# On the bottom of spectra under coverings

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## Abstract

For a Riemannian covering \(M_1\rightarrow M_0\) of connected Riemannian manifolds with respective fundamental groups \(\Gamma _1\subseteq \Gamma _0\), we show that the bottoms of the spectra of \(M_0\) and \(M_1\) coincide if the right action of \(\Gamma _0\) on \(\Gamma _1\backslash \Gamma _0\) is amenable.

## Keywords

Bottom of spectrum Amenable covering## Mathematics Subject Classification

58J50 35P15 53C99## 1 Introduction

In this article, we study the behaviour under coverings of the bottom of the spectrum of Schrödinger operators on Riemannian manifolds.

Let *M* be a connected Riemannian manifold, not necessarily complete, and \(V:M\rightarrow \mathbb {R}\) be a smooth potential with associated *Schrödinger operator* \(\Delta +V\). We consider \(\Delta +V\) as an unbounded symmetric operator in the space \(L^2(M)\) of square integrable functions on *M* with domain \(C^\infty _c(M)\), the space of smooth functions on *M* with compact support.

*M*with compact support in

*M*, we call

*Rayleigh quotient of*

*f*. We let

*f*runs through all non-vanishing Lipschitz continuous functions on

*M*with compact support in

*M*. If \(\lambda _0(M,V)>-\infty \), then \(\Delta +V\) is bounded from below on \(C^\infty _c(M)\) and \(\lambda _0(M,V)\) is equal to the bottom of the spectrum of the

*Friedrichs extension of*\(\Delta +V\). If \(\lambda _0(M,V)=-\infty \), then the spectrum of any self-adjoint extension of \(\Delta +V\) is not bounded from below.

Recall that \(\Delta +V\) is essentially self-adjoint on \(C^\infty _c(M)\) if *M* is complete and \(\inf V>-\infty \). Then the unique self-adjoint extension of \(\Delta +V\) is its closure. In the case where *M* is the interior of a complete Riemannian manifold *N* with smooth boundary and where *V* extends smoothly to the boundary of *N*, \(\lambda _0(M,V)\) is equal to the bottom of the Dirichlet spectrum of \(\Delta +V\) on *N*.

In the case of the *Laplacian*, that is, \(V=0\), we also write \(\lambda _0(M)\) and call it the *bottom of the spectrum of* *M*. It is well known that \(\lambda _0(M)\) is the supremum over all \(\lambda \in \mathbb {R}\) such that there is a positive smooth \(\lambda \)-eigenfunction \(f:M\rightarrow \mathbb {R}\) (see, e.g., [3, Theorem 7], [4, Theorem 1], or [5, Theorem 2.1]). It is crucial that these eigenfunctions are not required to be square-integrable. In fact, \(\lambda _0(M)\) is exactly the border between the positive and the \(L^2\) spectrum of \(\Delta \) (see, e.g., [5, Theorem 2.2]).

Suppose now that *M* is simply connected and let \(\pi _0:M\rightarrow M_0\) and \(\pi _1:M\rightarrow M_1\) be Riemannian subcovers of *M*. Let \(\Gamma _0\) and \(\Gamma _1\) be the groups of covering transformations of \(\pi _0\) and \(\pi _1\), respectively, and assume that \(\Gamma _1\subseteq \Gamma _0\). Then the resulting Riemannian covering \(\pi :M_1\rightarrow M_0\) satisfies \(\pi \circ \pi _1=\pi _0\). Let \(V_0:M_0\rightarrow \mathbb {R}\) be a smooth potential and set \(V_1=V_0\circ \pi \).

Since the lift of a positive \(\lambda \)-eigenfunction of \(\Delta \) on \(M_0\) to \(M_1\) is a positive \(\lambda \)-eigenfunction of \(\Delta \), we always have \(\lambda _0(M_0)\le \lambda _0(M_1)\) by the above characterization of the bottom of the spectrum of \(\Delta \) by positive eigenfunctions. In Sect. 4, we present a short and elementary proof of the inequality which does not rely on the characterization of \(\lambda _0\) by positive eigenfunctions:

## Theorem 1.1

Brooks showed in [2, Theorem 1] that \(\lambda _0(M_0)=\lambda _0(M_1)\) in the case where \(M_0\) is complete, has *finite topological type*, and \(\pi \) is *normal* with *amenable* group \(\Gamma _1\backslash \Gamma _0\) of covering transformations. Bérard and Castillon extended this in [1, Theorem 1.1] to \(\lambda _0(M_0,V_0)=\lambda _0(M_1,V_1)\) in the case where \(M_0\) is complete, \(\pi _1(M_0)\) is finitely generated [this assumption occurs in point (1) of their Section 3.1], and the right action of \(\Gamma _0\) on \(\Gamma _1\backslash \Gamma _0\) is amenable. We generalize these results as follows:

## Theorem 1.2

*X*is said to be

*amenable*if there exists a \(\Gamma \)-invariant mean on \(L^\infty (X)\). This holds if and only if the action satisfies the

*Følner condition*: For any finite subset \(G\subseteq \Gamma \) and \(\varepsilon >0\), there exists a non-empty, finite subset \(F\subseteq X\), a

*Følner set*, such that

*amenable*if the right action of \(\Gamma \) on itself is amenable, and then any action of \(\Gamma \) is amenable.

