arXiv research

Formula for squeezing function on annuli disproves conjecture.

Topological obstructions to admissibility in $σ_k$-Loewner--Nirenberg problem

Flow approach solves Ricci equation boundary problem.

The paper examines the smoothness of solutions to a specific partial differential equation on smooth domains.

Solves Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.

Solves geometric problems using fully nonlinear equations and Morse theory.

Proves existence of smooth metrics with specific curvature properties.

Solves nonlinear problems on metric structures through eigenvalue counting.

The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.

New fractal spaces not quasisymmetric to Loewner spaces discovered.

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Further…

Two flow methods solve a problem on Riemannian manifolds, proving convergence to the Loewner-Nirenberg solution.

Maximal solution of a PDE shows boundary smoothness for certain domains.

We prove that C. Loewner's inequality for the torus is satisfied by all hyperelliptic surfaces X, as well. We first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from the Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on…

Loewner's theorem connects two curve properties via simple functions.

Loewner inequality proven for curved surfaces.

New formula connects Loewner energy to moving frames' renormalised energy.

We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence a…

The study proves surfaces with high genus have a specific inequality.

The paper proves existence of solutions to a Loewner-Nirenberg problem on Riemannian manifolds.

A carpet is a metric space which is homeomorphic to the standard Sierpiński carpet in $\mathbb{R}^2$, or equivalently, in $S^2$. A carpet is called thin if its Hausdorff dimension is $<2$. A metric space is called Q-Loewner if its $Q$-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a $(1,Q)$-Poin…

The paper classifies solutions to a specific equation and finds counterexamples to boundary estimates.

We study combinatorial modulus on boundaries of hyperbolic Coxeter groups. We give new examples of hyperbolic groups whose boundary satisfies a combinatorial version of the Loewner property, and prove Cannon's conjecture for Coxeter groups. We also establish some connections with l^p cohomology.

We show that Bonnesen's isoperimetic defect has a systolic analog for Loewner's torus inequality. The isosystolic defect is expressed in terms of the probabilistic variance of the conformal factor of the metric g with respect to the flat metric of unit area in the conformal class of g.

Two optimization problems for Loewner energy curves and their symmetries.

Quasispheres can be approximated by smooth spheres.

In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner prop…

We generalize optimal inequalities of C. Loewner and M. Gromov, by proving lower bounds for the total volume in terms of the homotopy systole and the stable systole. Our main tool is the construction of an area-decreasing map to the Jacobi torus, streamlining and generalizing the construction of the first author in col…

Study properties of solutions with singularities in the negative cone.

The Schwarzian action is linked to the area of Epstein curves in hyperbolic geometry.

We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a "defect term", similar …

We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l…

In this paper, the existence of C^1-umbilics with arbitrarily high indices is shown. This implies that more than C^1-regularity is required to prove Loewner's conjecture.

The paper introduces a frequency-domain estimator for low-order systems from noisy data.

New inequalities for matrix supermartingales converge under various conditions.

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

Let X be a closed manifold of dimension 2m >= 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small volume, such that every middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.

We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns out to be…

Study on hyperbolic manifolds and their boundary data, focusing on volume functions.

Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.

Studies projective geometry and partial differential equations prolongation.

Paper solves a class of differential equations with specific solutions.

The paper generalizes Monge-Ampère equations and their solutions in differential geometry.

DiffEqFlux.jl is a library for fusing neural networks and differential equations. In this work we describe differential equations from the viewpoint of data science and discuss the complementary nature between machine learning models and differential equations. We demonstrate the ability to incorporate DifferentialEqua…

Classifies scalar second-order PDEs with low-dimensional symmetry groups.

A new systolic inequality with a remainder for the real projective plane.

Neural differential equations combine deep learning and differential equations for modeling complex systems.

Neural networks can approximate complex stochastic equations well.