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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. The Liouville theorem for $$p$$-harmonic functions and quasiminimizers with finite energy

   https://doi.org/10.1007/s00209-020-02536-2  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (February 2021, Mathematische Zeitschrift)

   null (Ed.)

   Abstract We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite p th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $$p$$ p -Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $$\mathbf {R}$$ R , we characterize all locally doubling measures, supporting a local $$p$$ p -Poincaré inequality, for which there exist nonconstant quasiminimizers of finite $$p$$ p -energy, and show that a quasiminimizer is of finite $$p$$ p -energy if and only if it is bounded. As $$p$$ p -harmonic functions are quasiminimizers they are covered by these results. 

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3. Classical multidimensional scaling on metric measure spaces

   https://doi.org/10.1093/imaiai/iaae007  

   Lim, Sunhyuk; Mémoli, Facundo (May 2024, Information and Inference: A Journal of the IMA)

   Abstract We study a generalization of the classical multidimensional scaling procedure (cMDS) which is applicable in the setting of metric measure spaces. Metric measure spaces can be seen as natural ‘continuous limits’ of finite data sets. Given a metric measure space $${\mathcal{X}} = (X,d_{X},\mu _{X})$$, the generalized cMDS procedure involves studying an operator which may have infinite rank, a possibility which leads to studying its traceability. We establish that several continuous exemplar metric measure spaces such as spheres and tori (both with their respective geodesic metrics) induce traceable cMDS operators, a fact which allows us to obtain the complete characterization of the metrics induced by their resulting cMDS embeddings. To complement this, we also exhibit a metric measure space whose associated cMDS operator is not traceable. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov–Wasserstein distance. 

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4. On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

   https://doi.org/10.1007/s00229-025-01656-5  

   Gibara, Ryan; Kangasniemi, Ilmari; Shanmugalingam, Nageswari (July 2025, Manuscripta mathematica)

   We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure μ with μ(X) = ∞ and 0 < μ(B(x, r)) < ∞ for all x ∈ X and r ∈ (0, ∞). Our objective is to understand the relationship between the Dirichlet space D^(1,p)(X), defined using upper gradients, and the Newton-Sobolev space N^(1,p)(X)+ℝ, for 1 ≤ p < ∞. We show that when X is of uniformly locally p-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space ℍⁿ with n ≥ 2, these two spaces coincide precisely when 1 ≤ p ≤ n-1. We also provide additional characterizations of when a function in D^(1,p)(X) is in N^(1,p)(X)+ℝ in the case that the two spaces do not coincide. 

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5. Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces

   https://doi.org/10.1090/tran/8495  

   Eriksson-Bique, Sylvester; Gill, James T.; Lahti, Panu; Shanmugalingam, Nageswari (November 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincare inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at co-dimension 1 Hausdorff measure almost every measure-theoretic boundary point of a set E of finite perimeter, there is an asymptotic limit set (E)∞ corresponding to the asymptotic expansion of E and that every such asymptotic limit (E)∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of (E)∞ is Ahlfors co-dimension 1 regular. 

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