Introduces Reshetnyak's subharmonic distances theory.
problem No specific problem stated; focuses on theory introduction.
method No specific method mentioned; focuses on theory introduction.
result Provides an overview of Reshetnyak's subharmonic distances theory.
The paper extends Reshetnyak's theorem to Lorentzian length spaces with upper curvature bounds.
problem Characterizing upper curvature bounds in Lorentzian geometry.
method Analogous to Reshetnyak's theorem, using convex regions and 1-anti-Lipschitz maps.
result Characterization of upper curvature bounds via four-point configurations.
Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalizati…
We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map f:M→N between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theo…
Let Z be an Alexandrov space with curvature bounded below by -1 such that Z is homotopy equivalent to a real hyperbolic manifold M. It is known that the volume of Z is not smaller than the volume of M. If the volumes are equal, this short paper proves that the homotopy equivalence is homotopic to an isometric homeomorp…
New method glues Lorentzian spaces, preserving curvature bounds.
problem Creating new spaces from existing ones in Lorentzian geometry.
method Introducing an amalgamation process for Lorentzian pre-length spaces.
result Gluing preserves upper curvature bounds in spacetimes.
Quantifies Schur's theorem for curves in CAT(k) spaces.
problem Quantifying Schur's comparison theorem for curves in CAT(k) spaces.
method Comparison formula for curves in model planes, curvature measures, moment arm, and Reshetnyak's theorem.
result Sharpens and extends classical arm and bow lemmas and Riemannian analogues.
Notes on quasiregular maps between Riemannian manifolds, preserving Sobolev forms.
problem Extending quasiregular map theory from Euclidean to Riemannian manifolds.
method Recalling different approaches to first-order Sobolev spaces, showing equivalence, and transferring key theorems.
result Pull-backs with quasiregular maps preserve Sobolev differential forms of the conformal exponent.
A map between manifolds is an isometry if it's Lipschitz and scalar curvature bounded.
problem Characterizing maps between manifolds based on their scalar curvature and Lipschitz continuity.
method Spectral properties of Dirac operators and index theory for low regularity metrics and bundles.
result A 1-Lipschitz map between manifolds is an isometry if it has bounded scalar curvature.
The paper proves rigidity for shells in non-Euclidean spaces.
problem Proving rigidity for shells in non-Euclidean spaces.
method Analyzing a stretching plus bending functional of an elastic shell in a Riemannian manifold.
result A sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell.
This paper introduces Alexandrov's theory of singular surfaces and their curvature.
problem Understanding the geometry of singular surfaces with intrinsic metrics and curvature.
method Develops the theory of Alexandrov surfaces, focusing on their convergence and stability properties.
result Classifies compact Alexandrov surfaces using the conformal viewpoint introduced by Reshetnyak.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by M. Gromov in 1983, who proposed an argument toward the existence of L2-extremizers exploiting the theory of r-regularity developed by P. A. White and others by the 1950s. We propose to study the problem of syst…
New method constructs solution operators for PDEs with prescribed support properties.
problem Constructing solution operators for under/overdetermined PDEs with specific support properties.
method Using a recovery on curves condition and taking smooth averages over curves, we obtain integral solution operators and representation formulas.
result Our method leads to integral representation formulas for overdetermined PDEs and solution operators for underdetermined PDEs.