arXiv research

The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

The paper proves a Liouville theorem for specific harmonic maps with free boundary.

We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.

In this short paper we study $L_f^p$-Liouville property with $0<p<1$ for nonnegative $f$-subharmonic functions on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with $\mathrm{Ric}_f^m$ bounded below for $0<m\leq\infty$. We prove a sharp $L_f^p$-Liouville theorem when $0<m<\infty$. We also prove an $…

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, whi…

The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.

This note provides a new proof of the real analyticity of the Liouville map.

For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.

Loewner inequality proven for curved surfaces.

Sharp heat equation gradient estimates on compact manifolds.

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.

The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.

Sharp gradient estimates for a weighted p-Laplacian equation on metric measure spaces.

New proof for weak mixing in polygonal billiards.

We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^…

The operator realizing a Dehn twist in quantum Teichmuller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c>1. The completeness condition of the eigenvectors includes the integration measure which app…

The study defines Finsler metrics on special surfaces and constructs geodesic currents.

The paper studies Harnack inequalities on Finsler metric measure spaces.

Researchers prove constant solutions for a specific Finslerian equation.

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.

Classifies invariant measures on specific character varieties.

For smooth metric measure spaces $(M, g, e^{-f} dvol)$ we prove a Liuoville-type theorem when the Bakry-Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case $f$ is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spac…

The paper explores the $L^1$-Liouville property on graphs and its connections to stochastic completeness.

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of…

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(Δ_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^α…

We give exposition of a Liouville theorem established in \cite{Li3} which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications of the …

Study explores Liouville's theorem and SLP for harmonic functions on cones and surfaces.

Equal diagonal energies proven on Liouville surfaces.

A 'Liouville structure' is a structure isomorphic to a cotangent vector fibration. A Liouville structure is an essential ingredient of every variational formulation of a physical theory. For reasons of interpretation the Liouville structure can not be replaced by the corresponding cotangent fibration. We give a precise…

Article proves Liouville theorem for heat equation in super Ricci flow.

The Liouville theorem is proven for V T-harmonic map heat flow.

Study Liouville action for harmonic maps between Riemann surfaces.

We consider a complete noncompact smooth metric measure space $(M^n,g,e^{-f} dv)$ and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative $f$-subharmonic function with bounded weighted $L^1$ norm is constant.

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.

For convex co-compact hyperbolic manifolds $Γ\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $δ_Γ< n/2$, we show that the high-frequency Eisenstein series associated to a point $ξ$ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …

The paper classifies invariant operators and proves a Liouville theorem.

In this paper, we will address to the following parabolic equation $u_t=Δ_fu + F(u)$ on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equ…

Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.

For any oriented Seifert manifold X and compact connected Lie group G with finite center, we relate the Reidemeister density of the moduli space of representations of the fundamental group of X into G to the Liouville measure of some moduli spaces of representations of surface groups into G.

The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.

Study shows that deformed Liouville metrics on tori remain Liouville.

Proves a Liouville-type theorem for p-Laplacian on manifolds.

Study Liouville theorem for specific harmonic maps.

The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.

Optimal Liouville theorem for half-Euclidean space equations.