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Tristan Humbert

Publications and source records attributed to Tristan Humbert.

14 recordsLinked to original sources
arXiv2026

Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric

In 2004, Manning showed that the topological entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly decreasing along the normalized Ricci flow, and he asked if an analogous result holds in higher dimensions for metrics in a neighborhood of a hyperbolic metric. In this paper, we affirmatively answer this question. Namely, we show that the topological entropy of the geodesic flow of a closed Riemannian manifold that carries a hyperbolic metric is indeed strictly decreasing along the normalized Ricci flow starting from a metric of variable negative sectional curvature sufficiently close to the hyperbolic metric.

math.DG↗
arXiv2026

Monotonicity of the Liouville entropy along the Ricci flow on surfaces

We show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is eventually strictly increasing along the normalized Ricci flow (NRF). More precisely, we obtain a new expression for the derivative of the Liouville entropy along an arbitrary conformal deformation in dimension 2, and we prove it is positive in the direction of the NRF for 1/6-pinched metrics. This partially answers a question of Manning from 2004. In addition, we show that the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is strictly increasing along the NRF starting from any metric of non-constant negative curvature.

math.DS↗
arXiv2026

Generic twisted Pollicott--Ruelle resonances and zeta function at zero

For a connected orientable closed surface $(Σ,g)$ of genus $G$ with Anosov geodesic flow, we show the existence of an open subset $U_g$ of finite-dimensional irreducible representations of the fundamental group of its unit tangent bundle, whose complement has complex codimension at least one and such that for any $ρ\in U_g$, the twisted Ruelle zeta function $ζ_{g,ρ}(s)$ vanishes at $s=0$ to order ${\rm dim}(ρ)(2G-2)$ if $ρ$ factors through $π_1(Σ)$, and does not vanish otherwise. In the second case, we show that $ζ_{g,ρ}(0)$ is given by the Reidemeister--Turaev torsion, thus extending Fried's conjecture to a generic set of acyclic (but not necessarily unitary) representations. We also show that the order of vanishing of the untwisted zeta function is constant for an open and dense subset of Anosov metrics in the connected component of a hyperbolic $3$-metric. Our proofs rely on computing the dimensions of the spaces of generalized twisted Pollicott--Ruelle resonant states at zero.

math.DS↗
arXiv2025

Unstable entropy for Anosov diffeomorphisms on the 3-torus

For Anosov diffeomorphisms on the $3$-torus which are strongly partially hyperbolic with expanding center, we construct systems of strong unstable and center stable Margulis measures which are holonomy-invariant. This allows us to obtain a characterization of the measures of maximal unstable entropy in terms of their conditional measures along the strong unstable leaves. Moreover, we show that the Margulis systems identify with Pollicott-Ruelle (co)-resonant states for the action on 2-forms. This shows that the unstable topological entropy and a measure of maximal unstable entropy can be retrieved from the spectral approach.

math.DS↗
arXiv2025

Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3

Let $M$ be a compact manifold without boundary equipped with a Riemannian metric $g$ of negative curvature. In this paper, we introduce the marked Poincaré determinant (MPD), a homothety invariant of $g$ depending on differentiable periodic data of its geodesic flow. The MPD associates to each free homotopy class of closed curves in $M$ a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. We prove a local MPD rigidity result in dimension 3: if $g$ is sufficiently close to a hyperbolic metric $g_0$ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of $g_0$ is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.

math.DG↗
arXiv2025

Katok's entropy conjecture near real and complex hyperbolic metrics

We show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $n\geq 3$, there exists a neighborhood $\mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $g\in \mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if and only if $g$ and $g_0$ are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result for the hyperbolic rank and for metrics with $C^2$ Anosov foliations near complex hyperbolic metrics.

math.DS↗
arXiv2025

Degenerate flat bands in twisted bilayer graphene

We prove that in the chiral limit of the Bistritzer--MacDonald Hamiltonian, there exist magic angles at which the Hamiltonian exhibits flat bands of multiplicity four instead of two. We analyse the structure of Bloch functions associated with the bands of arbitrary multiplicity, compute the corresponding Chern number to be $ -1 $, and show that there exist infinitely many degenerate magic angles for a generic choice of tunnelling potential, including the Bistritzer--MacDonald potential. Moreover, we demonstrate for generic tunnelling potentials flat bands have only twofold or fourfold multiplicities.

math-ph↗
arXiv2025

Measure of maximal entropy for minimal Anosov actions

For a minimal Anosov $\mathbb R^κ$-action on a closed manifold, we study the measure of maximal entropy constructed by Carrasco and Rodriguez-Hertz in \cite{CarHer} and show that it fits into the theory of Ruelle-Taylor resonances introduced by Guedes Bonthonneau, Guillarmou, Hilgert, and Weich in \cite{GBGHW}. More precisely, we show that the topological entropy corresponds to the first Ruelle-Taylor resonance for the action on a certain bundle of forms and that the measure of maximal entropy can be retrieved as the distributional product of the corresponding resonant and co-resonant states. As a consequence, we prove a Bowen-type formula for the measure of maximal entropy and a counting result on the number of periodic torii.

math.DS↗
arXiv2025

Critical axis of Ruelle resonances for Anosov flow with a potential

We combine methods from microlocal analysis and dimension theory to study resonances with largest real part for an Anosov flow with smooth real valued potential. We show that the resonant states are closely related to special systems of measures supported on the stable manifolds introduced by Climenhaga. As a result, we relate the presence of the resonances on the critical axis to mixing properties of the flow with respect to certain equilibrium measures and show that these equilibrium measures can be reconstructed from the spectral theory of the Anosov flow.

math.DS↗
arXiv2024

New bounds for the number of connected components of fewnomial hypersurfaces

We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lfloor(d-1)/2\rfloor + 3$ connected components in the positive orthant of $\mathbb{R}^n$. We present an explicit polynomial in $2$ variables with $5$ monomials which defines a curve with three connected components in the positive orthant, showing that our bound is sharp for $d = 2$ (and any $n$). Our results hold for polynomials with real exponent vectors.

math.AG↗
arXiv2023

Chiral limit of twisted trilayer graphene

We initiate the mathematical study of the Bistritzer-MacDonald Hamiltonian for twisted trilayer graphene in the chiral limit (and beyond). We develop a spectral theoretic approach to investigate the presence of flat bands under specific magic parameters. This allows us to derive trace formulae that show that the tunnelling parameters that lead to flat bands are nowhere continuous as functions of the twisting angles.

math-ph↗
arXiv2023

Integrability in the chiral model of magic angles

Magic angles in the chiral model of twisted bilayer graphene are parameters for which the chiral version of the Bistritzer--MacDonald Hamiltonian exhibits a flat band at energy zero. We compute the sums over powers of (complex) magic angles and use that to show that the set of magic angles is infinite. We also provide a new proof of the existence of the first real magic angle, showing also that the corresponding flat band has minimal multiplicity for the simplest possible choice of potentials satisfying all symmetries. These results indicate (though do not prove) a hidden integrability of the chiral model.

math-ph↗
arXiv2023

Fine structure of flat bands in a chiral model of magic angles

We analyze symmetries of Bloch eigenfunctions at magic angles for the Tarnopolsky--Kruchkov--Vishwanath chiral model of the twisted bilayer graphene (TBG) following the framework introduced by Becker--Embree--Wittsten--Zworski. We show that vanishing of the first Bloch eigenvalue away from the Dirac points implies its vanishing at all momenta, that is the existence of a flat band. We also show how the multiplicity of the flat band is related to the nodal set of the Bloch eigenfunctions. We conclude with two numerical observations about the structure of flat bands.

math-ph↗