Recent zbMATH articles in MSC 30C62https://www.zbmath.org/atom/cc/30C622022-05-16T20:40:13.078697ZWerkzeugQuasiconformal Whitney partitionhttps://www.zbmath.org/1483.300532022-05-16T20:40:13.078697Z"Gol'dshtein, Vladimir"https://www.zbmath.org/authors/?q=ai:goldshtein.vladimir"Zobin, N."https://www.zbmath.org/authors/?q=ai:zobin.nahum|zobin.naum-mSummary: Whitney partition is a very important concept in modern geometric analysis. We discuss here a quasiconformal version of Whitney
partition that can be useful for Sobolev spaces.Beurling-Ahlfors extension by heat kernel, \(\mathrm{A}_{\infty}\)-weights for VMO, and vanishing Carleson measureshttps://www.zbmath.org/1483.300542022-05-16T20:40:13.078697Z"Wei, Huaying"https://www.zbmath.org/authors/?q=ai:wei.huaying"Matsuzaki, Katsuhiko"https://www.zbmath.org/authors/?q=ai:matsuzaki.katsuhiko.1|matsuzaki.katsuhikoAn increasing homeomorphism \(h\) of the real line \(\mathbb R\) onto itself is said to be quasisymmetric if there exists some \(M>0\) such that \[\frac 1M\le\frac{h(x+t)-h(x)}{h(x)-h(x-t))}\le M\] for all \(x\in\mathbb R\) and \(t>0\). In an important paper [Acta Math. 96, 125--142 (1956; Zbl 0072.29602)] \textit{A. Beurling} and \textit{L. V. Ahlfors} constructed a quasiconformal homeomorphism \(f\) of the upper half plane \(\mathbb U=\{z=x+iy: y>0\}\) onto itself which has boundary values \(h\) when \(h\) is a quasisymmetric homeomorphism. A variant of the Beurling-Ahlfors extension was investigated by \textit{R. A. Fefferman} et al. [Ann. Math. (2) 134, No. 1, 65--124 (1991; Zbl 0770.35014)] when the quasisymmetric homeomorphism \(h\) is induced by an \(A^{\infty}\) weight. Precisely, when \(h\) is locally absolutely continuous on the real line such that \(h'\) is an \(A^{\infty}\) weight, Fefferman et al. [loc. cit.] showed that \(h\) can be extended to a quasiconformal mapping \(F\) of the upper half plane, which is a variant of the Beurling-Ahlfors extension by the heat kernel, such that the complex dilatation \(\mu\) of \(F\) induces a Carleson measure \(|\mu(x+iy)|^2/y\). In this paper, the authors give a rather detailed exposition of this result, and show that the the complex dilatation \(\mu\) of \(F\) induces a vanishing Carleson measure \(|\mu(x+iy)|^2/y\) under the additional assumption that \(\log h'\) is a VMO function. This answers a question raised recently by the reviewer [Ann. Fenn. Math. 47, No. 1, 57--82 (2022; Zbl 1482.30057)].
Reviewer: Yuliang Shen (Suzhou)The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometryhttps://www.zbmath.org/1483.300842022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Danciger, Jeffrey"https://www.zbmath.org/authors/?q=ai:danciger.jeffrey"Maloni, Sara"https://www.zbmath.org/authors/?q=ai:maloni.sara"Schlenker, Jean-Marc"https://www.zbmath.org/authors/?q=ai:schlenker.jean-marcA theorem by Alexandrov and Pogorelov says that any smooth Riemannian metric on the 2-sphere with curvature \(K>-1\) coincides with the induced metric on the boundary of some compact convex subset of hyperbolic 3-space with smooth boundary and, furthermore, that this compact convex subset is unique up to a global isometry of hyperbolic 3-space. In the paper under review, the authors study a generalization of this result to unbounded convex subsets of hyperbolic 3-space, more especially to convex subsets bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, they supplement the notion of induced metric on the boundary of the convex set so that it includes a gluing map at infinity which records how the asymptotic geometries of the two surfaces fit together near the limiting Jordan curve. They restrict their study to the case where the induced metrics on the two bounding surfaces have constant curvature \(K \in [-1, 0)\) and were the Jordan curve at infinity is a quasicircle. In this case the gluing map becomes a quasisymmetric homeomorphism of the circle and the authors prove that for \(K\) in the given interval, any quasisymmetric map can be obtained as the gluing map at infinity along some quasicircle. They also obtain Lorentzian analogous of these results, in which hyperbolic 3-space is replaced by the 3-dimensional anti-de Sitter space \(\mathbb{A}d\mathbb{S}^3\), whose natural boundary is the Einstein space \(\mathrm{Ein}^{1,1}\), a conformal Lorentzian analogue of the Riemannian sphere. The authors say that their results may be viewed as universal versions of a conjecture of Thurston about the realization of metrics on boundaries of convex cores of quasifuchsian hyperbolic manifolds and of an analogue of this conjecture, due to Mess, in the setting of globally hyperbolic anti-de Sitter spacetimes.
Reviewer: Athanase Papadopoulos (Strasbourg)Completely monotone sequences and harmonic mappingshttps://www.zbmath.org/1483.310052022-05-16T20:40:13.078697Z"Long, Bo-Yong"https://www.zbmath.org/authors/?q=ai:long.boyong"Sugawa, Toshiyuki"https://www.zbmath.org/authors/?q=ai:sugawa.toshiyuki"Wang, Qi-Han"https://www.zbmath.org/authors/?q=ai:wang.qihanSummary: In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.