Loewner's theorem connects two curve properties via simple functions.
problem Understanding the rotation number of curves defined by smooth periodic functions.
method Elementary proof following Bol's work.
result Two curve properties have non-negative rotation number.
New fractal spaces not quasisymmetric to Loewner spaces discovered.
problem Finding new fractal spaces not quasisymmetric to Loewner spaces.
method Introduced iterated graph systems (IGS) to create new fractal spaces.
result Disproved Kleiner's conjecture about self-similar fractals.
Topological obstructions to admissibility in σk-Loewner--Nirenberg problem
problem Admissibility condition for σk-Loewner--Nirenberg problem method Exhibit topological obstructions
result Illustrate with examples
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 inequality proven for curved surfaces.
problem Proving Loewner's inequality for nonpositively curved surfaces.
method Combining Gauss-Bonnet formula with averaging argument using geodesic flow invariance.
result Found a disk with large total curvature around its center, leading to large area.
New formula connects Loewner energy to moving frames' renormalised energy.
problem Calculating Loewner energy of Jordan curves.
method Using renormalised energy of moving frames.
result Loewner energy as Kähler potential for Weil-Petersson space.
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
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.
A carpet is a metric space which is homeomorphic to the standard Sierpiński carpet in R2, or equivalently, in S2. 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 finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
problem Finding a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
method Defining a complex projective structure and using accessory parameters to characterize the curve.
result The accessory parameters are the residues of the quadratic differential comparing the projective structure to the trivial one.
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.
problem Optimizing Jordan curves and positive curves on boundary spaces.
method Using conformal welding and Möbius transformations.
result Symmetries between boundary spaces and pleated planes.
Solves Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.
problem Solving the Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.
method Analyzes fully nonlinear Loewner-Nirenberg problem and uses conformal metrics.
result Solves the σk-Loewner-Nirenberg problem for all k ≤ n/2. Quasispheres can be approximated by smooth spheres.
problem Characterizing quasispheres using geometric conditions.
method Proving every quasisphere is a limit of smooth spheres and providing necessary and sufficient conditions for uniform quasispheres.
result Every quasisphere can be approximated by uniform quasispheres that satisfy specific geometric conditions.
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…
Flow approach solves Ricci equation boundary problem.
problem Solving generalized Loewner-Nirenberg problem for σk-Ricci equation. method Flow approach to prove existence and uniqueness of solution.
result Solution converges to the boundary value as time goes to infinity.
Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
Two flow methods solve a problem on Riemannian manifolds, proving convergence to the Loewner-Nirenberg solution.
problem Solving the Loewner-Nirenberg problem on compact Riemannian manifolds with boundary.
method Direct flow and Yamabe flow approaches.
result Convergence of the flows to the solution of the Loewner-Nirenberg problem under various conditions.
Solutions to a specific problem are shown to be locally Lipschitz but not differentiable.
problem Locally Lipschitz viscosity solutions to the σk-Loewner-Nirenberg problem on annuli. method Analytical proof of regularity and non-differentiability.
result Solutions are $C^{1, rac{1}{k}}_{
m loc}$ in each of the annulus regions and have a jump in radial derivative.
The paper classifies solutions to a specific equation and finds counterexamples to boundary estimates.
problem Solutions to a conformally invariant equation of the form f(λ(−Aw))=21 in the upper half-space. method Novel application of the method of moving spheres with new estimates and regularity near the boundary.
result When μΓ+≤1, solutions form a one-parameter family, including the hyperbolic solution. It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if T is a Riemannian 2-torus with boundary in Rn, such that the boundary curve is a standard unit circle, then the length o…
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…
The paper proves existence of solutions to a Loewner-Nirenberg problem on Riemannian manifolds.
problem Existence of solutions to a specific nonlinear problem on Riemannian manifolds.
method Proves existence of viscosity solutions using approximating cones and limit of smooth solutions.
result Existence of a Lipschitz viscosity solution to the Loewner-Nirenberg problem.
We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermo…
The Schwarzian action is linked to the area of Epstein curves in hyperbolic geometry.
problem Understanding the relationship between the Schwarzian action and geometric properties of curves.
method Applying Epstein's construction to relate the Schwarzian action to the area of Epstein curves in hyperbolic disk.
result The Schwarzian action is equivalent to the logarithm of the bi-local observable in Schwarzian field theory.
