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1. Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

   https://doi.org/10.3390/math10030522  

   Behzadan, Ali; Holst, Michael (February 2022, Mathematics)

   In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<1, and 1<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here. 

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2. On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics

   https://doi.org/10.37256/cm.3220221278  

   Behzadan, Ali; Holst, Michael (March 2022, Contemporary Mathematics)

   The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim. 

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3. Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

   https://doi.org/10.1007/JHEP05(2021)085  

   Anderson, Lara B.; Gray, James; Larfors, Magdalena; Magill, Matthew; Schneider, Robin (May 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results used differential geometric methods to explain the origin of some of this structure [1, 2]. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle. 

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4. Locally homogeneous non-gradient quasi-Einstein 3-manifolds

   https://doi.org/10.1515/advgeom-2021-0036  

   Lim, Alice (January 2022, Advances in Geometry)

   Abstract In this paper, we classify the compact locally homogeneous non-gradient m -quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m -quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m -quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S 1 is the only compact manifold of any dimension which admits a metric which is nontrivially m -quasi Einstein and Einstein. 

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5. 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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