## Summer Term 2016

Please note that this term the seminars will be held on Thursdays, from 2pm to 3pm.

*José Miguel Manzano, ***Compact stable surfaces with constant mean curvature in Killing submersions***. Thursday 5th May, 2-3pm, Huxley 139.
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Abstract: A Killing submersion is a Riemannian submersion from an orientable 3-manifold to an orientable surface, such that the fibres of the submersion are the integral curves of a Killing vector field without zeroes. The interest of this family of structures is the fact that it represents a common framework for a vast family of 3-manifolds, including the simply-connected homogeneous ones and the warped products with 1-dimensional fibres, among others.

In the first part of this talk we will discuss existence and uniqueness of Killing submersions in terms of some geometric functions defined on the base surface, namely the Killing length and the bundle curvature. We will show how these two functions and the metric in the base encode the geometry and topology of the total space of the submersion. In the second part, we will prove that if the base is compact and the submersion admits a global section, then it also admits a global minimal section. This gives a complete solution to the Bernstein problem (i.e., the classification of entire graphs with constant mean curvature) when the base surface is assumed compact. Finally we will talk about some results on compact orientable stable surfaces with constant mean curvature immersed in the total space of a Killing submersion. In particular, if they exist, then either (a) the base is compact and it is one of the above global minimal sections, or (b) the fibres are compact and the surface is a constant mean curvature torus.

This is based upon a joint work with Ana M. Lerma. It is available online at http://arxiv.org/abs/1604.00542.

*Pieter Blue*. **Hidden symmetries and decay of fields outside black holes.*** Thursday 12th May, 2-3pm, Huxley 139 *

Abstract: I will discuss energy and Morawetz (or integrated local decay) estimates for fields outside black holes, in particular the Vlasov equation. This builds on earlier work for the wave and Maxwell equation. Much of the work on these problems in the last decade has used the vector-field method and its generalisations. One generalisation has focused on using symmetries, differential operators that take solutions of a PDE to solutions. In this context, a hidden symmetry is a symmetry that does not decompose into first-order symmetries coming from a smooth family of isometries of the underlying manifold. In this talk, I will build on applications of the vector-field method to the Vlasov equation to prove an integrated energy decay for the Vlasov equation outside a very slowly rotating Kerr black hole, and I will discuss some new features of the symmetry algebra for the Vlasov equation, which illustrate the difficulties in passing to pointwise-decay estimates for the Vlasov equation in this context.

*Jacobus Portegies, ***Intrinsic Flat and Gromov-Hausdorff convergence.*** Thursday 15th June, 2-3pm, Clore Lecture Theatre (Huxley, ground floor) – please note the exceptional date and room.
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Abstract: We show that for a noncollapsing sequence of closed oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.

*Gerasim Kokarev*. **Eigenvalue problems on minimal submanifolds: old and new.*** Thursday 16th June, 2-3pm, Huxley 139 *

Abstract: I will give a short survey of inequalities for Laplace eigenvalues on Euclidean domains, and discuss their versions for minimal submanifolds. I will report on the work in progress and will describe a number of new results, generalizing previous work by Li and Yau, and other authors.

*Elena Mäder-Baumdicker*. **Willmore minimizing Klein bottles in Euclidean space.*** Thursday 23rd June, 2-3pm, Huxley 139*

Abstract: I will present results concerning immersed Klein bottles in euclidean n-space with low Willmore energy. Together with P. Breuning and J. Hirsch I proved that there is a smooth embedded Klein bottle that minimizes the Willmore energy among immersed Klein bottles when n\geq 4. I will shortly explain that the minimizer is probably already known: Lawson’s bipolar \tilde\tau_{3,1}-Klein bottle, a minimal Klein bottle in S^4. If n=4, there are three distinct homotopy classes of immersed Klein bottles that are regularly homotopic to an embedding. One contains the above mentioned minimizer. The other two are characterized by the property of having Euler normal number +4 or -4. I will explain that the minimum of the Willmore energy in these two classes is 8\pi. Furthermore, there are infinity many distinct embedded surfaces minimizing the Willmore energy in these classes. The proof is based on the twistor theory of the Euclidean four-space.

There will be no seminars on 19/05, 26/05, 02/06 and 09/06.