We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and Bcycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section. We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa’s mirror conjecture with bundles in this special case.

This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Khler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2\pi . We also put all our technical results together to complete the proof of the main theorem.

We prove higher regularity properties of inverse mean curvature flow in Euclidean space: A sharp lower bound for the mean curvature is derived for star-shaped surfaces, independently of the initial mean curvature. It is also shown that solutions to the inverse mean curvature flow are smooth if the mean curvature is bounded from below. As a consequence we show that weak solutions of the inverse mean curvature flow are smooth for large times,beginning from the first time where a surface in the evolution is star-shaped.

Let (M, !) be a compact symplectic manifold, and L a compact embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono [8] construct Lagrangian Floer cohomology, yielding groups HF(L, b;nov) for one or HF􀀀 (L1, b1), (L2, b2);nov
for two Lagrangians, where b, b1, b2 are choices of bounding cochains, and exist if and only if L,L1,L2 have unobstructed Floer cohomology. These are independent of choices up to isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, !), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend [8] to immersed Lagrangians  : L ! M, with transverse self-intersections. In the embedded case, Floer cohomology HF(L, b;nov) is a modified, ‘quantized’ version of singular homology Hn−(L;nov) over the Novikov ring nov. In our immersed case, HF(L, b;nov) turns out to be a quantized version of Hn−(L;nov)  L (p−,p+)2R nov · (p−, p+), where R =  (p−, p+) : p−, p+ 2 L, p− 6= p+, (p−) = (p+) is a set of two extra generators for each self-intersection point of L, and (p−, p+) has degree (p−,p+) 2 Z, an index depending on how L intersects itself at (p−) = (p+). The theory becomes simpler and more powerful for graded Lagrangians in Calabi–Yau manifolds, when we can work over a smaller Novikov ring CY. The proofs involve associating a gapped filtered A1 algebra over 0 nov or 0 CY to  : L ! M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called AN,0 algebras for N = 0, 1, 2, . . ..

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.