Joint with Analysis seminar.

I will discuss joint work with Jason Lotay from arXiv:2002.06444. We show how the three Bryant-Salamon G2 manifolds can be viewed as coassociative fibrations. In all cases the coassociative fibres are invariant under a 3-dimensional group and are thus of cohomogeneity one, In general there are both generic smooth fibres and degenerate singular fibres. The induced Riemannian geometry on the fibres turns out to exhibit asymptotically conical and conically singular behaviour. In some cases we also explicitly determine the induced hypersymplectic structure. In all three cases we show that the "flat limits" of these coassociative fibrations are well-known calibrated fibrations of Euclidean space. Finally, we establish connections with the multimoment maps of Madsen-Swann, the new compact construction of G2 manifolds of Joyce-Karigiannis, and recent work of Donaldson involving vanishing cycles and "thimbles".
 

Motivated by considerations from the mirror symmetry and
Landau-Ginzburg model, we consider Witten deformation on noncompact
manifolds.

Witten deformation is a deformation of the de Rham complex introduced by
Witten in an influential paper and has had many important applications,
mostly on compact manifolds. We will discuss some recent work with my
student Junrong Yan on the spectral theory of Witten Laplacian, the
cohomology of the deformation as well as its index theory.

A joint seminar with the Geometry & Topology Seminar series.

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C^0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.

The well-known Yau’s uniformization conjecture states that any
complete noncompact Kaehler manifold with positive bisectional curvature is
bi-holomorphic to the complex Euclidean space. The conjecture for the case
of maximal volume growth has been recently confirmed by G. Liu. In this
talk, we will consider the conjecture for manifolds with non-maximal volume
growth. We will show that the finiteness of the first Chern number is an
essential condition to solve Yau’s conjecture by using algebraic embedding
method. Furthermore, we can verify the finiteness in the case of minimal
volume growth. In particular, we obtain a partial answer to Yau’s
uniformization conjecture on complex two-dimensional Kaehler manifolds with
minimal volume growth. This is a joint work with Bing-Long Chen.