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Riemannian and Lorentzian Calderón problem under Magnetic Perturbation

Yuchao Yi

TL;DR

This work analyzes the Calderón problem in both Riemannian and Lorentzian settings under a family of magnetic/electromagnetic perturbations. It leverages Runge approximation in the Riemannian case to focus on interior points and prove metric recovery without gauge obstruction, while in the Lorentzian case it uses microlocal analysis to relate perturbations of potentials and metrics to the behavior of null-geodesics, enabling reconstruction of the conformal class. The results establish that a coefficient-parametric data model—perturbations of $A$ in the Riemannian setting and perturbations of $A$ or $g$ in the Lorentzian setting—suffices to recover interior geometric information from boundary measurements, with connections to Aharonov–Bohm-type phase effects. Overall, the paper extends Calderón-type boundary value problem results beyond single-map data, providing interior reconstruction under generic perturbations and highlighting robust techniques that do not require strong structural constraints on the base metrics.

Abstract

We study both the Riemannian and Lorentzian Calderón problem when a family of Dirichlet-to-Neumann maps are given for an open set of magnetic/electromagnetic potentials. For the Riemannian version, by allowing small perturbations of the magnetic potential, we use the Runge Approximation Theorem to show that the metric can be uniquely determined. There is no gauge equivalence in this case. For the Lorentzian version, we use microlocal analysis to construct the trajectory of null-geodesics via generic perturbations of the electromagnetic potential, hence the conformal class of the metric can be constructed. Moreover, we also show, in the Lorentzian case, the same result can be obtained using generic perturbations of the metric itself.

Riemannian and Lorentzian Calderón problem under Magnetic Perturbation

TL;DR

This work analyzes the Calderón problem in both Riemannian and Lorentzian settings under a family of magnetic/electromagnetic perturbations. It leverages Runge approximation in the Riemannian case to focus on interior points and prove metric recovery without gauge obstruction, while in the Lorentzian case it uses microlocal analysis to relate perturbations of potentials and metrics to the behavior of null-geodesics, enabling reconstruction of the conformal class. The results establish that a coefficient-parametric data model—perturbations of in the Riemannian setting and perturbations of or in the Lorentzian setting—suffices to recover interior geometric information from boundary measurements, with connections to Aharonov–Bohm-type phase effects. Overall, the paper extends Calderón-type boundary value problem results beyond single-map data, providing interior reconstruction under generic perturbations and highlighting robust techniques that do not require strong structural constraints on the base metrics.

Abstract

We study both the Riemannian and Lorentzian Calderón problem when a family of Dirichlet-to-Neumann maps are given for an open set of magnetic/electromagnetic potentials. For the Riemannian version, by allowing small perturbations of the magnetic potential, we use the Runge Approximation Theorem to show that the metric can be uniquely determined. There is no gauge equivalence in this case. For the Lorentzian version, we use microlocal analysis to construct the trajectory of null-geodesics via generic perturbations of the electromagnetic potential, hence the conformal class of the metric can be constructed. Moreover, we also show, in the Lorentzian case, the same result can be obtained using generic perturbations of the metric itself.

Paper Structure

This paper contains 11 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. History
  3. Idea of the proofs
  4. Outline of the paper
  5. Riemannian Calderón problem
  6. Preliminaries
  7. Perturbation of the Magnetic Potential
  8. Lorentzian Calderón problem
  9. Preliminaries
  10. Perturbation of The Electromagnetic Potential
  11. Perturbation of The Metric

Key Result

Theorem 1.1

Let $M$ be a smooth connected compact manifold with boundary of dimension $n \geq 3$, and $g_1, g_2$ are two Riemannian metrics on $M$. Suppose $\Lambda_{g_1,A} = \Lambda_{g_2, A}$ for all $A$ in a neighborhood of 0 with respect to $C^k$ topology for some $k \geq 0$, then $g_1 = g_2$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1: LLS20
  • Lemma 2.2
  • proof
  • Proposition 2.3: LU89
  • proof : Proof of Theorem \ref{['thm: riemannian perturb A']}:
  • ...and 7 more