In this paper, we derive two subgradient estimates of the CR heat equation in a closed pseudohermitian 3-manifold which are served as the CR version of the Li-Yau gradient estimate. With its applications, we first get a subgradient estimate of the logarithm of the positive solution of the CR heat equation. Secondly, we have the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formulae for the CR heat equation.

Let $M$ be a closed surface diffeomorphic to $S^2$ endowed with a Riemannian metric. Denote the diameter of $M$ by $d$. We prove that for every $x \in M$ and every positive integer $k$ there exist $k$ distinct geodesic loops based at $x$ of length $\le 20kd$.

In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat Kähler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interest.

The $n$-dimensional pair of pants is defined to be the complement of $n + 2$ generic hyperplanes in $\mathbb{CP}n$. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism $A_\infty$ algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with $n+2$ generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model $(\mathbb{C}^{n+2},W)$, where $W = z_1...z_{n+2}$. We show that the endomorphism $A_\infty$ algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.