of these results rely on the following basic idea: If {g_{t}} is a one-parameter family of negatively curved metrics with the same Laplace spectrum, then the lengths of closed geodesics are also identical by the Duistermaat-Guillemin trace formula [8]. Thus, the integrals of \frac{d}{dt}g_{t} along any closed geodesic are zero. This condition is then used to prove that {g_{t}} is the orbit of a fixed metric under the action of a one-parameter group of diffeomorphisms.

A crucial ingredient of their proofs is that closed manifolds with strictly negative sectional curvatures have a dense orbit of the geodesic flow in the unit tangent bundle. This argument fails in the case of noncompact manifolds.

Gordon and Szabo [19] already constructed examples of compact manifolds with boundary that have strictly negative sectional curvatures but admit nontrivial isospectral deformations.

The proof of Theorem 1.1 uses the Riemannian submersion method (see, e.g., [10], [9], [13], [30], [31], [32]). The present paper generalizes previous results of Gordon and the first author [11], who constructed continuous families of compactly supported perturbations of the Euclidean metric on ℝ^{n} with the same scattering phase. Although discrete sets of metrics with the same scattering data have been previously constructed by a number of authors (see [1], [2], [3], [4], [19], [35], [36]), the results of [11] and the present paper are the first examples of continuous families of non-isometric metrics with the same scattering data. Moreover, the metrics in [11] and the present paper are not even locally isometric.

We conclude with a brief sketch of the contents. In §2, we recall some basic facts regarding the spectral shift function and resolvent resonances. In §3, we give a version of the Riemannian submersion method adapted to scattering theory. In §4 we show how to construct examples—among others, those which yield the proof of Theorem 1.1; see remarks 4.8(ii) and 4.9, and in §5 we show that these examples are nontrivial (i.e., the metrics in the continuous isophasal/isopolar families which we construct are mutually non-isometric).

2. Scattering Theory

2.1. The Spectral Shift. Let (X, g_{0}) be a complete Riemannian manifold, and let g be another complete metric on X. For simplicity we will suppose that the volume forms of g and g_{0} are equal so that the respective Laplacians Δ and Δ_{0} act on the same Hilbert spaces. We wish to define the spectral shift function associated to the pair (Δ, Δ_{0}).

The following theorem is a standard consequence of the Birman-Krein theory of the spectral shift (see, for example, [34], chapter 8 for discussion and references, and see particularly Theorem 1 of §8.9.2).

Theorem 2.1. Suppose that H and H_{0} are positive self-adjoint operators on a Hilbert space, and that (H + c)^{−k} − (H_{0} + c)^{−k} is a trace-class operator for some c > 0 and some positive integer k. Then there is a real-valued measurable and locally integrable function λ ↦ ξ (λ, H, H_{0}) with the following properties:

(i) ξ (λ, H, H_{0}) = 0 if λ < 0, (ii) For any real-valued function f ∈ C^{2}(ℝ) that obeys the estimate

sup_{λ≥1} |λ^{1+ε} · \frac{d}{dλ} (λ^{k+1}f′(λ))| < ∞

for some ε > 0, the formula

(2.1)  Tr [f(H) − f(H_{0})] = − ∫ ξ(λ, H, H_{0}) f′(λ) dλ

holds.

Remark 2.2. (i) The function ξ( · , H, H_{0}) is called the spectral shift function associated with H and H_{0}.

(ii) Note that the class of admissible functions f includes the function f(λ) = exp(−tλ). With this choice of f we see that the formal Laplace transform of the “measure” dξ is Tr (exp(−tH) − exp(−tH_{0})), a relative heat trace.

Remark 2.3. The spectral shift function is a natural analogue of the counting function for eigenvalues of the Laplacian on a compact manifold. If (X, g) is a compact Riemannian manifold, H is the Laplacian on (X, g), {λ_{k}}_{k=0}^{∞} are the eigenvalues of H, and

N(λ) = # {λ_{k} : λ_{k} ≤ λ} ,

the ‘trace formula’

Tr (f(H)) = − ∫ f′(λ)N(λ) dλ

holds. In the non-compact case it is necessary to ‘renormalize’ by the Laplacian of a reference metric in order to obtain a well-defined trace formula.

The following result is easily deduced from [5] and guarantees that the trace condition in Theorem 2.1 is satisfied by the pair (H, H_{0}) = (Δ, Δ_{0}) under certain assumptions which will always be satisfied in our later constructions (see sections 3 and 4).

Theorem 2.4. Let X be a smooth manifold of dimension n and let K be a compact subset of X. Suppose that g_{0} and g are complete metrics on X with the property that g = g_{0} on X\K and the volume forms of g and g_{0} coincide. Let Δ be the Laplacian on (X, g) and let Δ_{0} be the Laplacian on (X, g_{0}). Then, for any c > 0 and k > n/2, the operator

(Δ + c)^{−k} − (Δ_{0} + c)^{−k}

is a trace-class operator. In particular, there is a spectral shift function ξ( · , Δ, Δ_{0}).

Definition 2.5. Let g_{0}, g_{1}, g_{2} be complete metrics on a manifold X such that both (Δ_{1}, Δ_{0}) and (Δ_{2}, Δ_{0}) satisfy the conditions of Theorem 2.1. We call g_{1} and g_{2} isophasal (with respect to g_{0}) if the associated spectral shift functions coincide almost everywhere.

The spectral shift function is connected to the scattering phase by the celebrated Birman-Krein formula (see, for example [34], §8.4). Although we will not use the Birman-Krein formula explicitly, we recall some basic ideas to motivate the terminology of Definition 2.5. For a pair of operators (H, H_{0}) on a Hilbert space H satisfying the hypotheses of Theorem 2.1, let

W^{±} = s- lim_{t→±∞} exp(itH) exp (−itH_{0}) P_{a.c.}(H_{0})

where s-lim means limit in the strong operator topology on H, and P_{a.c.}(H_{0}) projects onto the absolutely continuous spectral subspace for the operator H_{0}. The trace-class theory of scattering asserts that these limits exist and that Ran W^{+} = Ran W^{−} = H_{a.c.}(H), the absolutely continuous spectral subspace for the operator H. Thus the scattering operator S = (W^{+})^{∗} W^{−} is well-defined; owing to the intertwining relation exp(isH)W^{±} = W^{±} exp (isH_{0}), the operator S commutes with H_{0}. If H = ∫^{⊕} 𝔥_{λ} dλ is a direct integral decomposition of H into spectral subspaces for H_{0}, then S = ∫^{⊕} S(λ) dλ where S(λ) is a unitary operator of determinant class on 𝔥_{λ} (that is, S(λ) − I is a trace-class operator so that the operator determinant det S(λ) is well-defined). The Birman-Krein formula asserts that

det S(λ) = exp(2πiξ (λ, H, H_{0}))

for almost every λ. The quantity arg det S(λ) is referred to as the scattering phase, so it is clear that equality of spectral shifts implies equality of scattering phases.

