Spectral Geometry

Overview:

Let M be a closed Riemannian manifold and let Δ denote its Laplace operator acting on smooth functions on M. It is a self-adjoint, positive and elliptic differential operator which has a pure point spectrum

0<=λ_1<=λ_2<=... -> infinity

There is also an orthonormal basis \varphi_i , i=1,2,... of L^2 (M) consisting of eigenfunctions of Δ. The spectrum is an isometry invariant of M. Manifolds with the same spectrum, with multiplicities taken into account, are called isospectral. Isometric manifolds are isospectral. Spectral geometry's first goal is to answer the following question:

how much of the geometry and topology of M can be recovered from its spectrum?

For example, is it true that isospectral manifolds are isometric? The essence of this question was captured in the title of a famous article by Marc Kac [8], Can one hear the shape of a drum?

The first giant step in this direction was taken by Hermann Weyl in 1911 [19]. Addressing a conjecture advanced by physicist Lorentz and Sommerfeld around 1910 [9,17], he showed that the dimension and volume of a bounded domain M in R^n, n=2, 3 with smooth enough boundary, is determined by its Dirichlet or Neumann spectrum. In fact he showed that if N(λ)=#{λ_i<= λ} is the eigenvalue counting function, then one has an asymptotic formula

where $\omega_n$ is the volume of the unit ball in R^n. This result, now known as Weyl's Law, and its generalization to closed Riemannian manifolds is often paraphrased as saying:

One can hear the dimension and volume of a Riemannian manifold.

It is now known that one can also hear the total scalar curvature of a closed Riemannian manifold. This kind of results are typically referred to as inverse spectral problems. Results in this direction are both positive and negative.

On the positive side, an infinite sequence of spectral invariants, known as DeWitt--Gilkey--Seeley coefficients can be defined and can be computed, at least in principle, which give information about the geometry of a manifold which solely depends on its spectrum [7]. On the negative side, starting with Milnor's counterexample of 1964 [12], it is known that there are isospectral manifolds which are not isometric. Other milestones in this direction are results by Sunada [18] and Gordon-Web-Wolpert [5]. It was known for sometime that there are non-isomorphic number fields which have the same zeta function. Motivated by this, in [18] Sunada managed to give a general method for constructing isospectral but not isometric manifolds with dimensions bigger than two. In [5], Gordon-Web-Wolpert mangled to construct planar isospectral domains with piecewise linear boundaries which are not isometric.

Time permitting I shall also discuss a sample of direct spectral problems: knowing a manifold geometry, what one can say about its spectrum?

Another set of ideas in spectral geometry concerns with different types of trace formulae and applications to number theory, quantum physics, and quantum chaos. In some sense this even goes back to the very origins of the Weyl's law in quantum mechanics and in deriving Planck's radiation formula [1,9]. Time permitting, this will be touched in some detail in this course.

Topics to be covered:

I am planing to go over the following topics:

• Spectral decomposition of L2(M); discreteness of the spectrum, and its finite multiplicity. This needs some Sobolev space theory, functional analysis, and spectral theorem for compact operators.
• Weyl's law for bounded domains in Rn. I shall give a proof very close to Weyl's original proof based on Max-Min principle for eigenvalues of Δ, and the domain monotonicity of eigenvalues.
• Examples: Lattices and at tori, round spheres and spherical harmonics, projective spaces. Milnor's counter example: two lattices in R16 which are isospectral but not isometric. Theta functions and Modular forms for lattices. Gauss circle problem.
• The Minakshisundaram- Pleijel asymptotic expansion for the kernel of the heat operator e^(-tΔ) :
• The scalar functions ai, are invariantly depened on M, are isometry invariants, and their values ai(x) can be explicitly computed in terms of the curvature tensor of M at x and its covariant derivatives. I shall compute the rst three terms. Using a Tauberian theorem of Hardy-Littlewood-Karamata, Weyl's law follows after computing a0(x): a1(x) is 1/6 of the scalar curvature. It follows that the total scalar curvature is determined by spectrum of Δ, i.e.one can hear the total scalar curvature of M. Heat equation proof of Gauss- Bonnet theorem. The formulas for ai(x), for i >3, are exceedingly more and more complicated and it seems at present they are explicitly computed only up to i = 10.
• Spectral zeta function \zeta_Δ(s). Proof of analytic continuation, structure of poles and residues, and special values. Via the Mellin transform, this is more or less equivalent to heat trace asymptotic expansion, but the spectral zeta function has its own merits, and is indispensable, e.g. for determinants of Laplacian and analytic torsion.
• Trace formulae. This is a refinement of Weyl's law and a meeting point for number theory, spectral geometry, and physics in the form of correspondence principles. Poisson summation formula and applications to at tori: one can hear the lengths of closed geodesics of at tori. Semiclassical approximation, Born-Sommerfeld quantization rules and correspondence principles. Selberg trace formula. Relations between trace formulae and Riemann hypothesis. time permiting I shall also look at corresponding results for compact topological groups through examples.

