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Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature

Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the assumptions and conclusions localized near infinity.


Introduction and statement of results
Let (X, g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a > b > 0 such that −a 2 < K < −b 2 for all sectional curvatures K. Here bounded geometry is used in the sense of Shubin, [7, Appendix 1], namely that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of ∆ on X; here ∆ is the positive Laplacian of g. That is, fix some o ∈ X, and let r(p) = d(p, o), p ∈ X, be the distance function. Then: Theorem 1.1. Suppose that (X, g) is as above. If (∆−λ)ψ = 0 and ψ ∈ e −αr L 2 (X) for all α, then ψ is identically 0.
Since the curvature assumptions imply exponential volume growth, and due to elliptic regularity, the L 2 norm may be replaced by any L p , or indeed Sobolev, norm. This result strengthens Mazzeo's unique continuation theorem at infinity [4] by eliminating the asymptotic curvature assumption (4) there.
As shown below, the negative curvature assumption enters via the strict uniform convexity of the geodesic spheres centered at o, much as in the work of Mazzeo [4]. Thus, as observed by Rafe Mazzeo, the arguments go through equally well if X is replaced by a manifold M which is the union of a 'core' M 0 (not necessarily compact) and a product manifold M 1 = (1, ∞) r × N , with a Riemannian metric g = dr 2 + k(r, .), k a metric on N (r) = {r} × N , M 1 having bounded geometry, provided that the second fundamental form of N (r) is strictly positive, uniformly in r. Indeed, the assumptions on ψ only need to be imposed on M 1 , see Remark 2.4. Following [7, Appendix 1] we remark that an equivalent formulation of the definition of a manifold of bounded geometry is the requirement that the injectivity radius is bounded below by a positive constant r inj , and that the transition functions between intersecting geodesic normal coordinate charts (called canonical coordinates in [7]) of radius < r inj /2, say, are C ∞ with uniformly bounded derivatives (with bound independent of the base points). We let Diff(X) denote the algebra of differential operators corresponding to the bounded geometry, called the algebra of C ∞ -bounded differential operators in [ coordinates, A ∈ Diff m (X) has the form |α|≤m a α (x)D α x , with ∂ β a α uniformly bounded, with bound C |β| independent of the canonical coordinate chart, for all multiindices β. We also write H k (X) for the L 2 -based Sobolev spaces below.
If E is a vector bundle of bounded geometry, in the sense of [7, Appendix 1], then Theorem 1.1 is also valid for ∆ replaced by any second order differential operator P ∈ Diff 2 (X, E) acting on sections of E with scalar principal symbol equal to that of ∆, i.e. the metric function on T * X. Indeed, we can even localize at infinity, i.e. assume P u = 0 only near infinity, and obtain the conclusion that u is 0 near infinity.
Theorem 1.2. Let (X, g) be as above. Suppose P ∈ Diff 2 (X, E), σ 2 (P ) = g Id, where g denotes the metric function on T * X. If P ψ ∈ C ∞ c (X, E) and ψ ∈ e −αr L 2 (X, E) for all α then ψ ∈ C ∞ c (X, E). Remark 1.3. This theorem, together with the standard unique continuation result, [3, Theorem 17.2.1], implies that if P ψ = 0 on X then ψ = 0 on X, just as in Theorem 1.1.
In addition, the Sobolev order of the assumptions on ψ and P ψ is immaterial. In fact, as discussed in Remark 2.4, the argument localizes near infinity, hence we may assume P ψ ∈ C −∞ c (X, E), and then elliptic regularity allows us to conclude that if ψ is in an exponentially weighted Sobolev space near infinity (with possibly a negative exponent) then it is in the corresponding weighted L 2 -space.
We also remark on the bounded geometry hypotheses, more precisely on the assumptions on covariant derivatives. The results of Anderson and Schoen [1] on harmonic functions on negatively curved spaces are results below the continuous spectrum. Thus, these are elliptic problems even at infinity, in a rather strong sense -stronger than just the uniform ellipticity on manifolds with bounded geometry discussed below. In particular, the notion of positivity and the maximum principle are available, and can be used to eliminate conditions on covariant derivatives. However, for eigenfunctions embedded in the continuous spectrum such tools are unavailable. Indeed, for λ large, ∆ − λ can be seen to lose 'strong' ellipticity (so e.g. it is not Fredholm on L 2 (X)), and is in many ways (micro)hyperbolic, at least in settings with an additional structure (cf. the discussion in [5] in asymptotically flat spaces). In such a setting commutator estimates are very natural, and have a long tradition in PDEs; this explains the role of the assumption on the covariant derivatives.