2.2. Resolvent Resonances. Resolvent resonances are defined for many classes of non-compact Riemannian manifolds, and have been extensively studied for one particular class, namely the asymptotically hyperbolic manifolds (see, for example, [20], [22], [23] for spectral theory and meromorphic continuation of Green’s function and [17], [18] for bounds on the distribution of resonances). Let (X, g) be a conformally compact Riemannian manifold, that is, X is the interior of a smooth manifold \overline{X} with boundary, and there is a defining function ρ for ∂\overline{X} so that h = ρ^{2}g extends to a smooth, nondegenerate metric on \overline{X} (the condition that ρ be a defining function for ∂\overline{X} means that ρ ∈ C^{∞} (\overline{X}), ρ ≥ 0, ρ^{−1}(0) = ∂\overline{X}, and for each x ∈ ∂\overline{X}, dρ(x) ≠ 0). Such a manifold (X, g) is called asymptotically hyperbolic if, in addition, |dρ(x)|_{h} → 1 as x → ∂\overline{X}. An elementary example is the open unit ball in ℝ^{n} with the metric g = ρ^{−2}h, where h is the Euclidean metric restricted to the open unit ball in ℝ^{n} and ρ(x) = \frac{1}{2}(1 − |x|^{2}).

If (X, g) is asymptotically hyperbolic and dim X = n, then the Laplacian Δ_{g} has at most finitely many L^{2}-eigenvalues of finite multiplicity in the interval [0, (n− 1)^{2}/4) [22] and absolutely continuous spectrum in the interval [(n − 1)^{2}/4, ∞) (this follows from the meromorphic continuation of the resolvent kernel established in [23]). It follows that the resolvent operator

R̃(z) = (Δ_{g} − z)^{−1}

is a meromorphic function of z on the cut plane ℂ\[(n − 1)^{2}/4, ∞) with at most finitely many poles of finite rank at the eigenvalues (if any) of Δ_{g}. Viewed as an operator from L^{2} (X, g) to itself, the resolvent R̃(z) becomes unbounded as Im(z) → 0 and Re(z) approaches any number λ ∈ [(n − 1)^{2}/4, ∞) owing to the presence of the absolutely continuous spectrum. On the other hand, the operator R(z) has a Schwarz kernel (Green’s function) with respect to the Riemannian measure on (X, g) which continues meromorphically in z to a double cover of the cut plane. This meromorphic continuation can be captured by viewing R̃(z) as a map from a smaller space than L^{2}(X, g), namely the L^{2} functions of compact support, to a larger space, the locally square integrable functions on X. Poles of the meromorphically continued Green’s function are called resolvent resonances and physically represent “almost bound states.” As with eigenvalues of the Laplacian, resolvent resonances determine the expansion in “normal modes” of solutions for the wave equation but unlike eigenvalues they represent “normal modes” whose energy in any bounded region decays with time (see, for example, [6], [33], and [37] for recent work on such “resonance wave expansions”). The energy decay is a consequence of the non-compactness of the underlying manifold.

To make these notions more precise, let

R(s) = (Δ − s(n − 1 − s))^{−1}

for Re(s) > (n − 1)/2. The map s ↦ s (n − 1 − s) maps this region onto the cut plane ℂ\[(n − 1)^{2} /4, ∞). In the half-plane Re(s) > (n − 1) /2, R(s) is a meromorphic operator-valued function with poles at those real numbers ζ > (n − 1)/2 for which ζ(n − 1 − ζ) is an eigenvalue of Δ. Denote by L_{comp}^{2} (X, g) the compactly supported, square-integrable functions on (X, g) and by L_{loc}^{2} (X, g) the space of measurable, locally square-integrable functions on (X, g). Mazzeo and Melrose [23] proved:

Theorem 2.6. Suppose that (X, g) is an asymptotically hyperbolic manifold of dimension n. Then R(s), viewed as an operator from L_{comp}^{2}(X, g) to L_{loc}^{2}(X, g), has a meromorphic continuation to the complex s-plane whose poles do not accumulate in the complex s-plane with the possible exception of the points

{(n − 1 − j) /2 : j = 1, 2, ⋯ } .

The operator R(s) has no poles on the line Re(s) = (n − 1)/2 with the possible exception of the point s = (n − 1)/2, where the Laurent expansion has finite polar part. If ζ is a pole of R(s) and ζ is not an exceptional point as above, then the resolvent R(s) has a Laurent series with finite polar part of the form

(2.2)  ∑_{j=1}^{N_{ζ}} A_{j} (s − ζ)^{−j}

where the A_{j} are finite-rank operators from L_{comp}^{2} (X, g) to L_{loc}^{2} (X, g).

Remark 2.7. Guillopé-Zworski [18] showed that there are no exceptional points if (X, g) has constant sectional curvatures outside a compact subset of X, and Guillarmou [14] has given necessary and sufficient conditions for the meromorphically continued resolvent operator R(s) to have essential singularities at the exceptional points. For a careful analysis of the relationship between resolvent poles and poles of the meromorphically continued scattering operator, see [15].

Note that, for poles ζ of R(s) with s > (n − 1) /2, R(s) has polar part

A_{1} (n − 1 − 2ζ)^{−1} (s − ζ)^{−1}

where A_{1} is the orthogonal projection onto the eigenspace of Δ with eigenvalue ζ (n − 1 − ζ) (the factor (n − 1 − 2ζ) is the derivative of the map s ↦ s (n − 1 − s) at s = ζ). Thus, dim (Ran A_{1}) is the multiplicity of the eigenvalue associated to ζ. More generally, the multiplicity of a resolvent pole is the number

m_{ζ} = dim (Ran P_{ζ}) .

where

P_{ζ} = ∫_{|s−ζ|<ε} (2s − n + 1) R(s) ds

and ε > 0 is chosen so small that there are no other resolvent resonances in the disc of size ε centered at ζ. Note that, if ζ is a pole of the resolvent corresponding to an L^{2}-eigenvalue of the Laplacian, m_{ζ} is the dimension of the corresponding eigenspace.

Definition 2.8. Two asymptotically hyperbolic manifolds are called isopolar if the associated Laplacians on functions have the same eigenvalues and resolvent poles with multiplicity.

