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Let T  (Y, ) be a transverse knot which is the binding of some open book, (T, ), for the ambient contact manifold (Y, ). In this paper, we show that the transverse invariant bT(T ) 2[HFK(−Y,K), defined in [LOSSz09], is nonvanishing for such transverse knots. This is true regardless of whether or not  is tight. We also prove a vanishing theorem for the invariants L and T. As a corollary, we show that if (T, ) is an open book with connected binding, then the complement of T has no Giroux torsion.

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime (Wang and Yau in Phys Rev Lett 102 (2): 021101, 2009; Commun Math Phys 288 (3): 919942, 2009), the anti-de Sitter spacetime (Chen et al. in Commun Anal Geom, 2016. arXiv: 1603.02975), or the Schwarzschild spacetime (Chen et al. in Adv Theor Math Phys 22 (1): 123, 2018). In each case, the reference spacetime admits a conformal KillingYano 2-form which facilitates the application of the Minkowski formula in Wang et al.(J Differ Geom 105 (2): 249290, 2017) to estimate the quasi-local energy. As a consequence of the positive mass theorems in Liu and Yau (J Am Math Soc 19 (1): 181204, 2006) and Shi and Tam (Class Quantum Gravity 24 (9): 23572366, 2007) and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).

We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highest-degree asymptotics in formulas of Vafa-Intriligator type. In particular, we explicitly evalu- ate all intersection numbers appearing in the Verlinde formula.Our results are in agreement with previous computations of Wit- ten, Jeffrey-Kirwan and Liu. Moreover, we prove the vanishing of certain intersections on a suitable Quot scheme, which can be interpreted as giving equations between counts of maps to the Grassmannian.