In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rigid Calabi-Yau manifolds poses a serious challenge to the conjecture. The proposed mirror partners for them are higher dimensional generalized Calabi-Yau manifolds. For example, the mirror partner for a certain K3 surface is a cubic fourfold and its Fano variety of lines is birational to the Hilbert scheme of two points on the K3. This hyperk¨ahler manifold can be interpreted as the SYZ mirror of the K3 by considering singular special Lagrangian tori. We also compare the geometries between a CY and its associated generalized CY. In particular we present a new construction of Lagrangian submanifolds.

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.

Let M⊂CN be a generic real-analytic submanifold of finite type, M′⊂CN′ be a real-analytic set, and p∈M, where we assume that N,N′⩾2. Let H:(CN,p)→CN′ be a formal holomorphic mapping sending M into M′, and let EM′ denote the set of points in M′ through which there passes a complex-analytic subvariety of positive dimension contained in M′. We show that, if H does not send M into EM′, then H must be convergent. As a consequence, we derive the convergence of all formal holomorphic mappings when M′ does not contain any complex-analytic subvariety of positive dimension, answering by this a long-standing open question in the field. More generally, we establish necessary conditions for the existence of divergent formal maps, even when the target real-analytic set is foliated by complex-analytic subvarieties, allowing us to settle additional convergence problems such as e.g. for transversal formal maps between Levi-non-degenerate hypersurfaces and for formal maps with range in the tube over the light cone.

Given a compact orientable surface, we determine a complete set of relations for a function defined on the set of all homotopy classes of simple loops to be a geometric intersection number function. As a consequence, Thurston’s space of measured laminations and Thurston’s compactification of the Teichm¨uller space are described by a set of explicit equations. These equations are polynomials in the max-plus semi-ring structure on the real numbers. It shows that Thurston’s theory of measured laminations is within the domain of tropical geometry.

By the integral method, we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\mathbb{R}^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in [9] and [1] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [1].