3. Isoscattering via Effective Torus Actions

In this section, we prove two theorems (Theorems 3.2 and 3.3) which adapt Theorem 1.4 of [32] to the setting of scattering theory on a complete, non-compact Riemannian manifold. The first gives a criterion for constructing isophasal manifolds and the second gives a criterion for constructing isopolar manifolds.

To adapt the proof of Theorem 1.4 of [32] to the scattering context, it will be helpful to recall how the Laplacian is defined via quadratic forms on a closed manifold, a manifold with boundary, and an open manifold. In all cases we begin with the positive quadratic form

(3.1)  𝔮 (u, v) = ∫_{X} ∇u · ∇v dvol_{g}

for u and v belonging to a suitably chosen linear space 𝔔 of smooth functions, and we introduce the auxiliary inner product

(3.2)  ⟨u, v⟩ = 𝔮(u, v) + (u, v)_{L^{2}(X,g)}

on 𝔔. The Laplacian is then defined as the Friedrichs extension associated to the quadratic form 𝔮 (see, for example, Reed-Simon [28], §X.3, Theorem X.23; the operator A referred to there is in our case the Laplace operator on 𝔔). The Friedrichs extension has the property that its operator domain 𝔇 is contained in the form domain of the closure of 𝔮. By the Form Representation Theorem (see for example Kato [21], Theorem VI.2.1), a vector φ belongs to 𝔇 if and only if there is a vector χ ∈ L^{2} (X, g) with the property that

(3.3)  𝔮 (φ, η) = (χ, η)

for all η ∈ 𝔔, where the right-hand side denotes the L^{2}-inner product. In this case Δφ = χ.

(1) If X is a closed manifold, the closure of the form 𝔮 with domain 𝔔 = C^{∞}(X) has form domain H^{1} (X, g), a Hilbert space with inner product given by (3.2), and the self-adjoint operator associated to 𝔮 is the usual Laplacian on functions.

(2) If X is a compact manifold with boundary, as in [32], the quadratic form (3.1) with domain 𝔔 = C_{0}^{∞}(X) closes to the quadratic form associated with the Laplacian on X with Dirichlet boundary conditions, while the form (3.1) with domain 𝔔 = C^{∞}(X) closes to the quadratic form of the Laplacian with Neumann boundary conditions on X. The domain of the closure of 𝔮 is the space H_{0}^{1}(X, g) in the case of Dirichlet boundary conditions, and H^{1} (X, g) in the case of Neumann boundary conditions.

(3) If X is an open manifold and (X, g_{0}) is complete, the quadratic form 𝔮 with domain 𝔔 = C_{0}^{∞}(X) is closable and the domain of its closure is again denoted H^{1} (X, g). The Laplacian on X is the unique positive operator associated to the closure of 𝔮 by the Friedrichs construction.

In [32], manifolds (X, g) and (X′, g′) (which may be closed manifolds or compact manifolds with boundary) are shown to be isospectral by constructing an isometry Q from H_{∗}^{1} (X, g) to H_{∗}^{1}(X′, g′) which extends to a unitary map from L^{2} (X, g) to L^{2} (X′, g′) (here H_{∗}^{1} (X, g) denotes one of H_{0}^{1} (X, g) or H^{1} (X, g) and the same choice is made for H_{∗}^{1} (X′, g′)). Here we observe that the same construction works if (X, g) and (X′, g′) are open manifolds, with some additional hypotheses to insure that scattering data are preserved.

We first recall Theorem 1.4 of [32] (with trivial notational changes). By a torus we mean a nontrivial, compact, connected abelian Lie group. Suppose that T is a torus acting smoothly and effectively by isometries on a complete Riemannian manifold (X, g), and let X̂ denote the union of principal orbits of T. Then X̂ is an open and dense submanifold of X. We denote by g^{T} the unique Riemannian metric on X̂/T such that the canonical projection π : (X̂, g) → (X̂/T, g^{T}) is a Riemannian submersion.

Theorem 3.1. [32] Let T be a torus which acts effectively on two compact connected Riemannian manifolds (X_{1}, g_{1}) and (X_{2}, g_{2}) by isometries. For each subtorus W ⊂ T of codimension one, suppose there exists a T -equivariant diffeomorphism F_{W} : X_{1} → X_{2} which satisfies:

(i) F_{W}^{∗} dvol_{2} = dvol_{1} and (ii) F_{W} induces an isometry \overline{F}_{W} between the quotient manifolds (X̂_{1}/W, g_{1}^{W})

and (X̂_{2}/W, g_{2}^{W}).

Then (X_{1}, g_{1}) and (X_{2}, g_{2}) are isospectral; if the manifolds have boundary then they are Dirichlet and Neumann isospectral.

To construct examples of manifolds with the same scattering phase, we consider a noncompact complete Riemannian manifold (X, g_{0}) and metrics g_{1} and g_{2} (on the same smooth manifold X) which are compactly supported perturbations of g_{0} For simplicity, we suppose that g_{1} and g_{2} coincide with g_{0} outside a fixed compact subset of X. The metric g_{0} serves as a “background” or “comparison” metric. The following Theorem gives conditions under which the scattering phases of two pairs (Δ_{1}, Δ_{0}) and (Δ_{2}, Δ_{0}) are the same. In order to guarantee that the torus action is compatible with the background metric g_{0}, we add an hypothesis about the map F_{W}.

Theorem 3.2. Let X be a smooth Riemannian manifold carrying complete metrics g_{0}, g_{1}, and g_{2}, where the g_{i} are compactly supported perturbations of g_{0}. Suppose that the volume forms of g_{0}, g_{1}, g_{2} are equal, so that the Laplacians Δ_{0}, Δ_{1}, and Δ_{2} act on the same Hilbert space. Let T be a torus which acts effectively by isometries on (X, g_{1}) and (X, g_{2}). For each subtorus W ⊂ T of codimension one, suppose that there exists a T -equivariant diffeomorphism F_{W} of X with the following properties:

(i) F_{W}^{∗} (dvol_{0}) = dvol_{0}, (ii) F_{W} induces an isometry \overline{F}_{W} between the quotient manifolds (X̂/W, g_{1}^{W})

and (X̂/W, g_{2}^{W}), and (iii) F_{W}^{∗} : C^{∞}(X) → C^{∞}(X) commutes with Δ_{0}. Then g_{1} and g_{2} are isophasal with respect to g_{0} (see Definition 2.5).

