[Paper Review] Inverse spectral theory and the Minkowski problem for the surface of revolution

Hiroshi Isozaki, Evgeny Korotyaev|arXiv (Cornell University)|May 17, 2016

Geometric Analysis and Curvature Flows3 citations

TL;DR

This paper establishes an analytic isomorphism between spectral data and the radius function for rotationally symmetric manifolds, solving the inverse spectral problem for surfaces of revolution. It proves that the radius profile can be uniquely reconstructed from eigenvalues and norming constants, extending the classical Minkowski problem to this geometric setting via inverse spectral methods on warped product manifolds.

ABSTRACT

We solve the inverse spectral problem for rotationally symmetric manifolds, which include the class of surfaces of revolution, by giving an analytic isomorphism from the space of spectral data onto the space of functions describing the radius of rotation. An analogue of the Minkowski problem is also solved.

Motivation & Objective

• To solve the inverse spectral problem for rotationally symmetric manifolds, including surfaces of revolution, by reconstructing the radius of revolution from spectral data.
• To establish an analytic isomorphism between the space of spectral data and the space of functions describing the radius of rotation.
• To provide a solution to an analogue of the Minkowski problem in the context of surfaces of revolution using inverse spectral theory.
• To characterize the range of the spectral data mapping for Sturm-Liouville operators arising from the Laplace-Beltrami operator on warped product manifolds.

Proposed method

• Reduces the inverse spectral problem on a surface of revolution to a one-dimensional Sturm-Liouville problem via coordinate transformation and metric decomposition.
• Uses the Laplace-Beltrami operator in warped product form $ \Delta_M = \frac{1}{r^m} \partial_x(r^m \partial_x) + \frac{\Delta_Y}{r^2} $, decomposing the problem into orthogonal subspaces indexed by eigenvalues of $ \Delta_Y $.
• Introduces a transformation using $ \varrho = r^{m/2} $ and defines $ q(x) $ via $ \varrho' / \varrho = q_0 + q(x) $, reducing the problem to a Schrödinger-type operator with potential $ u = E_\nu / r^2 $.
• Applies real-analytic isomorphism theorems (Theorems 4.3–4.5) to show that the spectral data—eigenvalues and norming constants—uniquely determine the function $ q(x) $, and hence $ r(x) $.
• Uses spectral asymptotics to determine the total length $ t_0 $, normalizing the domain to $ [0,1] $, and applies inverse theory for Sturm-Liouville operators with Dirichlet, mixed, or Robin boundary conditions.
• Establishes that the potential $ u = E_\nu e^{-4Q/m} $ satisfies Condition U, enabling the application of known inverse spectral results to the transformed operator.

Experimental results

Research questions

• RQ1 Can the radius function of a surface of revolution be uniquely reconstructed from its spectral data, including eigenvalues and norming constants?
• RQ2 Is there a canonical isomorphism between the space of spectral data and the space of radius functions for rotationally symmetric manifolds?
• RQ3 To what extent can the classical Minkowski problem—reconstructing a convex surface from its surface area measure—be generalized to the inverse spectral setting on surfaces of revolution?
• RQ4 How do different boundary conditions (Dirichlet, mixed, Robin) affect the inverse spectral reconstruction of the radius function?
• RQ5 What is the role of the eigenvalues of the base manifold $ Y $ in determining the spectral data of the full Laplacian on the warped product manifold?

Key findings

• The mapping from the function $ q \in \mathscr{W}_1^0 $ to the spectral data $ \{ \widetilde{\mu}_n(q), \kappa_n(q) \} $ is a real-analytic isomorphism, ensuring unique and smooth reconstruction of the radius function from spectral data.
• For Dirichlet boundary conditions, the spectral mapping $ \widetilde{\mu} $ is a real-analytic isomorphism between $ \mathscr{W}_0^{1,\text{odd}} $ and $ \mathcal{M}_1 $, with $ \mu_n^0 = (\pi n)^2 $.
• For mixed boundary conditions, the mapping $ \Psi: q \mapsto \{ \widetilde{\mu}_n(q,b), \chi_{n-1}(q,b) - \chi_{n-1}^0 \} $ is a real-analytic isomorphism onto $ \mathcal{M}_1 \times \ell^2_1 $, with $ \mu_n^0 = (\pi n + \frac{1}{2})^2 + 2b $.
• For Robin boundary conditions, the mapping $ \Psi_{a,b} $ is a real-analytic isomorphism between $ \mathscr{W}_1^0 $ and $ \mathcal{M}_1 \times \ell^2_1 $, with $ \mu_n^0 = (\pi n)^2 + 2(a+b) $.
• An explicit identity is derived: $ b = \sum_{n=0}^\infty \left( 2 - \frac{e^{\chi_n(q,b)}}{ |\partial w / \partial \lambda (\mu_n, q, b)| } \right) $, linking the boundary parameter $ b $ to spectral data.
• The solution is robust under normalization: the total length $ t_0 $ is recovered from spectral asymptotics, allowing reduction to the unit interval $ [0,1] $ without loss of generality.