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 prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if $K_X$ (resp., $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature, and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti, and B. Weinkove.

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the kth mean curvature, for k greater than 2, as we construct the counterexamples for all k greater than 2. Our proof relies on a new geometric argument which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete noncompact asymptotically flat hypersurfaces with nonnegative scalar curvature are weakly mean convex and prove the positive mass theorem for such hypersurfaces in all dimensions.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less than 2 plays an important role in many problems in low-dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli space of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff [7] when the singular set has no vertices, and by Weiß [32] when the cone angles are less than . We prove here an infinitesimal rigidity result valid for cone angles less than 2, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author [22]; see also the concurrent work by Weiß [33].

Mean curvature ows of hypersurfaces have been extensively studied and there are various dierent approaches and many beautiful results. However, relatively little is known about mean curvature ows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature ows in general dimensions and co-dimensions. The related techniques in the so called \blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature ows of two-dimensional surfaces in four-dimensional ambient spaces.