Related papers: Curvature bounds on length-minimizing discs

We prove that a H-surface M in H^2xR, |H| <= 1/2, inherits the symmetries of its boundary when the boundary is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one,…

We consider curvature-induced resonances in a planar two-dimensional Dirichlet tube of a width $ d $. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as $ d\to 0+ $, provided…

We prove the existence of locally distance increasing maps with it controllable small curvatures

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are,…

Stabilizing defects in liquid-crystal systems is crucial for many physical processes and applications ranging from functionalizing liquid-crystal textures to recently reported command of chaotic behaviors of active matters. In this work, we…

We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterised by the local excess of curvature on the target manifold. We give a universal estimate for curvature…

Upper bounds are derived for codes in Stiefel and Grassmann manifolds with given minimal chordal distance. They stem from upper bounds for codes in products of unit spheres and projective spaces. The new bounds are asymptotically better…

We consider quasiconformal mappings of the unit disk that have a planar extension which have $p$-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this…

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that…

In this short note we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about next steps.

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical…

We relate the (non)existence of lower scalar curvature bounds to the existence of certain distance-decreasing maps. We also give a sufficient condition for the existence of a limiting scalar curvature measure in the backward limit of a…

In this work, we use the warped \( \mu \)-bubble method to study the consequences of a spectral curvature bound. In particular, with a lower spectral Ricci curvature bound and a lower spectral scalar curvature bound, we show that the band…

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic…

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded…

We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known…

Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence…

The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains with infinite mass boundary conditions. This bound is given in terms of a conformal…

We extend Struwe's result (Acta Math., 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in $(0, \pi)$. Specifically, let $\Sigma$ be a surface in $\mathbb{R}^3$…

We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous…