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Sebastián Muñoz-Thon

Publications and source records attributed to Sebastián Muñoz-Thon.

12 recordsLinked to original sources
arXiv2026

Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem

We prove local rigidity of every smooth conformally Euclidean metric for the anisotropic Calderón problem on smooth compact domains $M\subset\mathbb R^n$, $n\ge3$. Any smooth Riemannian metric $g$ sufficiently close to a fixed background $g_0=e^{2c}\mathsf e$ in $H^s(M)$, with integer $s>n/2+1$, having the same Dirichlet-to-Neumann map satisfies $g=Φ^*g_0$ for a smooth diffeomorphism $Φ$ fixing the boundary pointwise.

math.AP↗
arXiv2026

(In)stability results for the Light Ray Transform

We prove quantitative instability results for the light ray transform in the Minkowski setting, and locally in the smooth and Gevrey categories. In the smooth Sobolev setting, we show that no modulus of continuity can be better than $t^α$ for any $α\in(0,1)$, while in the Gevrey setting we obtain an explicit logarithmic lower bound. On the other hand, using the relation between stationary geometries and magnetic(-potential) systems, we obtain a stability result for "time moments" of the light ray transform.

math.AP↗
arXiv2026

Heat propagation on abelian covers of principal bundles

We study the long-time asymptotics of heat kernels on Abelian covers of compact manifolds and on Abelian covers of principal bundles. For Abelian covers, we establish three complementary asymptotic expansions: a distributional expansion describing correlations of the heat semigroup, a local pointwise expansion of the heat kernel on compact subsets, and a global expansion valid on the natural diffusive scale, revealing the Gaussian profile governing heat propagation. We then extend these results to horizontal heat kernels on principal bundles under natural holonomy and curvature assumptions. In this setting, we show that the leading asymptotics are entirely determined by the geometry of the underlying Abelian cover, while the nontrivial fiber modes decay exponentially fast due to a uniform spectral gap. The proof combines Floquet theory on Abelian covers with the Borel--Weil decomposition on principal bundles.

math.AP↗
arXiv2026

Marked magnetic action rigidity

An exact magnetic system over a closed manifold $M$ consists of a pair $(g,α)$, where $g$ is a Riemannian metric and $α$ is a 1-form encoding a magnetic field. In this context, we consider a generalization of the marked length rigidity conjecture: does the marked magnetic action spectrum of magnetic systems with Anosov magnetic flow determine the metric and the 1-form, up to a natural obstruction? In this article we answer this question in two settings: 1) locally for systems with close metrics and 1-forms and 2) for metrics in the same conformal class.

math.DS↗
arXiv2025

On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal Laplacian

In this short note, we use the relation obtained by Guillarmou--Guillopé and Chang--González between the generalized eigenvalue problem for asymptotically hyperbolic (AH) manifolds and the Conformal Laplacian, to obtain a new inverse scattering result: on an AH manifold of dimension $n+1$ with constant scalar curvature $-n(n+1)$, we show that the scattering matrix at energy $\frac{n+1}{2}$ determines the jet of the metric on the boundary, up to a diffeomorphism and conformal factor.

math.AP↗
arXiv2025

Guillarmou's Normal Operator for Magnetic and Thermostat Flows

Guillarmou's normal operator over a closed Anosov manifold is analogous to the classical normal operator of the geodesic X-ray transform over manifolds with boundary. In this paper, we generalize this normal operator, under some dynamical assumptions, to thermostat flows as well as to the case of the magnetic flows. In particular, we show that these generalized normal operators are elliptic pseudodifferential operators of order -1 in each case. As an application, we prove a stability estimate for the magnetic X-ray transform.

math.AP↗
arXiv2024

A Calderón's problem for harmonic maps

We study a version of Calderón's problem for harmonic maps between Riemannian manifolds. By using the higher linearization method, we first show that the Dirichlet-to-Neumann map determines the metric on the domain up to a natural gauge in three cases: on surfaces, on analytic manifolds, and in conformally transversally anisotropic manifolds on a fixed conformal class with injective ray transform on the transversal manifold. Next, using higher linearizations we obtain integral identities that allows us to show that the metrics on the target have the same jets at one point. In particular, if the target is analytic, the metrics are equal. We also prove an energy rigidity result, in the sense that the Dirichlet energies of harmonic maps determines the Dirichlet-to-Neumann map.

math.AP↗
arXiv2024

Scattering rigidity for standard stationary manifolds via timelike geodesics

We study the scattering rigidity problem for standard stationary manifolds using timelike geodesics with a fixed momentum. Taking advantage of the symmetry of this manifolds, we use Hamiltonian reduction to show that this problem is related to scattering rigidity for $\mathcal{MP}$-systems, a problem studied before. This gives several new rigidity results (up to some gauge) for this kind of Lorentzian manifolds.

math.DG↗
arXiv2024

The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity

We consider an $\mathcal{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathcal{MP}$-systems, we study the $\mathcal{MP}$-ray transform in order to obtain new boundary rigidity results for $\mathcal{MP}$-systems. We show that there is an explicit relation between the $\mathcal{MP}$-ray transform and the magnetic one, which allow us to apply results from [DPSU07] to our case. Regarding rigidity, we show that there exists a generic set $\mathcal{G}^{m}$ of simple $\mathcal{MP}$-systems, which is open and dense, such that any two $\mathcal{MP}$-systems close to an element in it and having the same boundary action function, must be $k$-gauge equivalent.

math.DG↗
arXiv2023

The boundary and scattering rigidity problems for simple MP-systems

We consider a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. This is called an $\mathcal{MP}$-system. On simple $\mathcal{MP}$-systems, we consider both the boundary rigidity problem and the scattering rigidity problem. Under the assumption of knowing the boundary action functions or the scattering relations for only one energy level, we prove that we can recover the $\mathcal{MP}$-system up to a gauge. To obtain the results, we reduce the $\mathcal{MP}$-system to the corresponding magnetic system and we apply the results from [DPSU07] on simple magnetic systems.

math.DG↗