Proof. The space H = L^{2}(X, g_{0}) admits a decomposition

(3.4)  H = H_{0} ⊕ ⨁_{W ⊂T} (H_{W} ⊖ H_{0})

where H_{0} denotes the T -invariant functions, H_{W} denotes the W -invariant functions, and the direct sum goes over subtori W of T having codimension one. Denote one of the various maps F_{W} by F_{0}. The proof of Theorem 1.4 in [32] goes over without change to show that there is an isometry Q from H to H given by

Q = F_{0}^{∗ −1} ⊕ ⨁_{W ⊂T} F_{W}^{∗ −1}

which restricts to a unitary map from H^{1} (X, g_{1}) to H^{1} (X, g_{2}).

We claim that (a) Q commutes with (Δ_{0} − z)^{−1} and (b) the identity

Q (Δ_{1} − z)^{−1} = (Δ_{2} − z)^{−1} Q

holds for any z ∈ ℂ\ℝ. Claim (a) follows directly from hypothesis (iii) and the definition of Q. To prove claim (b), we note that, since Q is an isometry from H^{1} (X, g_{1}) to H^{1} (X, g_{2}), we have

(3.5)  𝔮_{2}(Qφ, ψ) = 𝔮_{1} (φ, Q^{−1}ψ)

for any φ in the form domain of Δ_{1} and ψ belonging to the form domain of Δ_{2}. We will use this relation and the form representation theorem (see (3.3) and the surrounding remarks) to prove that if φ belongs to 𝔇(Δ_{1}) then Qφ ∈ 𝔇(Δ_{2}) and Δ_{2} (Qφ) = Q (Δ_{1}φ). Since φ ∈ 𝔇(Δ_{1}), there is a vector χ ∈ L^{2}(X, g_{1}) with the property that

𝔮_{1} (φ, η) = (χ, η)_{L^{2}(X,g_{1})}

for any η belonging to the form domain of Δ_{1}; by definition, Δ_{1}φ = χ. It follows from (3.5) that for any ζ in the form domain of Δ_{2},

𝔮_{2}(Qφ, ζ) = 𝔮_{1}(φ, Q^{−1}ζ)
= (χ, Q^{−1}ζ)_{L^{2}(X,g_{1})}
= (Qχ, ζ)_{L^{2}(X,g_{2})}

so that Qφ ∈ 𝔇(Δ_{2}) and Δ_{2} (Qφ) = Qχ as claimed. We now conclude that Q (Δ_{1} − z)^{−1} = (Δ_{2} − z)^{−1} Q, proving claim (b). From claims (a) and (b) we deduce by differentiation in z that

Q (Δ_{0} − z)^{−m} = (Δ_{0} − z)^{−m} Q

and

Q (Δ_{1} − z)^{−m} = (Δ_{2} − z)^{−m} Q

for any integer m and any z ∈ ℂ\ℝ. By the Stone-Weierstrass Theorem, polynomials in (x ± i)^{−1} are dense in the Banach algebra C_{∞}(ℝ) of continuous functions vanishing at infinity. It now follows that for any f ∈ C_{∞}(ℝ), Qf(Δ_{1}) = f(Δ_{2})Q and Qf (Δ_{0}) = f (Δ_{0}) Q.

For t > 0, let f_{t} ∈ C_{∞}(ℝ) with f_{t}(x) = exp(−tx) for x ≥ 0. Since

Tr (f_{t}(Δ_{1}) − f_{t} (Δ_{0})) = Tr (Q [f_{t}(Δ_{1}) − f_{t} (Δ_{0})] Q^{∗})
= Tr (f_{t}(Δ_{2}) − f_{t} (Δ_{0})) ,

we conclude from Theorems 2.1 and 2.4 , Remark 2.2, and the Krein trace formula (2.1) that for any t > 0,

∫ ξ(λ, Δ_{1}, Δ_{0}) exp (−λt) dλ = ∫ ξ(λ, Δ_{2}, Δ_{0}) exp (−λt) dλ.

We again appeal to the Stone-Weierstrass theorem to argue that polynomials in exp (−λt) for t > 0 are dense in the continuous functions on ℝ^{+} that vanish at infinity, so that for any continuous function f of compact support in (0, ∞) we have

∫ ξ(λ, Δ_{1}, Δ_{0}) f(λ) dλ = ∫ ξ(λ, Δ_{2}, Δ_{0}) f(λ) dλ.

It now follows that ξ(λ, Δ_{1}, Δ_{0}) = ξ(λ, Δ_{2}, Δ_{0}) almost everywhere. □

The next theorem concerns resolvent resonances. To discuss these, we impose an additional assumption on the map F_{W} which allows us to construct an intertwining operator Q for the meromorphically continued resolvents.

Theorem 3.3. Let g_{1} and g_{2} be asymptotically hyperbolic metrics on a manifold X. Suppose that the volume forms of g_{1}, g_{2} are equal, so that the Laplacians Δ_{1} and Δ_{2} act on the same Hilbert space. Let T be a torus which acts effectively by isometries on (X, g_{1}) and (X, g_{2}). For each subtorus W ⊂ T of codimension one, suppose that there exists a T -equivariant diffeomorphism F_{W} of X with the following properties:

(i) F_{W}^{∗} (dvol) = dvol, where dvol := dvol_{1} = dvol_{2}.

(ii) F_{W} induces an isometry \overline{F}_{W} between the quotient manifolds (X̂/W, g_{1}^{W})

and (X̂/W, g_{2}^{W}), and (iii) If C is a compact subset of X, then ∪_{W ⊂T} F_{W} (C) and ∪_{W ⊂T} F_{W}^{−1}(C) are

also compact, where the union goes over subtori of T having codimension one.

Then (X, g_{1}) and (X, g_{2}) are isopolar (see Definition 2.8).

Proof. First, we argue as in the proof of Theorem 3.2 that, under assumptions (i) and (ii), there is a unitary operator Q which intertwines the resolvents of Δ_{1} and Δ_{2}. Next, we show that the operator Q preserves the space L_{comp}^{2} (X) (since the volume forms of g_{1} and g_{2} are equal, we denote either of the spaces L_{comp}^{2} (X, g_{1}) or L_{comp}^{2}(X, g_{2}) by L_{comp}^{2}(X)). To see this, suppose that f has support contained in a fixed subset C of X, which we will assume without loss is T -invariant. The projection of f onto the subspace H_{0} of T -invariant functions is the average ∫_{T} f (z · x) dμ_{T} (z), where dμ_{T} is normalized Haar measure on T, and so is clearly supported in C. A similar formula computes the projection of f onto the subspace H_{W} of W -invariant functions for any subtorus W of T. It now follows from assumption (iii) that Qf and Q^{−1}f have support in a fixed compact subset of X. Hence, Q and Q^{−1} preserve the functions of compact support on X and, since Q is a unitary mapping from L^{2}(X, g_{1}) to L^{2}(X, g_{2}), they also preserve the space L_{comp}^{2}(X). For j = 1, 2, let

R_{j}(s) = (Δ_{j} − s (n − 1 − s))^{−1}

For all u, v ∈ L^{2}(X) and all s with Re(s) > (n − 1)/2 we have (Qu, QR_{1}(s)v) = (Qu, R_{2}(s)Qv) and thus

(3.6)  (u, R_{1}(s)v) = (Qu, R_{2}(s)Qv) .

