Leicester mini workshop, August 2018

Browse the talks and speakers who participated in our mini-workshop at the University of Leicester.

Huy The Nguyen

Cylindrical Estimates for High Codimension Mean Curvature Flow

We study high codimension mean curvature flow of a submanifold of dimension $n$ in Euclidean space $\R^{n+k}, k \geq 2$ subject to a certain quadratic curvature condition. This condition extends the notion of two-convexity for hypersurfaces to high codimension submanifolds. We analyse singularity formation in the mean curvature flow of high codimension by directly proving a pointwise gradient estimate. We then show that near a singularity the surface is quantitatively cylindrical.

Katsuhiro Moriya

The spinor representation of a surface in an Euclidean space of arbitrary dimension

We write a surface in an Euclidean space of arbitrary dimension by a section of a spinor bundle. We calculate curvatures of this surface in terms of the section and define a transformation on surfaces.

Eduardo Mota Sanchez

Heun equations and CMC surfaces

The class of 2nd order ODEs called 'Heun equations' can be used to construct new families of CMC surfaces using the DPW method. Each ODE in this class has different number and type of singularities which defines what kind of CMC surface we get. Regular singularities in the Heun equation correspond to Delaunay ends in the surface, while irregular singularities give us an irregular behaviour in that end.

Ben Sharp

Bubbling analysis for free-boundary minimal hypersurfaces 

We will discuss some recent results in the analysis of degenerating sequences of free-boundary minimal hypersurfaces (FBMH), with a view to gaining qualitative (and quantitative) relationships between their Morse indices, geometry and topology. A FBMH is a manifold with boundary which is a critical point of the area functional under the sole constraint that its boundary must lie along the boundary of the ambient space. Thus the mean curvature vanishes on the interior and they meet the boundary orthogonally. The Morse index is (roughly speaking) the number of local directions one can push the hypersurface to decrease area. I will present joint works with L. Ambrozio, A. Carlotto and R. Buzano.