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For a hyperbolic link K in the thickened torus, we show there is a decomposition of the complement of a link L , obtained from augmenting K , into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [D. Futer and F. Guéritaud, From angled triangulations to hyperbolic structures, in Interactions between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporary Mathematics, Vol. 541 (American Mathematical Society, Providence, RI, 2011), pp. 159–182.], this shows that L is hyperbolic. [ABSTRACT FROM AUTHOR]

Let (T k , h k) = (S r 1 1 × S r 2 1 × ⋯ × S r k 1 , d t 1 2 + d t 2 2 + ⋯ + d t k 2) be flat tori, r k ≥ ⋯ ≥ r 2 ≥ r 1 > 0 and (R n , g E) the Euclidean space with the flat metric. We compute the isoperimetric profile of (T 2 × R n , h 2 + g E) , 2 ≤ n ≤ 5 , for small and big values of the volume. These computations give explicit lower bounds for the isoperimetric profile of T 2 × R n . We also note that similar estimates for (T k × R n , h k + g E) , 2 ≤ k ≤ 5 , 2 ≤ n ≤ 7 - k , may be computed, provided estimates for (T k - 1 × R n , h k - 1 + g E) exist. We compute this explicitly for k = 3 . We use symmetrization techniques for product manifolds, based on work of Ros (Global theory of minimal surfaces (Proc. Clay Mathematics Institute Summer School, 2001). American Mathematical Society, Providence, 2005) and Morgan (Ann Glob Anal Geom 30:73–79, 2006). [ABSTRACT FROM AUTHOR]

This paper studies convergence to equilibrium for the spatially inhomogeneous linear relaxation Boltzmann equation in Boltzmann entropy and related entropy functionals, the p- entropies. Villani [Hypocoercivity, in Memoirs of the American Mathematical Society, Vol. 202, American Mathematical Society, Providence, RI, 2006] proved entropic hypocoercivity for a class of PDEs in a Hformander sum-of-squares form. It was an open question to prove such a result for an operator which does not share this form. We prove a closed entropy-entropy production inequality fa la Villani which implies exponentially fast convergence to equilibrium for the linear Boltzmann equation with a quantitative rate. The key new idea appearing in our proof is the use of a total derivative of the entropy of a projection of our solution to compensate for an error term which appears when using nonlinear entropies. We also extend the proofs for hypocoercivity for the linear relaxation Boltzmann to the case of entropy functionals. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of the SYZ transform on a semiflat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and the extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multisections defined by Arinkin and Joyce [Fukaya category and Fourier transform, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2001] and Leung, Yau, and Zaslow [Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319-1341]. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold (M, ω), under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho and Joyce [J. Differential Geom. 86 (2010), no. 3, 831-500, Question 13.15]. Furthermore, if M admits a Lagrangian torus fibration over an integral affine manifold, we prove, under some additional assumptions, that this new equivalence is mirror to an isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform. [ABSTRACT FROM AUTHOR]