This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains. We prove a “low energy” result in all dimensions, in the sense that if normalized energy in a large ball is small enough, then the normalized energy in any interior ball must also be small.

Let $f : \sum_1 \to \sum_2$ be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of $f$ can be viewed as a Lagrangian submanifold in $ \sum_1 \times \sum_2$. This article discusses a canonical way to deform $f$ along area preserving diffeomorphisms. This deformation process is realized through the mean curvature flow of the graph of $f$ in $ \sum_1 \times \sum_2$. It is proved that the flow exists for all time and the map converges to a canonical map. In particular, this gives a new proof of the classical topological results that $O(3)$ is a deformation retract of the diffeomorphism group of $S^2$ and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group .

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

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.

We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.