arXiv research

Generative model improved using Liouville PDE-based sliced-Wasserstein flow.

Study proves unique compactification of hyperbolic space.

Proves stability of convex disks close to round caps.

Paper solves curvature prescription problem on surfaces with boundary.

Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are…

Construct Lax pairs for BKM equations and related integrable hierarchies.

Study rough volatility models using path-dependent PDEs and fractional Brownian motions.

A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in $\bbR^n$, or $\bbS^n$, $n\in \bbN$, which are distributed according to the configurational microcanonical measure $δ(E-H)$, or rather some regularization thereof, where H is the configu…

Let $f$ be a smooth plurisubharmonic function which solves $$ \det(f_{i\bar j})=1\;\;\;\;\;\;\mbox{in }Ω\subset \mathbb C^n.$$ Suppose that the metric $ω_{f}=\sqrt{-1}f_{i\bar j}dz_{i}\wedge d\bar z_{j}$ is complete and $f$ satisfies the growth condition $$ C^{-1}(1+|z|^2)\leq f\leq C(1+ |z|^2),\;\;\;\; as\;\;\; |z|\to…

Local Neural Operators enable efficient system-level analysis of complex PDEs.

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a disk with two corners admits a conformal metric with constant Gauss curv…

We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate ell…

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

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 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.

The paper classifies invariant operators and proves a Liouville theorem.

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

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.

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

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

Study Liouville theorems for harmonic maps along ancient super Ricci flows.

Liouville domains have become central objects in symplectic and contact geometry. However, the auxiliary data they involve --- namely, Liouville forms --- and the non-compactness of their completions generate some inconvenience. The notion of ideal Liouville domains is designed to suppress these awkward aspects and to …

The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.

The paper extends Liouville's theorem to calibrated geometries in various dimensions.

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

Liouville entropy increases strictly along Ricci flow on surfaces.

Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.

New method shows some 3D shapes can't be filled in certain ways.

The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.

Researchers found a Weyl law for Liouville quantum gravity eigenvalues.

Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.

We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the cla…

Arnold-Liouville systems cannot be bi-Hamiltonian generically.

Ancient Lagrangian flows get limited convex solutions.

Study proves Liouville theorem for specific curvature equations with boundary conditions.

The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.