Now we assume that u and v — and hence, also, Qu and Qv — belong to L_{comp}^{2}(X). Then, by Theorem 2.6, the functions on each side of (3.6) admit a meromorphic continuation to the complex s-plane. If ζ is not an exceptional point, the resolvents R_{1}(s) and R_{2}(s) have finite polar parts at s = ζ whose coefficients are finite-rank operators from L_{comp}^{2}(X) to L_{loc}^{2}(X) (see (2.2)). For j = 1, 2, let

P_{ζ,j} = ∫_{|s−ζ|<ε} (2s − n + 1) R_{j}(s) ds

ISOSCATTERING DEFORMATIONS FOR MANIFOLDS OF NEGATIVE CURVATURE 11

where ε > 0 is so small that ζ is the only resonance in the disc of radius ε about ζ. Recall that the multiplicity of the resonance ζ for Δ_{j} is given by m_{ζ,j} = dim (Ran P_{ζ,j}). From (3.6) we conclude that

(3.7)  (u, P_{ζ,1}v) = (Qu, P_{ζ,2}Qv)

for all u, v ∈ L_{comp}^{2}(X). Let v_{1}, …, v_{m_{ζ,1}} ∈ L_{comp}^{2}(X) be such that the corresponding functions w_{j} := P_{ζ,1}v_{j} ∈ L_{loc}^{2}(X) constitute a basis of Ran P_{ζ,1}. By an elementary argument, there exist u_{1}, …, u_{m_{ζ,1}} ∈ L_{comp}^{2}(X) such that the matrix ((u_{i}, w_{j})) is invertible. By (3.7) it follows that ((Qu_{i}, P_{ζ,2}Qv_{j})) is invertible. Thus the functions P_{ζ,2}Qv_{j} are linearly independent in L_{loc}^{2}(X). We conclude m_{ζ,1} ≤ m_{ζ,2}. Replacing u, v by Q^{−1}u, Q^{−1}v in (3.7), we analogously obtain m_{ζ,2} ≤ m_{ζ,1}. □

4. Applications

In Theorems 4.3, 4.7 and Remarks 4.8, 4.10 we will formulate a succesive series of specializations of Theorems 3.2 and 3.3, leading up to the examples which give us the proof of Theorem 1.1.

Let (X, g_{0}) be a complete Riemannian manifold, and let T ⊂ Isom(X, g_{0}) be a torus with Lie algebra 𝔱. For Z ∈ 𝔱, we denote the induced vectorfield on X by Z^{∗}.

Definition 4.1. (i) A 𝔱-valued 1-form λ on X is called admissible (with respect to the T -action) if λ is T -invariant and horizontal, i.e., vanishes on vectors tangent to the T -orbits.

(ii) With each admissible λ on X we associate a Riemannian metric g_{λ} on X defined by

g_{λ}(X, Y) = g_{0}(X + λ(X)^{∗}, Y + λ(Y)^{∗}).

(iii) Two admissible 𝔱-valued 1-forms λ, λ′ on X are called isospectral (with respect to g_{0} and the T -action) if for each μ ∈ 𝔱^{∗} there exists a T -equivariant isometry F_{μ} ∈ Isom(X, g_{0}) such that μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′).

Remark 4.2. (i) If λ, λ′ are admissible and isospectral, then so are ψλ, ψλ′, where ψ is any smooth function on X which is invariant under T and for which the F_{μ} in the above definition can be chosen such that ψ is invariant under each of them.

(ii) If λ is admissible and compactly supported then g_{λ} is a compactly supported perturbation of the metric g_{0} on X, and the volume elements of g_{λ} and g_{0} coincide. Moreover, g_{0}^{T} = g_{λ}^{T} on X̂/T.

Theorem 4.3. Let λ, λ′ be admissible, isospectral, and compactly supported 𝔱-valued 1-forms on (X, g_{0}), where g_{0} is a complete Riemannian metric on X. Then g_{λ} and g_{λ′} are isophasal with respect to g_{0}. If, moreover, (X, g_{0}) — and hence also (X, g_{λ}) and (X, g_{λ′}) — are asymptotically hyperbolic, then (X, g_{λ}) and (X, g_{λ′}) are isopolar.

Proof. We appeal to Theorems 3.2 and 3.3, respectively. Let W ⊂ T be a subtorus of codimension one. Choose μ ∈ 𝔱^{∗} such that ker μ = T_{e}W, and let F_{W} := F_{μ}, where F_{μ} is chosen as in Definition 4.1(iii). Since F_{μ} is an isometry of g_{0}, it commutes with Δ_{0} and satisfies F_{μ}^{∗} dvol_{g_{λ′}} = dvol_{g_{λ}} because of dvol_{g_{λ}} = dvol_{g_{0}} = dvol_{g_{λ′}}. We claim that F_{μ} induces an isometry from (X̂/W, g_{λ}^{W}) to (X̂/W, g_{λ′}^{W}). Denote the space of vectors in T_{x}X tangent to the W -orbit through x by 𝔴_{x}. Let V ∈ T_{x}X be any vector which is g_{λ}-orthogonal to 𝔴_{x}. Then V = V_{0} − λ(V_{0}) for some V_{0} ∈ T_{x}X which is g_{0}-orthogonal to 𝔴_{x}. We have F_{μ∗}V = F_{μ∗}(V_{0} − λ(V_{0})^{∗}) = F_{μ∗}V_{0} − λ(V_{0})^{∗}, and by the choice of F_{μ} this is, up to an error in 𝔴_{F_{μ}x}, the same as V′ := F_{μ∗}V_{0} − λ′(F_{μ∗}V_{0})^{∗}. But F_{μ∗}V_{0} is g_{0}-orthogonal to 𝔴_{F_{μ}(x)} ; thus V′ is the projection of F_{μ∗}V_{0} to the g_{λ′}-orthogonal complement of 𝔴_{F_{μ}(x)}. Our claim now follows from ∥V′∥_{g_{λ′}} = ∥F_{μ∗}V_{0}∥_{g_{0}} = ∥V_{0}∥_{g_{0}} = ∥V ∥_{g_{λ}} .