Solves geometric problems using fully nonlinear equations and Morse theory.
problem Geometric problems, specifically Loewner-Nirenberg and Yamabe problems.
method Investigates structure of fully nonlinear equations and applies Morse theory techniques.
result Constructs admissible metrics under weak conditions and demonstrates topological obstructions.
Maximal solution of a PDE shows boundary smoothness for certain domains.
problem Boundary behavior of solutions to a specific PDE.
method Reduction to a nonlinear Fuchsian elliptic PDE.
result Hyperbolic radius is smooth up to the boundary.
Solves nonlinear problems on metric structures through eigenvalue counting.
problem Nonlinear equations on metric structures
method Counting large eigenvalues of linearized operators
result Solves fully nonlinear Loewner-Nirenberg and Yamabe problems
The paper examines the smoothness of solutions to a specific partial differential equation on smooth domains.
problem Regularity of viscosity solutions of the σk-Loewner-Nirenberg problem. method Analysis of the Schouten tensor and geometric measure theory.
result The first (n−1) derivatives of $d^{ rac{n-2}{2}} u$ are Hölder continuous in a specific region, and u is smooth in another region. We consider the problem of finding on a given Euclidean domain Ω of dimension n≥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…
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.
Formula for squeezing function on annuli disproves conjecture.
problem Proving squeezing function formula for annuli.
method Schottky-Klein prime function and Loewner differential equation.
result Formula for squeezing function on annuli established.
Study on metrics on manifolds with specific curvature properties.
problem Existence of conformal metrics with negative constant scalar curvature and negative constant mean curvature.
method Construction of metrics on smooth manifolds with solid cones removed, proving existence under certain conditions.
result Existence of such metrics if and only if the dimension condition d>(n-2)/2.
The paper introduces a frequency-domain estimator for low-order systems from noisy data.
problem Estimating frequency responses of low-order systems from noisy measurements.
method Uses a quadratic data-fitting term regularized by the nuclear norm of a Loewner matrix, subject to a convex stability constraint.
result Proves a finite-sample error bound and extends it to all frequencies through rational interpolation.
New inequalities for matrix supermartingales converge under various conditions.
problem Convergence and maximal inequalities of supermartingales in positive semidefinite matrices.
method Developed new concentration inequalities for matrix supermartingales.
result New inequalities for matrix supermartingales under different tail conditions.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
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.
problem Determining the hyperbolic metric from boundary data of convex co-compact hyperbolic manifolds.
method Analysis of volume functions and their relation to boundary data, using first variations.
result New connections with physics and probability theory, with open questions remaining.
Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.
problem Determining the volume of conformal metrics on planar domains with circular boundaries.
method Extending Epstein maps to conformal metrics, defining W-volume, using Schottky uniformization and Loewner energy.
result Shows a bound on the renormalized volume of Schottky uniformization and provides a realization of Loewner energy.
The paper proves Ricci flow convergence on compact manifolds with boundary.
problem Analyzing Ricci flow on compact manifolds with boundary conditions.
method Normalized Ricci flow with prescribed mean curvature on boundary.
result The solution converges to a complete hyperbolic metric with sectional curvature < -1.
The paper extends a Ricci flow result for compact manifolds with boundary.
problem Analyzing Ricci flow on compact manifolds with boundary.
method Normalized Ricci flow with specific boundary conditions.
result The flow converges to a complete hyperbolic metric as to∞. A new systolic inequality with a remainder for the real projective plane.
problem Proving a stronger systolic inequality for the real projective plane.
method Developing a new systolic inequality with a remainder term.
result A stronger systolic inequality with a remainder for the real projective plane.
In 1972, Marcel Berger defined a metric invariant that captures the `size' of k-dimensional homology of a Riemannian manifold. This invariant came to be called the k-dimensional SYSTOLE. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, ins…
Given a centrally symmetric convex body K⊂Rd and a positive number λ, we consider, among all ellipsoids E⊂Rd of volume λ, those that best approximate K with respect to the symmetric difference metric, or equivalently that maximize the volume of E∩K: these are the maxi…
This paper deals with the conformal deformation of the standard metric in a domain on the sphere to a complete metric with the constant scalar curvature. The problem of description of domains allowing such deformation originates in the works of Loewner and Nirenberg, and Schoen and Yau concerned with the locally confor…
This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a confor…