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Mihajlo Cekić

Publications and source records attributed to Mihajlo Cekić.

At least 19 recordsLinked to original sources
arXiv2026

The anisotropic Calderón problem for Riemannian metrics at high fixed frequency

We study an inverse boundary value problem for the Helmholtz equation on a smooth compact non-trapping Riemannian manifold with strictly convex boundary. We prove that, given two such metrics, for sufficiently large but fixed frequency $λ$, equality of their Dirichlet-to-Neumann maps implies equality of their lens data, up to a smooth boundary-fixing diffeomorphism. This establishes a high-frequency bridge between the Calderón problem and lens rigidity. We also prove a quantitative version of this result uniformly valid in a suitable bounded set of Riemannian metrics. Crucially, we show that Gaussian beam type solutions concentrating on maximal geodesics, split near the boundary into outgoing/incoming parts, such that the phase of the outgoing part contains information on exit point, direction, and travel time. The recovery of lens data then proceeds by a careful stationary phase analysis and a boundary integral identity.

math.AP↗
arXiv2026

Quasi-Fuchsian flows and the coupled vortex equations

We provide an alternative construction of the quasi-Fuchsian flows introduced by Ghys in \cite{Ghys-92}. Our approach is based on the coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of the Blaschke metric uniquely determined by a conformal class and a holomorphic quadratic differential. We also give formulas for the marked length spectrum of a quasi-Fuchsian flow in the thermostat parametrization.

math.DS↗
arXiv2025

Magnetic Steklov problem on surfaces

The magnetic Dirichlet-to-Neumann map encodes the voltage-to-current measurements under the influence of a magnetic field. In the case of surfaces, we provide precise spectral asymptotics expansion (up to arbitrary polynomial power) for the eigenvalues of this map. Moreover, we consider the inverse spectral problem and from the expansion we show that the spectrum of the magnetic Dirichlet-to-Neumann map, in favourable situations, uniquely determines the number and the length of boundary components, the parallel transport and the magnetic flux along boundary components. In general, we show that the situation complicates compared to the case when there is no magnetic field. For instance, there are plenty of examples where the expansion does $\textit{not}$ detect the number of boundary components, and this phenomenon is thoroughly studied in the paper.

math.AP↗
arXiv2025

Calderón problem for systems via complex parallel transport

We consider the Calderón problem for systems with unknown zeroth and first order terms, and improve on previously known results. More precisely, let $(M, g)$ be a compact Riemannian manifold with boundary, let $A$ be a connection matrix on $E = M \times \mathbb{C}^r$ and let $Q$ be a matrix potential. Let $Λ_{A, Q}$ be the Dirichlet-to-Neumann map of the associated connection Laplacian with a potential. Under the assumption that $(M, g)$ is isometrically contained in the interior of $(\mathbb{R}^2 \times M_0, c(e \oplus g_0))$, where $(M_0, g_0)$ is an arbitrary compact Riemannian manifold with boundary, $e$ is the Euclidean metric on $\mathbb{R}^2$, and $c > 0$, we show that $Λ_{A, Q}$ uniquely determines $(A, Q)$ up to natural gauge invariances. Moreover, we introduce new concepts of complex ray transform and complex parallel transport problem, and study their fundamental properties and relations to the Calderón problem.

math.AP↗
arXiv2025

Resonant forms at zero for dissipative Anosov flows

We study resonant differential forms at zero for transitive Anosov flows on $3$-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen $3$-forms, $Ω_{\text{SRB}}^{\pm}$, and the cohomology classes $[ι_{X}Ω_{\text{SRB}}^{\pm}]$ (where $X$ is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant $1$-forms. When both classes vanish we associate to the flow a $\textit{helicity}$ that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant $1$-forms and SRB-measures as well as the local geometry of the maps $X\mapsto [ι_{X}Ω_{\text{SRB}}^{\pm}]$ near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant $1$-forms at zero, we show that $[ι_{X}Ω_{\text{SRB}}^{\pm}]=0$ and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is $-χ(M)$, the same as in the geodesic flow case. In contrast, we show that if $(M,g)$ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic $1$-form has a Ruelle zeta function whose order of vanishing at zero is $-χ(M)-1$.