We have now shown that F_{μ} satisfies all conditions of Theorem 3.2, as well as conditions (i) and (ii) of Theorem 3.3. It remains to check condition (iii) of the latter. Let C ⊂ X be compact, and denote by D the maximal g_{0}-distance between points of C and points of the compact support supp(λ) of λ. For any μ ∈ 𝔱^{∗} with μ ∘ λ ≠ 0, the set F_{μ}(C) is then contained in {x ∈ X | d(x, F_{μ}(supp(μ ∘ λ)) ≤ D} = {x ∈ X | d(x, supp(μ ∘ λ′)) ≤ D} ⊂ {x ∈ X | d(x, supp(λ′)) ≤ D}. Moreover, for any μ with μ ∘ λ = 0 we necessarily have μ ∘ λ′ = 0 and can thus assume F_{μ} = Id. Hence each F_{μ} maps C to the union of C and the closed D-neighborhood of supp(λ′), which is compact. Thus, the maps F_{μ} satisfy the required condition; the same argument with reversed roles of λ and λ′ works for the maps F_{μ}^{−1}. □

In the following, let H be a compact connected semisimple Lie group with Lie algebra 𝔥, and let the Lie algebra 𝔱 of T be endowed with a fixed (auxiliary) Euclidean inner product.

Definition 4.4. (i) Two linear maps j, j′ : 𝔱 → 𝔥 are called isospectral if for each Z ∈ 𝔱 there is a_{Z} ∈ H such that j′_{Z} = Ad_{a_{Z}} (j_{Z}). Here j_{Z} denotes j(Z), and similarly for j′.

(ii) j and j′ are called equivalent if there is Φ ∈ Aut(𝔥) and C ∈ O(𝔱) such that j′_{Z} = Φ(j_{C(Z)}) for all Z ∈ 𝔱.

Remark 4.5. Let 𝔱 = ℝ^{2}, equipped with the standard metric, and denote by J the vector space of all linear maps from 𝔱 to 𝔥.

(i) [13] If 𝔥 = 𝔰𝔬(m), where m is any positive integer other than 1, 2, 3, 4, or 6, then there is a Zariski open subset O of J such that each j ∈ O belongs to a d-parameter family of isospectral, inequivalent elements of J. Here d ≥ m(m − 1)/2 − [m/2]([m/2] + 2) > 1. For m = 6, there exist at least 1-parameter families in J with these properties.

(ii) [32] If 𝔥 = 𝔰𝔲(m), where m ≥ 3, then there is a Zariski open subset O of J such that each j ∈ O belongs to a continuous family of isospectral, inequivalent elements of J.

(iii) [27] Recently, Emily Proctor established results analogous to those in (i) (multiparameter families) for 𝔰𝔲(m ≥ 5) and 𝔰𝔭(m ≥ 2).

Now suppose that H × T acts on a complete connected Riemannian manifold (X, g_{0}) by isometries.

Definition 4.6. For each linear map j : 𝔱 → 𝔥 define an associated 𝔱-valued 1-form λ^{j} on X by

⟨λ^{j}(Y), Z⟩ = g_{0}(j_{Z}^{∗}(x), Y)

for all Z ∈ 𝔱 and Y ∈ T_{x}X, x ∈ X, where ⟨ , ⟩ is the given inner product on 𝔱. Here again, the star denotes the induced vectorfield on X.

Theorem 4.7. Let H × T act on (X, g_{0}) by isometries. Let j, j′ : 𝔱 → 𝔥 be isospectral, and let ψ ∈ C^{∞}(X) be compactly supported and invariant under H × T.

(i) If, for each x ∈ X, the H-orbit through x and the T -orbit through x meet g_{0}-orthogonally in x, then the 𝔱-valued 1-forms ψλ^{j} and ψλ^{j′} satisfy the conditions of Theorem 4.3.

(ii) Let {Z_{1} , …, Z_{k}} be a basis of 𝔱, and denote by λ_{h}^{j}, λ_{h}^{j′} the horizontal part of vλ^{j}, vλ^{j′}, respectively, where v := ∥Z_{1}^{∗} ∧ … ∧ Z_{k}^{∗}∥^{2} and the norm on ⋀^{k} T_{x}M is the usual Euclidean norm induced by g_{0}. Then the 𝔱-valued 1-forms ψλ_{h}^{j} and ψλ_{h}^{j′} satisfy the conditions of Theorem 4.3.

In particular, the associated pairs of metrics g_{ψλ^{j}} and g_{ψλ^{j′}} in (i) (respectively g_{ψλ_{h}^{j}} and g_{ψλ_{h}^{j′}} in (ii)) are isophasal with respect to g_{0}; if (X, g_{0}) is asymptotically hyperbolic, then they are also isopolar.

Proof. (i) The assumption on the H- and T -orbits implies that ψλ^{j} and ψλ^{j′} are horizontal. Obviously, they are also T -invariant: Note that ψ is so by assumption, and the vectorfields j_{Z}^{∗} are T -invariant vectorfields on X since the actions of H and T commute. Moreover, the two 1-forms are compactly supported since so is ψ. It remains to show that ψλ^{j} and ψλ^{j′} are isospectral in the sense of Definition 4.1(iii). Let μ ∈ 𝔱^{∗}, and let Z ∈ 𝔱 be the dual vector with respect to the inner product on 𝔱. Choose a_{Z} ∈ H such that j′_{Z} = Ad_{a_{Z}} (j_{Z}). A straightforward calculation shows that the T -equivariant isometry F_{μ} := a_{Z} of (X, g_{0}) satisfies μ∘λ^{j} = F_{μ}^{∗}(μ∘λ^{j′}). The same remains true for ψλ^{j} and ψλ^{j′} since ψ is H-invariant.

(ii) First of all, it is easy to see that λ_{h}^{j} and λ_{h}^{j′} are indeed smooth; in fact, if Y is a vectorfield on X then λ_{h}^{j}(Y) equals

vλ^{j}(Y) − ∑_{i=1}^{k}⟨Z_{1}^{∗} ∧ … ∧ Z_{i−1}^{∗} ∧ Y ∧ Z_{i+1}^{∗} ∧ … ∧ Z_{k}^{∗}, Z_{1}^{∗} ∧ … ∧ Z_{k}^{∗}⟩λ^{j}(Z_{i}^{∗}).