In comparison with the results of Brooks, Bérard, and Castillon, the main point of Theorem 1.2 is that we do not need any assumptions on metric and topology of \(M_0\). A main new point of our arguments is that we adopt our constructions more carefully to the different competitors for \(\lambda _0\) separately.

## 2 Fundamental domains and partitions of unity

Choose a complete Riemannian metric *h* on \(M_0\). In what follows, geodesics, distances, and metric balls in \(M_0\), \(M_1\), and *M* are taken with respect to *h* and its lifts to \(M_1\) and *M*, respectively.

*x*in \(M_0\). For any \(y\in \pi ^{-1}(x)\), let

*fundamental domain*of \(\pi \) centered at

*y*. Then \(D_y\) is closed in \(M_1\), the boundary \(\partial D_y\) of \(D_y\) has measure zero in \(M_1\), and \(\pi :D_y{\setminus }\partial D_y\rightarrow M_0{\setminus } C\) is an isometry, where

*C*is a subset of the cut locus \(\mathrm{Cut}(x)\) of

*x*in \(M_0\). Recall that \(\mathrm{Cut}(x)\) is of measure zero. Moreover, \(M_1=\cup _{y\in \pi ^{-1}(x)}D_y\), \(y\in \pi ^{-1}(x)\).

## Lemma 2.1

For any \(\rho >0\), there is an integer \(N(\rho )\) such that any *z* in \(M_1\) is contained in at most \(N(\rho )\) metric balls \(B(y,\rho )\), \(y\in \pi ^{-1}(x)\).

## Proof

Let \(z\in B(y_1,\rho )\cap B(y_2,\rho )\) with \(y_1\ne y_2\) in \(\pi ^{-1}(x)\) and \(\gamma _1,\gamma _2:[0,1]\rightarrow M_1\) be minimal geodesics from \(y_1\) to *z* and \(y_2\) to *z*, respectively. Then \(\sigma _1=\pi \circ \gamma _1\) and \(\sigma _2=\pi \circ \gamma _2\) are geodesic segments from *x* to \(\pi (z)\). Since \(y_1\ne y_2\), \(\sigma _1\) and \(\sigma _2\) are not homotopic relative to \(\{0,1\}\). Hence, if *z* lies in in the intersection of *n* pairwise different balls \(B(y_i,\rho )\) with \(y_1,\dots ,y_n\in \pi ^{-1}(x)\), then the concatenations \(\sigma _1^{-1}*\sigma _i\) represent *n* pairwise different homotopy classes of loops at *x* of length at most \(2\rho \). Hence *n* is at most equal to the number \(N(\rho )\) of homotopy classes of loops at *x* with representatives of length at most \(2\rho \). \(\square \)

## Lemma 2.2

If \(K\subseteq M_0\) is compact, then \(\pi ^{-1}(K)\cap D_y\) is compact. More precisely, if \(K\subseteq B(x,r)\), then \(\pi ^{-1}(K)\cap D_y\subseteq B(y,r)\).

## Proof

*x*. Let \(\gamma \) be the lift of \(\gamma _0\) to \(M_1\) starting in

*z*. Then \(\gamma \) is a minimal geodesic from

*z*to some point \(y'\in \pi ^{-1}(x)\). Since \(z\in D_y\), this implies

Let \(K\subseteq M_0\) be a compact subset and choose \(r>0\) such that \(K\subseteq B(x,r)\). Let \(\psi :\mathbb {R}\rightarrow \mathbb {R}\) be the function which is equal to 1 on \((-\infty ,r]\), to \(t+1-r\) for \(r\le t\le r+1\), and to 0 on \([r+1,\infty ]\). For \(y\in \pi ^{-1}(x)\), let \(\psi _y=\psi _y(z)=\psi (d(z,y))\). Note that \(\psi _y=1\) on \(\pi ^{-1}(K)\cap D_y\) and that \(\mathrm{supp}\,\psi _y=\bar{B}(y,r+1)\).

## Lemma 2.3

Any *z* in \(M_1\) is contained in the support of at most \(N(r+1)\) of the functions \(\psi _y\), \(y\in \pi ^{-1}(x)\).

## Proof

This is clear from Lemma 2.1 since \(\mathrm{supp}\,\psi _y\) is contained in the ball \(B(y,r+1)\). \(\square \)

## Lemma 2.4

The functions \(\varphi _y\), \(y\in \pi ^{-1}(x)\), are Lipschitz continuous with Lipschitz constant \(3N(r+1)\).

