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1. Kähler–Einstein metrics near an isolated log-canonical singularity

   https://doi.org/10.1515/crelle-2022-0095  

   Datar, Ved; Fu, Xin; Song, Jian (March 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity.Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry. 

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2. Fractional Yamabe Problem on Locally Flat Conformal Infinities of Poincaré-Einstein Manifolds

   https://doi.org/10.1093/imrn/rnad195  

   Mayer, Martin; Ndiaye, Cheikh_Birahim (August 2023, International Mathematics Research Notices)

   Abstract We study the fractional Yamabe problem first considered by Gonzalez-Qing [36] on the conformal infinity $$(M^{n}, \;[h])$$ of a Poincaré-Einstein manifold $$(X^{n+1}, \;g^{+})$$ with either $n=2$ or $$n\geq 3$$ and $$(M^{n}, \;[h])$$ locally flat, namely $(M, h),$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits a local situation and also a global one. The latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either $n=2$ or of dimension $$n\geq 3$$ and which are locally flat, and hence the minimizing technique of Aubin [4] and Schoen [48] in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau [49], which is not known to hold. Using the algebraic topological argument of Bahri-Coron [8], we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n=2$ or of dimension $$n\geq 3$$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature. 

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3. Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

   https://doi.org/10.1017/etds.2021.42  

   AVILA, A.; VIANA, MARCELO; WILKINSON, A. (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact. 

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4. Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

   https://doi.org/10.1515/crelle-2021-0028  

   Cao, Junyan; Guenancia, Henri; Păun, Mihai (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric inducesa semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p . We also propose a conjectural generalization of this result for relativetwisted Kähler–Einstein metrics. Then we show that our conjecture holds trueif the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension). 

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5. Nearly-Doubling Spaces of Persistence Diagrams

   Sheehy, Donald R. (January 2022, 38th International Symposium on Computational Geometry (SoCG 2022))

   Goaoc, Xavier and (Ed.)

   The space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension. Because many metric search algorithms and data structures have bounds that depend on the dimension of the search space, the high-dimensionality makes it difficult to analyze and compare asymptotic running times of metric search algorithms on this space. We introduce the notion of nearly-doubling metrics, those that are Gromov-Hausdorff close to metric spaces of bounded doubling dimension and prove that bounded $$k$$-point persistence diagrams are nearly-doubling. This allows us to prove that in some ways, persistence diagrams can be expected to behave like a doubling metric space. We prove our results in great generality, studying a large class of quotient metrics (of which the persistence plane is just one example). We also prove bounds on the dimension of the $$k$$-point bottleneck space over such metrics. The notion of being nearly-doubling in this Gromov-Hausdorff sense is likely of more general interest. Some algorithms that have a dependence on the dimension can be analyzed in terms of the dimension of the nearby metric rather than that of the metric itself. We give a specific example of this phenomenon by analyzing an algorithm to compute metric nets, a useful operation on persistence diagrams. 

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