math.DS↗
arXiv2024

On the ergodicity of the frame flow on even-dimensional manifolds

It is known that the frame flow on a closed $n$-dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ is odd and $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, we show that: if $n \equiv 2$ mod $4$ or $n=4$, the frame flow is ergodic if the manifold is $\sim 0.3$-pinched, if $n \equiv 0$ mod $4$, it is ergodic if the manifold is $\sim 0.6$-pinched. In the three dimensions $n=7,8,134$, the respective pinching bounds that we need in order to prove ergodicity are $0.4962...$, $0.6212...$, and $0.5788...$. This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that $0.25$-pinched even-dimensional manifolds have an ergodic frame flow.

math.DS↗
arXiv2024

Semiclassical analysis on principal bundles

Let $G$ be a compact Lie group. We introduce a semiclassical framework, called Borel-Weil calculus, to investigate $G$-equivariant (pseudo)differential operators acting on $G$-principal bundles over closed manifolds. In this calculus, the semiclassical parameters correspond to the highest roots in the Weyl chamber of the group $G$ that parametrize irreducible representations, and operators are pseudodifferential in the base variable, with values in Toeplitz operators on the flag manifold associated to the group. This monograph unfolds two main applications of our calculus. Firstly, in the realm of dynamical systems, we obtain explicit sufficient conditions for rapid mixing of volume-preserving partially hyperbolic flows obtained as extensions of an Anosov flow to a $G$-principal bundle (for an arbitrary $G$). In particular, when $G = \mathrm{U}(1)$, we prove that the flow on the extension is rapid mixing whenever the Anosov flow is not jointly integrable, and the circle bundle is not torsion. When $G$ is semisimple, we prove that ergodicity of the extension is equivalent to rapid mixing. Secondly, we study the spectral theory of sub-elliptic Laplacians obtained as horizontal Laplacians of a $G$-equivariant connection on a principal bundle. When $G$ is semisimple, we prove that the horizontal Laplacian is globally hypoelliptic as soon as the connection has a dense holonomy group in $G$. Notably, this result encompasses all flat bundles with a dense monodromy group in $G$. We also prove a quantum ergodicity result for flat (and in some situations non-flat) principal bundles under a suitable ergodicity assumption. We believe that this monograph will serve as a cornerstone for future investigations applying the Borel-Weil calculus across different fields.

math.AP↗
arXiv2024

Stability estimates for the Holonomy Inverse Problem

On a Riemannian manifold $(M, g)$ with Anosov geodesic flow, the problem of recovering a connection from the knowledge of traces of its holonomies along primitive closed geodesics is known as the holonomy inverse problem. In this paper, we prove Hölder type stability estimates for this inverse problem: 1) locally, near generic connections; 2) globally, for line bundles, and for vector bundles satisfying a certain low-rank assumption over negatively curved base $(M, g)$. The proofs are based on a combination of microlocal analysis along with a new non-Abelian approximate Livsic Theorem in hyperbolic dynamics.