Who shall benefit from this course?

Graduate students and faculty working in Riemannian geometry, symplectic geometry, geometric analysis and PDE's, mathematical and theoretical physics, number theory, and noncommutative geometry should nd this material quite relevant and useful.

Textbooks:

There will be no textbook and I shall follow my own notes. The following texts are however standard and I shall use them throughout the course:

	Isac Chavel's book [3] is a good introductory text. Specially useful as it gives a proof of Weyl's law along the original lines by Weyl.
	The text by M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une vari'et'e riemannienne [2] is in my opinion the best account of the theory up to the year of its publication in 1971. I shall use it very often.
	Gilkey's book [7] is simply a gem and indispensable for workers in the eld. In particular it is the best source for DeWitt-Gilkey-Seeley theorem. Unlike the other books I discuss here, it uses pseudodifferential calculus in an essential manner, so one should rst absorb some heavy doses of nalysis before starting to benefit from this book.
	Steve Rosenberg's book [14] is a lucid, well written and very readable introduction to many ideas in the subject, including applications to index theory.

Weekly lectures:

Here is a breakdown of weekly lectures up to February 27, 2012.

• Lecture 1, Monday, Jan 16.
  A brief history of spectral geometry: Conjectures of Lorentz and Sommerfeld,
  Weyl's 1911 paper, asymptotic distribution of eigenvalues for Dirichlet
  problem in R2 and R3, one can hear the volume of a drum.
  Isospectral versus isometric domains.
  The physics of Laplacian: heat, wave and Schrodinger equations.
  Gradient, covariant derivative, divergence, and Laplacian for a Riemannian
  manifold.
  Green's theorems.
• Lecture 2, Monday, Jan 23.
  Dirichlet (energy) functional and Dirichlet's principle,
  Spectral theorem for compact selfadjoint operators,
  Enter 3 Hilbert spaces: L2(M);L2(M);H(M), Sobolev space,
  The Green operator,
  Rellich's embedding theorem and compactness of Green's operator,
  Poincare inequality.
• Lecture 3, Monday, Jan 30.
  Proof of the fundamental decomposition theorem for L2(M);
  Max-min theorem for eigenvalues,
  The Dirichlet princinple.
• Lecture 4, Monday, Feb 6.
  Domain monotonicity of eigenvalues,
  A reverse inequality for eigenvalues,
  Weyl's law for bounded domains in Rn: approximation from within and from outside by boxes.
• Lecture 5, Monday, Feb 13.
  Finish proof of Weyl's law for bounded domains in Rn
  Eigenvalues and eigenfunctions of flat tori Rn
  Milnor's counterexample: E8 \oplus E8 and E16 are isospectral but not isometric.
  Poisson summation formula.
  The theta function of a lattice \Theta_{\Gamma}(z), and a sucient condition for its modularity.
  Lattices in Rn, dual lattices, volume of a lattice,
  check Weyl's law for flat tori: counting lattice points in a ball.
• Lecture 6, Tuesday, Feb 20.
  Eigenvalues and eigenvalues for Dirichlet and Neumann problems for a Box.
  Finish proof of Milnor's counterexample.
  Basics of modular forms.
  Poisson summation formula, one can hear the lengths of closed geodesics of a flat torus.
  isospectral flat two tori are isometric.
  The trace formula for flat tori: what it entails?
  Spherical harmonics.
• Lecture 7, Tuesday, Feb 27.
  Spherical harmonics, eigenvalues and eigenfunctions of round sphere Laplacian,
  relation with represenation theory of SO(3).
  Applications of spherical harmonics to the hydrogen atom,
  Weyl's law and semiclassical approximation in quantum mechanics, correspondence
  principle, Examples of correspondence principle: harmonic oscillator,
  hydrogen atom (postponed),
  Asymptotic expansion of the heat kernel, fundamental solution of the heat
  operator, existence and uniqueness.

