We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

In this paper, we first derive the CR analogue of matrix Li–Yau–Hamilton inequality for the positive solution to the CR heat equation in a closed pseudohermitian (2n + 1)-manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li–Yau gradient estimate in the Heisenberg group. We apply this CR gradient estimate and extend the CR matrix Li–Yau–Hamilton inequality to the case of the Heisenberg group. As a consequence, we derive the Hessian comparison property for the Heisenberg group.

Hence the energy defines a functional on the space of Lipshitz maps between M and M'. Critical points of this functional are called harmonic maps. These maps were studied by Bochner, Morrey, Rauch, Eells and Sampson, Hartman, Uhlenbeck, Hamilton, Hildebrandt and others. The first fundamental result was due to Eells and Sampson [3] who proved that, in case M'has non-positive sectional curvature, each map from M to M'is homotopic to a harmonic map.(This result was then extended by Hamilton [8] to the case where both M and M'are allowed to have boundary.) Later Hartman [7] was able to prove the harmonic map is unique in each homotopy class if M'has strictly negative curvature. This last result of Hartman leads one to believe that harmonic maps between compact manifolds with negative curvature must enjoy a lot of nice properties. In fact, a few years ago, B. Lawson and the second author conjectured

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$ and prove a systolic inequality for all $M \neq S^3$. The sweepout of surfaces is generated from a min-max minimal surface. If further assuming a positive scalar curvature lower bound, we can get a diameter upper bound for the min-max surface.

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.