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1. New classes of quasigeodesic Anosov flows in $$3$$-manifolds

   https://doi.org/10.1017/etds.2024.89  

   CHANDA, ANINDYA; FENLEY, SÉRGIO R (July 2025, Ergodic Theory and Dynamical Systems)

   Abstract Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$$\mathbb {R}$$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic. 

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2. Cusp volumes of alternating knots on surfaces

   https://doi.org/10.2140/agt.2022.22.2467  

   Bavier, B. (November 2022, Algebraic geometric topology)

   We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3–manifolds. We show that, under mild hypothesis, their cusp area admits two-sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the 3–sphere. Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus can have arbitrarily small cusp density, in contrast to alternating knots on spheres whose cusp densities are bounded away from zero due to Lackenby and Purcell. 

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3. Bridge trisections of knotted surfaces in 4-manifolds

   https://doi.org/10.1073/pnas.1717171115  

   Meier, Jeffrey; Zupan, Alexander (October 2018, Proceedings of the National Academy of Sciences)

   We prove that every smoothly embedded surface in a 4-manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4-manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a generalized bridge trisection, extends the authors’ definition of bridge trisections for surfaces in S 4 . Using this construction, we give diagrammatic representations called shadow diagrams for knotted surfaces in 4-manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside ℂ ℙ 2 . Using these examples, we prove that there exist exotic 4-manifolds with ( g , 0 ) —trisections for certain values of g. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams. 

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4. Shapes of hyperbolic triangles and once-punctured torus groups

   https://doi.org/10.1007/s00209-021-02745-3  

   Kim, Sang-hyun; Koberda, Thomas; Lee, Jaejeong; Ohshika, Ken’ichi; Tan, Ser Peow; Gao, Xinghua (January 2021, Mathematische Zeitschrift)

   null (Ed.)

   Abstract Let $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds. 

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5. Minkowski Inequality in Cartan–Hadamard Manifolds

   https://doi.org/10.1093/imrn/rnad114  

   Ghomi, Mohammad; Spruck, Joel (June 2023, International Mathematics Research Notices)

   Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard $$3$$-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic $$3$$-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved $$3$$-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan–Hadamard manifolds of any dimension. 

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