We are very grateful to Rafe Mazzeo and Richard Melrose for numerous very helpful conversations, and for their interest in the present work.

The proofs
To make the argument more transparent, we write up the proof of Theorem 1.1, at each step pointing out any significant changes that are needed to prove Theorem 1.2. The proofs are a version of Carleman estimates, see (2.2) below, at least for self-adjoint operators, but we phrase these somewhat differently, in the spirit of operators with complex symbol and codimension 2 characteristic variety (which in this case is in the semiclassical limit) on which the Poisson bracket of the real and imaginary part of the (in this case, semiclassical) principal symbol is positive. This corresponds to non-solvability of the inhomogeneous PDE in the sense of [3,Section 26.4]; see also [10] for a recent discussion.
We consider eigenfunctions of ∆ that are superexponentially decaying: (∆ − λ)ψ = 0, and ψ ∈ e −αr L 2 (X) for all α, ψ L 2 (X) = 1 (for convenience). Note that λ is real by the self-adjointness of ∆. Moreover, ψ ∈ C ∞ (X) by standard elliptic regularity, and indeed ψ ∈ e −αr ′ H m (X) for all m, where r ′ is a smoothed version of r, changed only near the origin. It is convenient to assume that r ′ > 0, so inf X r ′ > 0.
For α real, we consider Here we need to use r ′ since r is not smooth at o. However, for notational simplicity, to avoid an additional compactly supported error term on almost every line, we ignore this, and simply add back a compactly supported error term in (2.7). Let , be the symmetric and skew-symmetric parts of P α . Thus, P α = Re P α + i Im P α , and Re P α , Im P α are symmetric. Note also that P α ψ α = 0 where ψ α = e αr ψ. Thus, Roughly speaking, this will give a contradiction provided the commutator is positive -although in the presence of error terms one needs to be a little more careful. We remark that this argument parallels the last part of the N -body argument of [8], showing exponential decay and unique continuation results for N -particle Hamiltonians with second order interactions, which in turn placed the work of Froese and Herbst [2] in potential scattering into this framework. However, in [8] (as in [2]) this is the simplest part of the argument; it is much more work to show that L 2 -eigenfunctions decay at a rate given by the next threshold above the eigenvalue λ -hence superexponentially in the absence of such thresholds.
We now relate our arguments to the usual Carleman-type arguments, at least if P 0 is symmetric (as is for self-adjoint operators in Diff 2 (X) with the same principal symbol as ∆). In those, one considers P α and P −α , with the same notation as above, and computes P α ψ α 2 ± P −α ψ α 2 . Since P 0 is symmetric, indeed self-adjoint, Thus, the usual Carleman argument breaks up (2.1) into two pieces, and is completely equivalent to (2.1). However, dividing up P α into its symmetric and skewsymmetric parts makes the calculations below more systematic, which is particularly apparent in how the double commutator appears in Re P α below. This double commutator, in turn, makes it clear why various terms, which one might expect by expanding out the squares P ±α ψ α 2 , do not appear in the evaluation of P α ψ α 2 ± P −α ψ α 2 . Due to the prominent role played by r, we work in Riemannian normal coordinates. So let g = dr 2 + k(r, .) be the metric on X, where k is the metric on the geodesic sphere of radius r, denoted by S(r), and let A(r, .) dr ∧ ω denote the volume element, ω being the standard volume form on the unit sphere. By the bounded geometry assumptions, ∂ r log A = −∆r ∈ C ∞ b (X) = Diff 0 (X) (see e.g. [9, Lemma 2.3] for the identity), i.e. is uniformly bounded with analogous conditions on the covariant derivatives. Then Here the expressions for Re P α and Im P α follow directly from the definition of the symmetric and skew-symmetric parts, using that ∆ and e ±αr are symmetric.
