We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in our previous paper [HL4], we define equations via closed subsets of the 2-jet bundle where each equation has a natural dual equation. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Amp`ere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain  X, we give geometric conditions which imply that the Dirichlet problem on is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then introduce two fundamental concepts. The first is the notion of a monotonicity cone M for F. If X carries a global Msubharmonic function, then weak comparison implies full comparison. The second notion is that of boundary F-convexity, which is defined in terms of the asymptotics of F and is used to define barriers. In combining these notions the Dirichlet problem becomes uniquely solvable as claimed. This article also introduces the notion of local affine jet-equivalence for subequations. It is used in treating the cases above, but gives results for a much broader spectrum of equations on manifolds, including inhomogeneous equations and the Calabi-Yau equation on almost complex hermitian manifolds. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to S4 or RP4 or quotients of S3 ×R by a cocompact fixed point free subgroup of the isometry group of the standard metric of S3 × R, or a connected sum of them.

We study isometric Lie group actions on symmetric spaces admitting a section, i.e., a submanifold that meets all orbits orthogonally at every intersection point. We classify such actions on the compact symmetric spaces with simple isometry group and rank greater than one. In particular, we show that these actions are hyperpolar, i.e., the sections are flat.

We study the action of the group of contact diffeomorphisms on CR deformations of compact three dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighborhood of a fixed embeddable strongly pseudoconvex CR structure.