We give a parameterization of the moduli space of marked Gromov hyperbolic convex domains of $\mathbb{R}\mathrm{P}^2$ by the space of bounded holomorphic cubic differentials on the disk.

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential.We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta, and Ono.

Let $\mathcal{p : X \to S}$ be a proper Kähler fibration and $\mathcal{E \to X}$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson (Curvature of vector bundles associated to holomorphic fibrations), by using basic Hodge theory, we derive several general curvature formulas for the direct image $\mathcal{p_* (K_{X/S} \otimes E)}$ for general Hermitian holomorphic vector bundle $\mathcal{E}$ in a simple way. A straightforward application is that, if the family $\mathcal{X \to S}$ is infinitesimally trivial and Hermitian vector bundle $\mathcal{E}$ is Nakano-negative along the base $\mathcal{S}$, then the direct image $\mathcal{p_* (K_{X/S} \otimes E)}$ is Nakano-negative. We also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first Chern classes and obtain that, if the Chern curvature of direct image $p_*(K_X \otimes E)$ —of a positive projectively flat family $(E, h(t))_{t \in \mathbb{D}} \to X$ —vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair $(X,E)$.

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface $(M, g)$, given a smooth function $f$ on $M$, there is a distribution in the Sobolev space $H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to $M$ is the given function $f$.