In the double commutator in the expression for Re P α above, changing ∆ by a first order operator would not alter the result, as commutation with a scalar reduces the order by 1. Thus, in view of (2.3), in the double commutator in Re P α all terms but ∂ 2 r give vanishing contribution, so we immediately see that We next compute the skew-symmetric part. This is The crucial estimate for this commutator that we need below is that there is c > 0 such that (2.4) [∆, 2∂ r + (∂ r log A)] ≥ c∆ S(r) + R, R ∈ Diff 1 (X); here R is symmetric and the inequality is understood in the sense of quadratic forms, e.g. with domain H 2 (X). Since the commutator is a priori in Diff 2 (X), this means that we merely need to calculate its principal symbol, which in turn only depends on the principal symbols of the commutants. Thus, with H g denoting the Hamilton vector field of g, and σ the canonical dual variable of r, with respect to the product decomposition (0, ∞) × S of X \ o, the principal symbol of ∂ r is σ 1 (∂ r ) = iσ, and σ 2 ([∆, 2∂ r ]) = 2H g σ.
It is convenient to rephrase this by noting that 2∂ r = −[∆, r] + R ′ , R ′ ∈ Diff 0 (X), so 2iσ = σ 1 (2∂ r ) = iH g r, and hence σ 2 ([∆, 2∂ r ]) = H 2 g r. The estimate we need then is that there is c > 0 such that (2.5) H 2 g r ≥ ck. Indeed, (2.5) implies (2.4), since for each x ∈ X, both sides of (2.5) are quadratic forms on T * X, depending smoothly on x, so their difference can be written as a ij (x)ξ i ξ j (ξ i are canonical dual variables of local coordinates x i ), with a ij a non-negative matrix. This in turn is the principal symbol of ij D * xi a ij (x)D xj , and To analyze (2.5), recall that arclength parameterized geodesics of g are projections to X of the integral curves of 1 2 H g inside S * X, the unit cosphere bundle of X. Thus, (2.5) tells us that r is strictly convex along geodesics tangent to S(r 0 ) at the point of contact. Equivalently, the Hessian ∇dr, which is the form on the fibers of T X dual to H 2 g r, is strictly positive on T S(r 0 ), uniformly as r 0 → ∞. As r = r 0 defines S(r 0 ), with |∇r| = 1, this Hessian equals the second fundamental form of S(r 0 ); hence (2.5) is also equivalent to the uniform convexity of the hypersurfaces S(r 0 ). Now, (2.5) follows immediately when the sectional curvatures of X are bounded above by a negative constant −b 2 , since by the Hessian comparison theorem (see e.g. [6, Theorem 1.1]), H 2 g r| T S(r 0 ) X ≥ H 2 g0 r| T S(r 0 ) X , where g 0 is the metric with constant negative sectional curvature −b 2 , and the right hand side is b coth br ≥ b (cf. [6, Equation (1.7)]).
We will consider α → ∞, but for notational reasons it is convenient to work in the semiclassical setting. Thus, let h = α −1 , h ∈ (0, 1], ∆ h = h 2 ∆, ∆ S(r),h = h 2 ∆ S(r) , and slightly abuse notation by writing P h = h 2 e r/h (∆ − λ)e −r/h , so We denote the space of semiclassical differential operators of order m by Diff m h (X). We recall that A ∈ Diff m h (X) means that, in the usual multiindex notation, A = |α|≤m a α (x)(hD x ) α locally; in our bounded geometry setting we still impose, as for standard differential operators, that for all multiindices β, ∂ β a α is bounded uniformly in all Riemannian normal coordinate charts of radius R (R less than half the injectivity radius, say), with bound only dependent on |β|. Then, weakening the above statements somewhat, in a way that still suffices below, (2.6) . We stated (2.6) in a weakened form to make it only depend on the principal symbol of ∆. Namely, if ∆ is replaced by any operator ∆ + Q, Q ∈ Diff 1 (X, E) (not necessarily symmetric), and , where we used that Re P h , Im P h have scalar principal symbols, hence so do Re P ′ h and Im P ′ h , giving the extra h (compared to the order of the product) and the lower order in the commutators. In other words, (2.6) still holds for P h replaced by P ′ h . In the above calculations we ignored a compact subset of X, so we need to add a compactly supported error. To avoid overburdening the notation, we write r for the smoothed out distance function, denoted by r ′ above, so for r sufficiently large, r(p) = d(p, o). Thus, we have shown that for some c > 0, R ′ 0 supported in r ≤ r 1 for some r 1 > 0, with the inequality holding in the sense of operators. Since ∆ S,h = ∆ h − (hD r ) 2 − h(D r log A)hD r , this estimate implies , R 4 having compact support in r ≤ r 1 . (In fact, for our purposes the compact support assumption is equivalent to assuming that R 4 is o(1) as r → ∞, as it can be absorbed in the first term for r sufficiently large.) We now show how to use (2.8) to prove unique continuation at infinity. To be systematic, we set this part up somewhat abstractly. Recall that P h ∈ Diff 2 h (X, E) is elliptic (or more precisely uniformly elliptic, both in X and in h) if there is C > 0 such that for all (x, ξ) ∈ T * X \ o, and for all h ∈ (0, 1], x , with |ξ| 2 x = g x (ξ, ξ) the length of ξ ∈ T * x X with respect to g and |.| is the operator norm of the matrix of σ 2,h (P h )(x, ξ) −1 in any (bounded geometry) trivialization of E.
