Biharmonic maps are maps between Riemannian manifolds which are critical points of the bi-energy. They are solutions of a system of 4thorder PDEs and they include harmonic maps and biharmonic functions as special cases. Biharmonic submanifolds (which include minimal submanifolds as special cases) are the images of biharmonic isometric immersions. The talk will review some problems, including classification of biharmonic submanifolds in space forms, biharmonic maps into spheres, biharmonic conformal maps, and unique continuation theorems, studied in this field and their progress since 2000. The talk also presents some recent work on equivariant biharmonic maps and the stability and index of biharmonic hypersurfaces in space forms.

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.