We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-type trace formulae.Comment: 40 page
For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern-Gauss-Bonnet formula. Relations between renormalized volume and the CR Q-curvature are also investigated.
Abstract. -To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M × (−1, 0).We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × (−1, 0) with respect to g and prove that the log term coefficient is independent of J (and any choice of contact form θ), i.e., is an invariant of the contact structure H.The approximately Einstein ACH metric g is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman's approximately Einstein complete Kähler metric g + on strictly pseudoconvex domains. The present work demonstrates that the CR-invariant log term coefficient in the asymptotic volume expansion of g + is in fact a contact invariant. We discuss some implications this may have for CR Q-curvature.The formal power series method of finding g is obstructed at finite order. We show that part of this obstruction is given as a one-form on H * . This is a new result peculiar to the partially integrable setting. (−1, 0).Nous considérons le développement asymptotique, en des puissances d'une fonction définissante spéciale, du volume de M × (−1, 0) par rapport à g. Nous démontrons que le coefficient du terme logarithmique est indépendant de J (et du choix de la forme de contact θ) ; par conséquent, c'est un invariant de la structure de contact H.La métrique presque d'Einstein ACH g est une généralisation de la métrique presque d'Einstein kählérienne complète g + de Fefferman sur les domaines strictement pseudoconvexes. Elle a également un comportement asymptotique similaire au bord. Le pré-sent travail démontre que le coefficient du terme logarithmique CR-invariant dans le développement asymptotique du volume de g + est, en fait, un invariant de contact. Nous traitons également quelques implications possibles pour la Q-courbure CR.La méthode de trouver g par le biais de séries formelles comporte une obstruction d'ordre fini. Nous démontrons que cette obstruction est partiellement donnée par une 1-forme sur H * . Ceci est un résultat nouveau particulier au contexte partiellement intégrable.
The motion of two spherical particles translating with equal velocities in an unbounded and vertically stratified viscous fluid is considered. The force experienced by each sphere is determined using the method of matched asymptotic expansions to small values of the stratification parameter α. The distance between the particle centres is assumed very much larger than the particle radius a and the particles are supposed to be sufficiently separated so that the second sphere is located in the outer region of expansion of the first sphere. The lift and the drag on the spheres are found up to first order of calculation and plotted against the distance between the spheres. The effect of inertia is studied through the variation in low values of the Reynolds number whose order has been taken as O(|α| 1/3 ). The velocity components are computed when the spheres are aligned along the axes. It has been found that the drag is increased due to stratification. The spheres are found to experience no lift when they are aligned in the horizontal plane. The presence of the second sphere is found to reduce the drag on the first sphere and the contributions of stratification and fluid inertia compensate each other exactly when the spheres are sufficiently close i.e. up to the order O(a|α| 1/3 ).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.