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1. 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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2. Boundary regularity and stability for spaces with Ricci bounded below

   https://doi.org/10.1007/s00222-021-01092-8  

   Bruè, Elia; Naber, Aaron; Semola, Daniele (May 2022, Inventiones mathematicae)

   Abstract This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X . 

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3. On Carrasco Piaggio's theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces

   Shanmugalingam, Nageswari (January 2023, Potentials and Partial Differential Equations - The Legacy of David R. Adams (series: Advances in Analysis and Geometry ))

   Xiao, Jie; Lenhart, Suzanne (Ed.)

   In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space. 

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4. Potential theory and quasisymmetric maps between compact Ahlfors regular metric measure spaces via Besov functions: preliminary

   Lehrback, Juha; Shanmugalingam, Nageswari (May 2024, Pute and Applied Functional Analysis)

   We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space.This approach is applicable even if the space does not support any Poincar´e inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric mea- sure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map. 

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5. On big pieces approximations of parabolic hypersurfaces

   https://doi.org/10.54330/afm.115417  

   Bortz, Simon; Hoffman, John; Hofmann, Steve; Luna-Garcia, Jose Luis; Nyström, Kaj (November 2021, Annales Fennici Mathematici)

   Let \(\Sigma\) be a closed subset of \(\mathbb{R}^{n+1}\) which is parabolic Ahlfors-David regular and assume that \(\Sigma\) satisfies a 2-sided corkscrew condition. Assume, in addition, that \(\Sigma\) is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that \(\Sigma\) satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that \(\Sigma\) contain suniform big pieces of Lip(1,1/2) graphs. When \(\Sigma\) is parabolic uniformly rectifiable the construction can be refined and in this case we prove that \(\Sigma\) contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if \(\Omega\subset\mathbb{R}^{n+1}\) is a connected component of \(\mathbb{R}^{n+1}\setminus\Sigma\) and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if \(\Omega\) is a one-sided parabolic chord arc domain, and if \(\Sigma\) is parabolic uniformly rectifiable, then \(\Omega\) is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition. 

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