In the ¯rst Heisenberg group H1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H1 the only stable C2 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature dimension inequality C D E(n, K), which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are the first kind of results in this direction for unbounded Laplacian on graphs.

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We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Khler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and SiuYau on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the Khler cone. First, we deform the metric along the SasakiRicci flow and obtain a limit SasakiRicci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the SasakiRicci soliton to a SasakiEinstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats

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, . . ..