## Proof

*N*]. Hence

As a consequence of Lemma 2.4, we get that \(\varphi _1=1-\sum \varphi _y\) is also Lipschitz continuous with Lipschitz constant \(6N(r+1)^2\).

## 3 Pulling up

Let *f* be a non-vanishing Lipschitz continuous function on \(M_0\) with compact support and let \(f_1=f\circ \pi \). We will construct a cutoff function \(\chi \) on \(M_1\) such that \(R(\chi f_1)\) is close to *R*(*f*).

*g*be the given Riemannian metric on \(M_0\) and

*h*be a complete background Riemannian metric on \(M_0\) as in Sect. 2. Then there is a constant \(A\ge 1\) such that

*f*. We continue to take distances and metric balls in \(M_0\), \(M_1\), and

*M*with respect to

*h*and its respective lifts to \(M_1\) and

*M*.

Fix a point *x* in \(M_0\). With \(K=\mathrm{supp}\,f\) and \(r>0\) such that \(K\subseteq B(x,r)\), we get a partition of unity with functions \(\varphi _1\) and \(\varphi _y\), \(y\in \pi ^{-1}(x)\), as above.

*x*under \(\pi _0\) and \(\pi \), respectively. Write \(\pi _0^{-1}(x)=\Gamma _0u\) as the union of \(\Gamma _1\)-orbits \(\Gamma _1gu\), where

*g*runs through a set

*R*of representatives of the right cosets of \(\Gamma _1\) in \(\Gamma _0\), that is, of the elements of \(\Gamma _1\backslash \Gamma _0\). Then \(\pi ^{-1}(x)=\{\pi _1(gu)\mid g\in R\}\). Let

*S*and

*T*are finite subsets of \(\Gamma _0\), hence also

*G*.

*G*and \(\varepsilon \) satisfying (1.3). Let

*R*is a set of representatives of the right cosets of \(\Gamma _1\) in \(\Gamma _0\), there exists a one-to-one correspondence between

*P*and \(P_1\), and hence

*Q*is

*g*on

*M*.

*f*. Therefore

*f*, but not on

*y*or the choice of \(\varepsilon \) and

*F*. With \(D=|G|N(2r+1)\), we obtain from (3.2) that

*R*(

*f*).

## Proof of Theorem 1.2

By Theorem 1.1, we have \(\lambda _0(M_0,V_0)\le \lambda _0(M_1,V_1)\). By (1.2), the bottom of the spectrum of Schrödinger operators is given by the infimum of corresponding Rayleigh quotients *R*(*f*) of Lipschitz continuous functions with compact support. The arguments above show that, for any such function *f* on \(M_0\) and any \(\delta >0\), there is a Lipschitz continuous function \(\chi f_1\) on \(M_1\) with compact support and Rayleigh quotient at most \(R(f)+\delta \). Therefore we also have \(\lambda _0(M_0,V_0)\ge \lambda _0(M_1,V_1)\).\(\square \)

## 4 Pushing down

*f*be a Lipschitz continuous function on \(M_1\) with compact support. Define the

*push down*\(f_0:M_0\rightarrow \mathbb {R}\) of

*f*by

*x*, where

*f*is differentiable at all \(y\in \pi ^{-1}(x)\) and \(f(y)\ne 0\) for some \(y\in \pi ^{-1}(x)\), and then

*x*, we get

## Proof of Theorem 1.1

For any non-vanishing Lipschitz continuous function *f* on \(M_1\) with compact support, the push down \(f_0\) as above is a Lipschitz continuous function on \(M_0\) with compact support and Rayleigh quotient \(R(f_0)\le R(f)\). The asserted inequality follows now from the characterization of the bottom of the spectrum by Rayleigh quotients as in (1.2). \(\square \)

## 5 Final remarks

It is well-known that any countable group is the fundamental group of a smooth four-manifold. (A variant of the usual argument for finitely presented groups, taking connected sums of \(S^1 \times S^3\) and performing surgeries, can be used to produce five-manifolds with fundamental group any countable group.) In particular, for a non-finitely generated, amenable group *G*, e.g., \(G=\bigoplus _{n \in \mathbb {N}} \mathbb {Z}\) or \(G=\mathbb {Q}\), there is a smooth manifold *M* with \(\pi _1(M)\cong G.\) In contrast to the results in [1, 2], our main result also applies to such examples.

Moreover, we do not assume \(\lambda _0(M_0,V_0)>-\infty .\) Given any non-compact manifold \(M_0\), it is indeed easy to construct a smooth potential \(V_0\) such that \(\lambda _0(M_0,V_0)=-\infty .\) In fact, it suffices that \(V_0(x)\) tends to \(-\infty \) sufficiently fast as \(x\rightarrow \infty \).

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society.

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