We study the moduli space M of torsion-free G2-structures on a fixed compact manifold M7, and define its associated universal intermediate Jacobian J . We define the Yukawa coupling and relate it to a natural pseudo-K¨ahler structure on J . We consider natural Chern–Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang–Mills connections), and also deformed Donaldson–Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel–Jacobi maps.

In this paper, we proved the mass angular momentum inequality\cite{D1}\cite{ChrusLiWe}\cite{SZ} for axisymmetric, asymptotically flat, vacuum constraint data sets with small trace. Given an initial data set with small trace, we construct a boost evolution spacetime of the Einstein vacuum equations as \cite{ChOM}. Then a perturbation method is used to solve the maximal surface equation in the spacetime under certain growing condition at infinity. When the initial data set is axisymmetric, we get an axisymmetric maximal graph with the same ADM mass and angular momentum as the given data. The inequality follows from the known results\cite{D1}\cite{ChrusLiWe}\cite{SZ} about the maximal graph.

We consider manifolds with conic singularities that are isometric to Rn outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author [23] to establish a “very weak” Huygens’ principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr¨odinger equation.

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Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.