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1. CR embeddability of quotients of the Rossi sphere via spectral theory

   https://doi.org/10.1142/S0129167X22500148  

   Bosch, Henry; Gonzales, Tyler; Spinelli, Kamryn; Udell, Gabe; Zeytuncu, Yunus E. (February 2022, International Journal of Mathematics)

   We look at the action of finite subgroups of [Formula: see text] on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of [Formula: see text] to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum. 

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2. The CR-Volume of Horizontal Submanifolds of Spheres

   https://doi.org/10.1093/imrn/rnaf044  

   Bernstein, Jacob; Bhattacharya, Arunima (March 2025, International Mathematics Research Notices)

   Abstract We study an analog in CR-geometry of the conformal volume of Li–Yau. In particular, to submanifolds of odd-dimensional spheres that are Legendrian or, more generally, horizontal with respect to the sphere’s standard CR-structure, we associate a quantity that is invariant under the CR-automorphisms of the sphere. We apply this concept to a corresponding notion of Willmore energy. 

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3. G2 manifolds with nodal singularities along circles

   https://doi.org/10.1007/s12220-019-002833-3  

   Chen, Gao (September 2019, The Journal of geometric analysis)

   This paper finds matching building blocks for the construction of a compact manifold with G2 holonomy and nodal singularities along circles using twisted connected sum method by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with G2 holonomy and isolated conical singularities with link S3×S3. 

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4. Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

   https://doi.org/10.1112/jlms.12922  

   Godoy, Yamile; Harrison, Michael; Salvai, Marcos (June 2024, Journal of the London Mathematical Society)

   Abstract Let be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points. 

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5. Equivariant hyperbolization of 3-manifolds via homology cobordisms

   Dave Auckly, Hee Jung (January 2023, Topology and its applications)

   The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Zp-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order. 

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