h (X, E) is elliptic and satisfies (2.8) for some c > 0. Suppose also that ψ ∈ e −αr L 2 (X, E) for all α. If then there exists R > 0 such that ψ vanishes when r > R.
Remark 2.2. To simplify notation, we drop the bundle E below. Its presence would not require any changes, except in the notation.
Proof. Let Ψ(X) the algebra of pseudodifferential operators corresponding to the bounded geometry, with uniform support, see [7, Appendix 1, Definition 3.1-3.2], denoted by U Ψ(X) there. The elements of Ψ 0 (X) are bounded on L 2 (X), and if A ∈ Ψ m (X) is elliptic, there is B ∈ Ψ −m (X) such that AB−Id, BA−Id ∈ Ψ −∞ (X), so elliptic regularity statements and estimates work as usual.
We also need the corresponding semiclassical space of operators Ψ h (X). These can be defined by modifying the definition of Ψ(X) exactly as if X were compact, i.e. defining Ψ m h (X) near the diagonal using the semiclassical quantization of symbols a, and globally as the sums of such operators and elements of Ψ −∞ h (X). The latter space consists of operators with smooth Schwartz kernel that decays rapidly off the diagonal as h → 0. More precisely, for R ∈ Ψ −∞ h (X) we require that its Schwartz kernel K satisfy K ∈ C ∞ ((0, 1]×X ×X), that there is C R > 0 such that K(x, y) = 0 if d(x, y) > C R , and for all N there is C N > 0 such that for all α, β with |α| ≤ N , |β| ≤ N , and for all h ∈ (0, 1], Since P h is an elliptic family, (2.9) and elliptic regularity give , C 1 independent of h ∈ (0, 1]. Correspondingly, we do not specify below which Sobolev norms we are taking. In general, the letter C, C ′ will be used denote a constant independent of h ∈ (0, 1], which may vary from line to line.
Remark 2.4. Note that in this formulation, if X is replaced by a manifold with several ends, one of which is of the product form eluded to in the introduction, our theorem holds locally on this end. That is, if P ψ vanishes on this end and ψ has superexponential decay there, then ψ vanishes on the end -hence globally by the standard unique continuation theorem if P ψ is identically zero. To prove this, we merely multiply by a cutoff function supported on this end, and apply the lemma to the resulting inhomogeneous problem.
Proof. The elliptic regularity estimate now becomes (2.17) ψ h H 2 h (X) ≤ C 1 ( ψ h L 2 (X) + P h ψ h L 2 (X) ), C 1 independent of h ∈ (0, 1], and we need to keep track of the second term on the right hand side. Correspondingly, R * j ψ h ≤ C( ψ h + P h ψ h ), j = 1, 2, 3, 4. Thus, on the right hand side of (2.11), we need to add Ch P h ψ h 2 , resp. Ch P h ψ h 2 , while on the right hand side of (2.12) we need to add Ch 2 P h ψ h 2 . Similarly, we need to add C ′ he 4r1/h P ψ 2 to the right hand side of (2.13). Thus, (2.14) becomes Since P h ψ h = e r/h h 2 P ψ, we have P h ψ h ≤ e r0/h h 2 P ψ . Let r 2 = max(r 0 , r 1 ). Thus, there exists h 0 > 0 such that for h ∈ (0, h 0 ), Taking R > 2r 2 , the proof is now finished as in Lemma 2.1, for (2.16) becomes 2 P ψ 2 + Ch ψ 2 ≥ c 2 C ′ he 2(R−2r2)/h , and the right hand side still goes to +∞, while the left hand side is bounded as h → 0.

Department of Mathematics, MIT and Northwestern University
Department of Mathematics, Northwestern University E-mail address: andras@math.mit.edu E-mail address: jwunsch@math.northwestern.edu