Now ψλ_{h}^{j} and ψλ_{h}^{j′} are horizontal by construction. The rest of the proof is the same as for (i); one has to keep in mind that the vectorfields Z_{1}^{∗}, …, Z_{k}^{∗} are H- and T -invariant. □

Remark 4.8. (i) Let (X, g_{0}) be a complete connected manifold admitting an effective action of O(m + 4) by isometries, where m ≥ 5. Consider the subgroup H × T := SO(m) × (SO(2) × SO(2)) ⊂ O(m + 4). Then one can use the action of H × T on X, along with the isospectral families of maps j : 𝔱 → 𝔥 from 4.5(i), to obtain continuous multiparameter families of isophasal metrics g_{ψλ_{h}^{j}} (with varying isospectral maps j) on X as applications of Theorem 4.7(ii); one only has to choose, in addition, a compactly supported H × T -invariant cut-off function ψ on X; for example, one can choose ψ to be even O(m + 4)-invariant and with support in a small neighbourhood of one of the O(m + 4)-orbits. If (X, g_{0}) is asymptotically hyperbolic, then the resulting metrics are also isopolar.

(ii) If, in the situation of (i), the H- and T -orbits meet orthogonally in (X, g_{0}), then one obtains applications of 4.7(i). For example, this is the case for the canonical action of O(n) on ℝ^{n} (n = m + 4), when ℝ^{n} is endowed with any complete O(n)-invariant metric g_{0}. In this case, ψ can be chosen, for example, to be any radial cut-off function on ℝ^{n}. One obtains continuous multiparameter families of isophasal metrics g_{ψλ^{j}} on ℝ^{n} (with varying j), each of which is a compact perturbation of some fixed O(n)-invariant metric. If (ℝ^{n}, g_{0}) is asymptotically hyperbolic, then these metrics are also isopolar.

Remark 4.9. (Proof of the Main Theorem) In section 5 we will show that the isophasal/isopolar families of metrics arising from the construction in 4.8(ii) — that is, from the canonical O(n)-action on ℝ^{n} and a radial cut-off function ψ — are generically nontrivial. The genericity condition will concern only the choice of isospectral j-maps, but neither the choice of rotational metric g_{0} nor the choice of radial cut-off function ψ. Thus by Remark 4.5(i) we obtain, for any complete O(n)-invariant metric g_{0} on R^{n} and any compactly supported radial cut-off function ψ, continuous multiparameter families of mutually nonisometric, isophasal metrics (also isopolar, if (ℝ^{n}, g_{0}) is asymptotically hyperbolic) coinciding with g_{0} outside the support K of ψ. This proves Theorem 1.1.

We conclude this section by considering several special cases of 4.8(ii) (including classes of metrics which are not asymptotically hyperbolic):

Remark 4.10. (i) Let (r, ω) ∈ ℝ_{+} × S^{n−1} denote polar coordinates on ℝ^{n}. Let h denote the round metric on S^{n−1}. Some canonical choices for (ℝ^{n}, g_{0}) are:

Hyperbolic space:  g_{0}_{(r,ω)} = r^{2}h + (1 + r^{2})^{−1}dr^{2}
Euclidean space:  g_{0}_{(r,ω)} = r^{2}h + dr^{2}
Paraboloid:  g_{0}_{(r,ω)} = r^{2}h + (1 + r^{2})dr^{2}
Hyperboloid:  g_{0}_{(r,ω)} = r^{2}h + (1 + \frac{1}{1+r^{2}} )dr^{2}

Thus for each of these metrics g_{0} we have continuous families of isophasal metrics g_{ψλ^{j}} on ℝ^{n} which agree with g_{0} outside a compact O(n)-invariant region. In the first example (hyperbolic space), these metris are also isopolar.

(ii) After normalizing the radial coordinate, a general O(n)-invariant, complete metric g_{0} on ℝ^{n} is given by

g_{0}_{(r,ω)} = f(r)h + dr^{2},

where f is the restriction to ℝ_{+} of any smooth, even, positive function satisfying f(r) ∼ r^{2} for r ↓ 0.

• If we choose f(r) to become constant for large r, then (ℝ^{n}, g_{0}) has a cylindrical end. If, moreover, we choose f(r) = r^{2} for all r in some initial interval (0, a) and choose ψ to have compact support away from 0, then (ℝ^{n}, g_{0}) as well as each (ℝ^{n}, g_{ψλ^{j}}) from an isophasal family as above will be isometric to a flat ball glued onto a standard cylinder with a compact transition region.

• If, instead, we choose f(r) = r^{−2} for r large, then (ℝ^{n}, g_{0}) as well as the isophasal manifolds (ℝ^{n}, g_{ψλ^{j}}) have a cuspidal end.

5. Non-Isometry

In this section we show that the continuous families of isophasal metrics on ℝ^{n} discussed in Remark 4.8(ii) are generically nontrivial.

Let H ×T = SO(m)×(SO(2)×SO(2)) ⊂ O(m+4) with m ≥ 5. Let 𝔱 ≅ ℝ^{2} be the Lie algebra of T, endowed with the standard Euclidean metric, and let 𝔥 = 𝔰𝔬(m). Consider the standard action of O(n) on ℝ^{n}, where n := m + 4, and the induced action of H × T. Let g_{0} be any complete O(n)-invariant metric on ℝ^{n}, and let 0 ≠ ψ ∈ C^{∞}(ℝ^{n}) be a smooth, radial, compactly supported function on ℝ^{n}. For any linear map j : 𝔱 → 𝔥, let λ^{j} be the associated 1-form on ℝ^{n} and consider the Riemannian metric g_{ψλ^{j}} on ℝ^{n} (recall Remark 4.8(ii)).

Theorem 5.1. Suppose that j, j′ : 𝔱 → 𝔥 are nonequivalent in the sense of Definition 4.4(ii), and that j′ is generic in the sense that j′(𝔱) has trivial centralizer in 𝔥. Then the Riemannian manifolds (ℝ^{n}, g_{ψλ^{j}}) and (ℝ^{n}, g_{ψλ^{j′}}) constructed as in Remark 4.8(ii) are nonisometric.

This theorem, applied to the continuous families of pairwise isospectral and nonequivalent maps j_{t} from Remark 4.5(i), shows that the associated isophasal Riemannian manifolds (ℝ^{n}, g_{ψλ^{j_{t}}}) are nonisometric provided that all j_{t}(𝔱) have trivial centralizer in 𝔥. But this condition is indeed generically satisfied.

To prove Theorem 5.1, we first need the following auxiliary result.

Lemma 5.2. Suppose that j(𝔱) has trivial centralizer in 𝔥. Then T is a maximal torus in Isom(ℝ^{n}, g_{ψλ^{j}}).

