In this paper, we give a proof to a statement in Perelman’s paper for finite extinction time of Ricci flow [5]. Our proof draws on different techniques from the one given in Morgan-Tian’s exposition [3] and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions.

We prove a version of homological mirror symmetry statement for toric Calabi-Yau 3-orbifolds, thus extending [30] to the case of orbifolds under the mirror symmetry setting considered in [13]. The B-model is the matrix factorization category for the toric Calabi-Yau 3-orbifold with a superpotential; while the A-model is a topologically defined Fukaya-type category on its mirror curve.

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of Kähler submanifolds of a fixed Kähler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth Kähler manifold satisfying the same curvature bounds, and admitting a holomorphic isometry in the same ambient space. We then apply this convergence result to prove that there are no holomorphic isometries of a non-compact complete Kähler manifold with asymptotically non-negative curvature into a finite dimensional complex projective space endowed with the Fubini-Study metric.