The inverse spectral problem for surfaces of revolution

We prove that isospectral simple analytic surfaces of revolution are isometric. 0 Introduction This article is concerned with the inverse spectral problem for metrics of revolution on S. We will assume that our metrics are real analytic and belong to a class R∗ of rotationally invariant metrics which are of ‘simple type’ and which satisfy some generic non-degeneracy conditions (see Definition (0.1)). In particular, we will assume they satisfy the generalized ‘simple length spectrum’ condition that the length functional on the loop space is a clean Bott-Morse function which takes on distinct values on distinct components of its critical set (up to orientation). Denoting by Spec(S, g) the spectrum of the Laplacian ∆g, our main result is the following: Theorem I Spec: R∗ → IR is 1-1. Thus, if (S, g), (S, h) are isospectral surfaces of revolution in R∗, then g is isometric to h. It would be very desirable to strengthen this result by removing the assumption that h ∈ R∗, thereby showing that metrics in R∗ are spectrally determined within the entire class of analytic metrics on S with simple length spectra. The only metric on S presently known to be spectrally determined in this sense is the standard one (which is known to be spectrally determined among all C∞ metrics). A metric h satisfying Spec(S, h) = Spec(S, g) for some g ∈ R∗ must have many properties in common with a surface of revolution of simple type; it would be interesting to explore whether it must necessarily be one. Let us now be more precise about the hypotheses. First, we will assume that there is an effective action of S by isometries of (S, g). The two fixed points will be denoted N,S and (r, θ) will denote geodesic polar coordinates centered at N , with θ = 0 some fixed meridian γM from N to S. The metric may then be written in the form g = dr + a(r)dθ where a : [0, L] → IR is defined by a(r) = 1 2π |Sr(N)|, with |Sr(N)| the length of the distance circle of radius r centered at N . For any smooth surface of revolution, the function a satisfies a(0) = a(L) = 0, a′(0) = 1, a′(L) = −1 and two such surfaces (S, gi) (i = 1, 2) are isometric Research partially supported by NSF grants #DMS-9404637 and #DMS-9703775.