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In this paper, we study the convergence of Calabi-Yau manifolds under K\"{a}hler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds., Comment: 54 pages

If a normalized K\"{a}hler-Ricci flow $g(t),t\in[0,\infty),$ on a compact K\"{a}hler $n$-manifold, $n\geq 3$, of positive first Chern class satisfies $g(t)\in 2\pi c_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will converge along a subsequence to a K\"{a}hler-Ricci soliton.

If a normalized K��hler-Ricci flow $g(t),t\in[0,\infty),$ on a compact K��hler $n$-manifold, $n\geq 3$, of positive first Chern class satisfies $g(t)\in 2��c_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will converge along a subsequence to a K��hler-Ricci soliton.

In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as perturbation of the Lagrangian skeleton of the exact symplectic manifold., 59 pages, 2 figures. Figures look better in ps file

This paper provides the technical details of gradient flow construction and related problems, which are essential for our construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces., 81 pages. Posting of an earlier paper

In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

H-minimal Lagrangian submanifolds in general K\"{a}hler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in K\"{a}hler manifolds to construct minimal Lagrangian torus in certain K\"{a}hler-Einstein manifolds with negative first Chern class., Comment: 13 pages. We find that the "harmonic Lagrangian" we introduced is equivalent to "H-minimal Lagangian" studied by Oh 10 years ago. Revisions are made accordingly. Some results are simplified and rearranged

In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit., A mistake in the proof of the previous lemma 3.4 is corrected. Some arguments and results are simplified and clarified

In this paper we prove that the K\"{a}hler-Einstein metrics for a toroidal canonical degeneration family of K\"{a}hler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete K\"{a}hler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also prove the incompleteness of the Weil-Peterson metric in this case., Comment: The assumption of simple in the toroidal degeneration is removed using base extension

In this paper we prove that the K\"{a}hler-Einstein metrics for a degeneration family of K\"{a}hler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete K\"{a}hler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil-Peterson metric in this case., Comment: minor correction in referencing

In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration., Comment: Revised version. Some clarification made and an example added

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric variety near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit $X_0$. In this paper, we will deal with the region near the smooth top dimensional torus fibres of $X_0$ and its mirror dual situation: the region near the 0-dimensional fibres of $X_0$., Comment: A picture is added and some arguments and results are simplified and clarified

In this paper we prove that the K��hler-Einstein metrics for a degeneration family of K��hler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete K��hler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil-Peterson metric in this case., minor correction in referencing

In this paper we prove that the K��hler-Einstein metrics for a toroidal canonical degeneration family of K��hler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete K��hler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also prove the incompleteness of the Weil-Peterson metric in this case., The assumption of simple in the toroidal degeneration is removed using base extension

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres)., Comment: 60 pages. More references will be added later

In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category., 54 pages, 12 figures. Updated version as published