Proof of Lemma 5.2. Let F_{t} be a 1-parameter family of isometries of (ℝ^{n}, g_{ψλ^{j}}) commuting with T. Then the F_{t} are T -equivariant and preserve the g_{ψλ^{j}} -horizontal distribution on the union ℝ̂^{n} of all principal T -orbits. Denoting by ω_{ψλ^{j}} the connection form on the principal T -bundle ℝ̂^{n} whose kernel is the g_{ψλ^{j}} -horizontal distribution, it follows that the F_{t} preserve ω_{ψλ^{j}} = ω_{0} + ψλ^{j} and consequently also preserve dω_{ψλ^{j}} = dω_{0} + d(ψλ^{j}) = d(ψλ^{j}); note that dω_{0} = 0 here since the g_{0}-horizontal distribution is integrable.

The support of d(ψλ^{j}) is exactly the support of ψ. In fact, for x ∈ ℝ^{n} with x ∉ {0} × ℝ^{4} and ψ(x) ≠ 0 one can find Z ∈ 𝔱 and X, Y ∈ T_{x}ℝ^{n} ≅ ℝ^{n} tangent to the O(m)-orbit through x such that ψ(x) · 2g_{0}(j_{Z}X, Y) ≠ 0 (using the genericity condition on j and m ≥ 5 > 4); note that this is, for vectors X, Y as chosen, the same as ⟨d(ψλ^{j})_{x}(X, Y), Z⟩, by the O(n)-invariance of g_{0} and ψ and the skewsymmetry of j_{Z}. We have now shown that the intersection of the support of ψ with the open dense set ℝ^{n}\({0} × ℝ^{4}) is contained in the support of d(ψλ^{j}); this implies that the support of ψ is contained in the support of d(ψλ^{j}). The converse is obvious.

Therefore, the F_{t} induce a 1-parameter family of isometries F̄_{t} on (ℝ̂^{n}/T, g_{0}^{T}) (recall that g_{0}^{T} = g_{ψλ^{j}}^{T}) which preserve the support of the induced function ψ̄ on ℝ̂^{n}/T. We identify ℝ̂^{n}/T with the open “quadrant”

Q = {(p, a, b) | p ∈ ℝ^{m}, a, b > 0} ⊂ ℝ^{m+2}.

Noting that g_{0}^{T} is the restriction to Q of an O(m + 2)-invariant metric on ℝ^{m+2}, and that supp ψ̄ is the intersection of Q with some compact O(m + 2)-invariant set, it follows that F̄_{t} is of the form (A_{t}, Id) with a 1-parameter family A_{t} in SO(m).

But F̄_{t} must preserve the form Ω_{ψλ^{j}} on ℝ̂^{n}/T induced by the T -invariant, horizontal, 𝔱-valued 2-form d(ψλ^{j}). It follows for all x ∈ ℝ^{n} in the support of ψ, all X, Y ∈ T_{x}ℝ^{n} tangent to the O(m)-orbit through x and all Z ∈ 𝔱:

ψ(A_{t}x) · 2(g_{0})_{A_{t}x}(j_{Z}A_{t}X, A_{t}Y) = ⟨(F_{t}^{∗}d(ψλ^{j}))_{x}(X, Y), Z⟩

= ⟨d(ψλ^{j})_{x}(X, Y), Z⟩ = ψ(x) · 2(g_{0})_{x}(j_{Z}X, Y)

and hence j_{Z} ≡ A_{t}j_{Z}A_{t}^{−1} for all Z ∈ 𝔱. From the genericity assumption on j we conclude A_{t} ≡ Id. Thus F_{t} restricts to a gauge transformation on the principal T -bundle ℝ̂^{n}. On the other hand, we saw above that F_{t}^{∗}ω_{ψλ^{j}} ≡ ω_{ψλ^{j}} . But a gauge transformation of ℝ̂^{n} which preserves a connection form must be an element of T. Thus F_{t} ∈ T for all t, as claimed. □

Proof of Theorem 5.1. Suppose there were an isometry F : (ℝ^{n}, g_{ψλ^{j}}) → (ℝ^{n}, g_{ψλ^{j′}}). By 5.2, T is a maximal torus in Isom(ℝ^{n}, g_{ψλ^{j′}}). Since all maximal tori are conjugate, we can assume (after possibly composing F with an isometry of g_{ψλ^{j′}}) that conjugation by F maps T ⊂ Isom(ℝ^{n}, g_{ψλ^{j}}) to T ⊂ Isom(ℝ^{n}, g_{ψλ^{j′}}). Then F maps T -orbits to T -orbits, and, letting τ denote the automorphism of 𝔱 induced by conjugation by F, we have F_{∗}(Z^{∗}) = τ(Z)^{∗} for all Z ∈ 𝔱.

Recall that g_{ψλ^{j}} , g_{ψλ^{j′}} , and g_{0} are all the same when restricted to the T - orbits in ℝ^{n}. So all principal T -orbits are rectangular tori whose shortest closed geodesics are integral curves parametrized on [0, 2π] of the vectorfields ±Z_{1}^{∗} or ±Z_{2}^{∗}, where {Z_{1}, Z_{2}} denotes the standard basis of 𝔱. It follows that τ preserves the set {±Z_{1}, ±Z_{2}}; in particular, τ ∈ O(𝔱).

Since F maps the g_{ψλ^{j}} -horizontal distribution on ℝ̂^{n} to the g_{ψλ^{j′}} -horizontal distribution, we have F^{∗}ω_{ψλ^{j′}} = τ ∘ ω_{ψλ^{j}} for the associated connection forms; in particular, F^{∗}d(ψλ^{j′}) = F^{∗}dω_{ψλ^{j′}} = τ ∘ dω_{ψλ^{j}} = τ ∘ d(ψλ^{j}). Using the facts that supp ψ = supp(d(ψλ^{j})) = supp(d(ψλ^{j′})) is O(n)-invariant and that g_{0}^{T} is the restriction of an O(m + 2)-invariant metric to the open quadrant Q representing ℝ̂^{n}/T, we now see, just as in the proof of Lemma 5.2, that F̄ is of the form (A, σ) ∈ O(m) × O(2) on the quadrant Q ⊂ ℝ^{m+2} (where σ is either Id or interchanges the two coordinates of ℝ^{2}).

By a similar calculation as in the proof of Lemma 5.2 and using τ ∈ O(𝔱), we conclude that j′_{Z} = Aj_{τ^{−1}(Z)}A^{−1} for all Z ∈ 𝔱, contradicting the nonequivalence assumption. □

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(Perry) Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506–0027, U. S. A.

E-mail address: perry@ms.uky.edu

(Schueth) Inst. f. Math., HU Berlin, 10099 Berlin, Germany

E-mail address: schueth@math.hu-berlin.de