Isospectrality problems on symmetric spaces

This research project concerns inverse spectral geometry. The area asks the extent to which the geometry and topology of a compact Riemannian manifold is determined by its spectral information. M. Kac [Ka66] described this area with the beautiful question “Can one hear the shape of a drum?”, appealing to the interpretation of a bounded planar domain as a drum and its spectrum encoding the characteristic frequencies of vibration. Let (M, g) be a compact Riemannian manifold without boundary. The most important object within the spectral information associated to (M, g), is the spectrum (the collection of eigenvalues counted with multiplicities) of the Laplace-Beltrami operator Δg on M. For simplicity, we denote it by Spec(M, g). We will always assume that Riemannian manifolds are compact without boundary. Two Riemannian manifolds (M, g) and (M’, g’) are called isospectral if Spec(M, g)= Spec(M’, g’). The dimension and the volume are spectral invariants, that is, two isospectral manifolds have necessarily the same dimension and volume. Very useful spectral invariants are the so-called heat invariants (see for instance [Gi]). However, the spectral information is quite far from determining the geometry and the topology of a Riemannian manifold. Milnor [Mi64] gave the first pair of non-isometric isospectral Riemannian manifolds (two 16-dimensional flat tori), showing that the isometry class of an arbitrary Riemannian manifold is not determined by its spectrum. Plenty of other examples have been found later (see [Go00] for more information). In the last years, the attention has focused on spectral uniqueness problems, that is, to show that certain Riemannian manifolds are spectrally distinguished in the sense that they cannot be isospectral to any other Riemannian manifold, maybe assumed in a restricted class (see [S21] for an excelente and recent survey on the subject). It is expected that certain ‘geometrically distinguished’ or ‘nice’ Riemannian manifolds are determined by the spectral information. This project concerns the class of ‘compact symmetric spaces’. Motivation The Laplace operator in Rn is a fundamental object in physics and in analysis. In thermodynamics it can be used to model the flow of heat on a domain. The extension of the Laplacian to an arbitrary Riemannian manifold (M, g) is given by The operator Δg, defined on the space of infinitely differentiable functions on M, is called the Laplace-Beltrami operator. When M is compact, it turns out that the eigenvalues of Δg are all non-negative and form a discrete set in R. The least positive eigenvalue of Δg is called the first eigenvalue or the fundamental tone of M. Shing-Tung Yau wrote about it: “While this constant has analytic importance, it also gives strong insight in the geometry of the manifold”. In the last 50 years, the interest of the relationship between the spectrum and the geometry of Riemannian manifolds has growth. It is expected that for distinguished Riemannian manifolds the spectrum determines the geometry and the topology completely. In particular, for symmetric spaces it is conjectured that a compact symmetric space is isospectral only to Riemannian manifolds on its isometry class. The problem of showing that a particular Riemannian manifold is spectrally unique among the huge class of all Riemannian manifolds is extremely difficult. It is only solved for a few very special cases. For instance, [Ta73] obtained the strongest result in this direction: every round sphere of dimension at most 6 can be distinguished by its spectrum. Outside of the setting of constant curvature, we are not aware of any examples of Riemannian metrics that are known to be spectrally isolated among arbitrary Riemannian metrics. Therefore, it is interesting to give global or local spectral uniqueness results within certain classes of Riemannian metrics. In this project we will focus in the class of homogeneous Riemannian manifolds. A Riemannian manifold (M, g) is called homogeneous if its isometry group acts transitively on M. Roughly speaking, this implies that M does not have distinguished points geometrically speaking. Since we are assuming always M compact, it turns out that Iso(M, g) is a compact Lie group. If G is any compact Lie group acting isometrically and transitively on (M, g), then M can be identified with G/K where K is the isotropy subgroup at some point m in M. Furthermore, the metric g is determined by g_eK, which corresponds to a K-invariant inner product on the vector space T_eK (G/K). The symmetries of (M,g), which forms a compact Lie group, provided tools from Lie theory to describe in a quite explicit way different geometric objects like the different notion of curvature. The same situation occurs with the Laplace-Beltrami operator, mostly thanks to the Peter and Weyl Theorem that gives an explicit orthogonal basis of in terms of the irreducible representation of G . The following general question is an important open problem in the area: Question 1 Is a compact Riemannian symmetric space uniquely determined by its spectrum among homogeneous Riemannian manifolds? It worth to mention that the spectrum cannot distinguish any homogeneous Riemannian manifold. Schueth [Sch01] constructed the first examples of continuous spectral deformations of left-invariant metrics on compact Lie groups. We recall that a compact Lie group endowed with a bi-invariant metric is a compact symmetric space. Schueth proved that the curve of isospectral metrics can be arbitrarily close to a bi-invariant metric. However, she also established that any non-trivial deformation cannot contain a bi-invariant metric, consequently, a bi-invariant metric is infinitesimally spectrally rigid, that is, one cannot continuously deform the bi-invariant metric without changing the spectrum. There are other local results. Gordon and Sutton [GS10] proved that in a compact simple Lie group G, any left-invariant naturally reductive metric on G is spectrally isolated in the set of left-invariant naturally reductive metrics on G with the topology induced by the left-invariant metrics, that is, As a continuation, Gordon, Schueth y Sutton [GSS10] showed that a bi-invariant metric on a compact Lie group G is spectrally isolated within the class of left-invariant metrics on G. An important step toward an answer of Question 1 is to know whether the symmetric space given by a compact Lie group G endowed with the bi-invariant metric is spectrally unique among the space of left-invariant metrics on G. Note that the cases are solved by Tanno [Ta73] for being spectrally distinguished among all Riemannian manifolds because they have constant curvature. Lauret [Lau19b] established the case Sp(n) for n greatest or equals to 2. In some restricted class of Riemannian homogeneous manifolds is possible to show that the Laplace spectrum distinguishes any element on it, rather than only the symmetric ones. Schmidt and Sutton, in an unpublished preprint [SS14] from 2014, established the global spectral uniqueness result for the class of left-invariant metrics on SU(2), and also on SO(3). That is, two isospectral left-invariant metrics on SU(2) (or SO(3)) are necessarily isometric. This result was extended in [LSS21] to the class of 3-dimensional elliptic locally homogeneous manifolds. Lauret [Lau19a] obtained an alternative proof for the mentioned result as a consequence of an explicit expression for the first eigenvalue of the Laplace-Beltrami operator for any left-invariant metric on these two groups in terms of the metric eigenvalues. Recently, Bettiol, Lauret and Piccione [BLP20a] obtained that the spectrum distinguishes any metric in the class of homogeneous metrics on compact rank one symmetric spaces (CROSSes), that is, More precisely, two homogeneous metrics on any two (possible equal) CROSSes are isospectral if and only if they are isometric. As a last motivation, we mention the application of spectral uniqueness results to physical chemistry included in [SS14, ] (see also [LSS21, .5]). Roughly speaking, the moment of inertia of a 3-dimensional rigid body W determines a left-invariant metric g_W on SO(3). It turns out that the geodesics of this metric define the free rotations of W around its center of mass. Moreover, when W is a molecule, Schrödinger’s equation implies that Spec(SO(3),g_w) describes the energy levels of the molecule. In this context the global spectral uniqueness of left-invariant metrics on SO(3) suggest the following phenomenon: The energy levels of a molecule determines its moment of inertia. More details can be found in the attached file "proyecto_revisado.pdf"