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1. Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

   https://doi.org/10.1007/s12220-020-00477-0  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (May 2021, The Journal of Geometric Analysis)

   null (Ed.)

   Abstract The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces” , Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p -Poincaré inequality, then the transformed measure is globally doubling and supports a global p -Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p -Poincaré inequality, carries nonconstant globally defined p -harmonic functions with finite p -energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain. 

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2. Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

   https://doi.org/10.4153/S0008414X19000567  

   Izzo, Alexander J.; Papathanasiou, Dimitris (February 2021, Canadian Journal of Mathematics)

   null (Ed.)

   Abstract We strengthen, in various directions, the theorem of Garnett that every $$\unicode[STIX]{x1D70E}$$ -compact, completely regular space $$X$$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $$X$$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $$X$$ of a Euclidean space, there is a compact set $$K$$ in some $$\mathbb{C}^{N}$$ so that $$\widehat{K}\backslash K$$ contains a Gleason part homeomorphic to  $$X$$ , and $$\widehat{K}$$ contains no analytic discs. 

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3. Polyhedral approximation and uniformization for non-length surfaces

   https://doi.org/10.4171/JEMS/1538  

   Ntalampekos, Dimitrios; Romney, Matthew (September 2024, Journal of the European Mathematical Society)

   We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain. 

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4. On Polish groups admitting non-essentially countable actions

   https://doi.org/10.1017/etds.2020.133  

   KECHRIS, ALEXANDER S.; MALICKI, MACIEJ; PANAGIOTOPOULOS, ARISTOTELIS; ZIELINSKI, JOSEPH (January 2022, Ergodic Theory and Dynamical Systems)

   Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable. 

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5. The set of bounded continuous nowhere locally uniformly continuous functions is not Borel

   Alexander J. Izzo (January 2022, Houston journal of mathematics)

   It is known that for $$X$$ a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ contains a dense $$G_\delta$$ set in the space $$C_b(X)$$ of all bounded continuous real-valued functions on $$X$$ in the supremum norm. Furthermore, when $$X$$ is separable, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ is itself a $$G_\delta$$ set. We show that in contrast, when $$X$$ is nonseparable, this set of functions is not even a Borel set. 

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