math.AP↗
arXiv2024

Generic injectivity of the X-ray transform

In dimensions $\geq 3$, we prove that the X-ray transform of symmetric tensors of arbitrary degree is generically injective with respect to the metric on closed Anosov manifolds and on manifolds with spherical strictly convex boundary, no conjugate points and a hyperbolic trapped set. This has two immediate corollaries: local spectral rigidity, and local marked length spectrum rigidity (building on earlier work by Guillarmou, Knieper and the second author [arXiv:1806.04218], [arXiv:1909.08666]), in a neighbourhood of a generic Anosov metric. In both cases, this is the first work going beyond the negatively curved assumption or dimension $2$. Our method, initiated in [arXiv:2008.09191] and fully developed in the present paper, is based on a perturbative argument of the $0$-eigenvalue of elliptic operators via microlocal analysis which turn the analytic problem of injectivity into an algebraic problem of representation theory. When the manifold is equipped with a Hermitian vector bundle together with a unitary connection, we also show that the twisted X-ray transform of symmetric tensors (with values in that bundle) is generically injective with respect to the connection. This property turns out to be crucial when solving the $\textit{holonomy inverse problem}$, as studied in a subsequent article [arXiv:2105.06376].

math.AP↗
arXiv2023

The Holonomy Inverse Problem

Let $(M,g)$ be a smooth Anosov Riemannian manifold and $\mathcal{C}^\sharp$ the set of its primitive closed geodesics. Given a Hermitian vector bundle $\mathcal{E}$ equipped with a unitary connection $\nabla^{\mathcal{E}}$, we define $\mathcal{T}^\sharp(\mathcal{E}, \nabla^{\mathcal{E}})$ as the sequence of traces of holonomies of $\nabla^{\mathcal{E}}$ along elements of $\mathcal{C}^\sharp$. This descends to a homomorphism on the additive moduli space $\mathbb{A}$ of connections up to gauge $\mathcal{T}^\sharp: (\mathbb{A}, \oplus) \to \ell^\infty(\mathcal{C}^\sharp)$, which we call the $\textit{primitive trace map}$. It is the restriction of the well-known $\textit{Wilson loop}$ operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map $\mathcal{T}^\sharp$ is locally injective near generic points of $\mathbb{A}$ when $\dim(M) \geq 3$. We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of $\mathbb{A}$ with the Pollicott-Ruelle resonance near zero of a certain natural transport operator.

math.DS↗
arXiv2023

Isospectral connections, ergodicity of frame flows, and polynomial maps between spheres

We show that on closed negatively curved Riemannian manifolds with simple length spectrum, the spectrum of the Bochner Laplacian determines both the isomorphism class of the vector bundle and the connection up to gauge under a low-rank assumption. We also show that flows of frames on low-rank frame bundles extending the geodesic flow in negative curvature are ergodic whenever the bundle admits no holonomy reduction. This is achieved by exhibiting a link between these problems and the classification of polynomial maps between spheres in real algebraic geometry.

math.DS↗
arXiv2023

Local lens rigidity for manifolds of Anosov type

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if $g_0$ is such a metric, then any metric $g$ sufficiently close to $g_0$ and with same lens data is isometric to $g_0$, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension $2$ and in any dimension, further assuming that the curvature is non-positive.

math.DG↗
arXiv2023

Correspondence between Pestov and Weitzenböck identities

The aim of this note is to establish the correspondence between the twisted localized Pestov identity on the unit tangent bundle of a Riemannian manifold and the Weitzenböck identity for twisted symmetric tensors on the manifold.

math.DG↗
arXiv2023

On the ergodicity of unitary frame flows on Kähler manifolds

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim_{\mathbb{C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm{U}(m)$-bundle $F_{\mathbb{C}}M$ of unitary frames. We show that if $m \geq 6$ is even, and $m \neq 28$, there exists $λ(m) \in (0, 1)$ such that if $(M, g, J)$ has negative $λ(m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $λ(m)$ satisfy $λ(6) = 0.9330...$, $\lim_{m \to +\infty} λ(m) = \tfrac{11}{12} = 0.9166...$, and $m \mapsto λ(m)$ is decreasing. This extends to the even-dimensional case the results of Brin-Gromov who proved ergodicity of the unitary frame flow on negatively-curved compact Kähler manifolds of odd complex dimension.

math.DS↗
arXiv2022

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $Σ$ with Betti number $b_1$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott-Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $SΣ$ with harmonic 1-forms on $Σ$.

math.DS↗