For a toric Calabi-Yau (CY) orbifold X whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of X to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in X bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of X. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of X – this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions – an open analogue of Ruan’s crepant resolution conjecture.

The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k . We also discuss some applications of our estimates.

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.

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Eigenvalues and capacitors. Let X be a Riemannian manifold and be the Laplace operator on X. It is well-known that if X is compact then the spectrum of is discrete and consists of an increasing sequence {k} k= 1 of the eigenvalues (counted with the multiplicities) where 1= 0 and k as k. Moreover, if n= dim X then Weyls asymptotic formula says that (1.1) k cn