We consider compact submanifolds of dimension n  2 in Rn+k, with nonzero mean curvature vector everywhere, and such that the full norm of the second fundamental form is bounded by a fixed multiple (depending on n) of the length of the mean curvature vector at every point. We prove that the mean curvature flow deforms such a submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in some (n + 1)- dimensional affine subspace of Rn+k.

This note is based on a series of four lectures the author gave at University of Montpellier, September 28-30, 2015. He started with the notion of mass in general relativity, gave a brief review of some known constructions of quasi-local mass, and then discussed the new quasi-local energy and quasi-local mass which Shing-Tung Yau and the author introduced in 2009. At the end, the proof of the positivity of quasi-local energy was sketched and a stability theorem of critical points of quasi-local energy by Po-Ning Chen, Shing-Tung Yau, and the author was discussed

Wedge product on deRham complex of a Riemannian manifold M can be pulled back to H^∗(M) via explicit homotopy, constructed using Green’s operator, to give higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of M. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya.

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX ) with curvature bounded above by a constant κ (κ 􏰣 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants of semistable sheaves on them. Our formula generalizes KawaiYoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of KatzKlemm-Vafa conjecture for stable pairs.