References:

[1] W. Arendt, R. Nittka, W. Peter, and F. Steiner; Weyls Law: Spectral Properties of the Laplacian in Mathematics and Physics, in Mathematical Analysis of Evolution, Information, and Complexity. Edited by Wolfgang Arendt and Wolfgang P. Schleich, 2009.
[2] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une vari'et'e riemannienne. (French) Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971.
[3] I. Chavel, Eigenvalues in Riemannian geometry. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984.
[4] C. Gordon, and D. Webb, You can't hear the shape of a drum, American Scientist 84 (JanuaryFebruary): 46-55.
[5] C. Gordon, D. Webb, and S. Wolpert, S. (1992), "Isospectral plane domains and surfaces via Riemannian orbifolds", Inventiones mathematicae 110 (1): 1-22, 1992.
[6] C. Gordon, D. Webb, and S. Wolpert, "One Cannot Hear the Shape of a Drum", Bulletin of the American Mathematical Society 27 (1), 1992.
[7] P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem.
[8] M. Kac, Can One Hear the Shape of a Drum? The American Mathematical
Monthly, 73(4), 1-23, 1966.
[9] H. A. Lorentz, Alte und neue Fragen der Physik. Physikal. Zeitschr., 11, 1234-1257, 1910.
[10] H. P. McKean, Selberg's trace formula as applied to a compact riemann surface,
[11] H. P. McKean, and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. Volume 1, Number 1-2, 43{69, 1967.
[12] J. Milnor, Eigenvalues of the Laplace Operator on Certain Manifolds. Proc. Nat. Acad. Sci. USA, 51(4),1964.
[13] S. Minakshisundaram, and A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math., 1, 242-256, 1949.
[14] S. Rosenberg, The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds (London Mathematical Society Student Texts).
[15] R. Seeley, Complex powers of an elliptic operator. Proc. Sympos. Pure Math., Vol. X, pp. 288307, AMS, 1967.
[16] I. M. Singer, Eigenvalues of the Laplacian and Invariants of Manifolds, Proceedings of ICM, 1974.
[17] A. Sommerfeld, Die Greensche Funktion der Schwingungsgleichung furein beliebiges Gebiet. Physikal. Zeitschr., 11, 1057{1066, 1910.
[18] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Of Math. 121 (1): 169{186, 1985.
[19] H.Weyl, Uber die asymptotische Verteilung der Eigenwerte. Nachrichtender Koniglichen Gesellschaft der Wissenschaften zu Gottingen. Mathem.-
physikal. Klasse, 110{117, 1911.
[20] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Mathematische Annalen, 71(4), 441{479, 1912.
[21] H. Weyl, Uber die Abhngigkeit der Eigenschwingungen einerMembran von deren Begrenzung. J. Reine Angew. Math., 141, 1-11, 1912.
[22] H. Weyl, Uber das Spektrum der Hohlraumstrahlung. J. Reine Angew. Math., 141, 163-181, 1912.
[23] H. Weyl, Uber die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgeometrie. J. Reine Angew. Math., 143, 177-202, 1913.
[24] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Krpers. Rend. Circ. Mat. Palermo, 39, 1-50, 1915.
[25] H.Weyl, H. Ramifications, old and new, of the eigenvalue problem. Bull. Amer. Math. Soc., 56(2), 115-139, 1950.
[26] E. Witt, Eine Identitat zwischen Modulformen zweiten Grades. Abh. Math. Sem. Hansischen Universitat, 14, 323-337, 1941.