Recently, following seminal works by A. Fraser and R. Schoen, there has been a renewed interest in minimal surfaces satisfying free boundary conditions. It is desirable to understand how the Morse index of such a surface, a number quantifying its degree of instability, is related to the topological complexity of the surface (genus, number of boundary components). In this talk, we will discuss some new results and open questions concerning free boundary minimal surfaces in the 3-ball with low index.
The question of the symmetry of the solutions of a PDE which is invariant under a transformation is a difficult question. If a PDE on the Euclidean space is invariant under rotations, when do we know that solutions, for instance the solutions which minimize the energy, inherit of radial symmetry ? This lecture is intended to review some problems and partial answers to these questions, with a special emphasis on problems at the interface of an elliptic point of view, interpreted as the description of stationary solutions, and a dynamical approach based on related equations of evolution.
In this talk we extend the theory of complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature to the wider class of elliptic special Weingarten surfaces of finite total curvature; in particular, we extend the seminal works of L. Jorge and W. Meeks and R. Schoen.Specifically, we extend the Jorge-Meeks formula relating the total curvature and the topology of the surface and we use it to classify planes as the only elliptic special Weingarten surfaces whose total curvature is less than $4 \pi$. Moreover, we show that a complete (connected), embedded outside a compact set, elliptic special Weingarten surface of minimal type in $\mathbb{R} ^3$ of finite total curvature and two ends is rotationally symmetric; in particular, it must be one of the rotational special catenoids described by R. Sa Earp and E. Toubiana. This answers in the positive a question posed in 1993 by R. Sa Earp. We also prove that the special catenoids are the only connected non-flat special Weingarten surfaces whose total curvature is less than $8 \pi$. This is a joint work with H. Mesa.
We give a rather general presentation about the Maxwell-Proca-Schrödinger and the Bopp-Podolsky-Schrödinger-Proca constructions/equations in the context of closed manifolds.
Kaluza-Klein theories are elegant scenarios for unifying gauge theories with gravitation through a generalization of General Relativity to a higher dimensional 'space-time'. An ad hoc supplementary assumption is however necessary for leading to a system of equations coupling the Einstein equations with the Yang-Mills one, namely to assume that the higher dimensional 'space-time' has the structure of a fiber bundle over some base manifold and that all fields can be identified with pull-back images of fields on the base manifold by the fibration.In this talk we present a new model based on a variational principle which leads to such a scenario without all these ad hoc assumptions. We will also discuss previously discovered models which are based on the same mechanism.
We show that the first eigenvalue of a closed Riemannian surface normalized by the area can be strictly increased by attaching a cylinder or a cross cap. As a consequence we obtain the existence of maximising metrics for the normalized first eigenvalue on any closed surface of fixed topological type.
The existence of Yamabe metrics, that is, metrics which minimize the Einstein-Hilbert functional in a conformal class, has been proven for compact smooth manifolds thanks to the celebrated work of Yamabe, Trudinger, Aubin and Schoen. When considering manifolds with singularities, the situation is quite different: while an existence result due to Akutagawa, Mazzeo and Carron is available, Viaclovsky had constructed in 2010 an example of 4-manifold, with one orbifold isolated singularity, for which a Yamabe metric does not exists. After briefly presenting the singularities we deal with, we will discuss a new non-existence result for a class of examples with non isolated singularities, not necessarily orbifold. This is based on a joint work with Kazuo Akutagawa.
The standard $L^2$ gradient flow for the harmonic map energy is the celebrated harmonic map flow, which is the original geometric flow introduced by Eells and Sampson in 1964. After giving a brief survey of the two-dimensional theory for this flow, we will take a look at the alternative gradient flow for this energy that was introduced jointly with Melanie Rupflin, which evolves both a map and the metric on its domain simultaneously. Now the gradient flow wants to converge to a minimal immersion rather than a harmonic map. This generalises the harmonic map flow from $S^2$ and a flow from $T^2$ introduced by Ding-Li-Liu. A theory for this flow has been developed over recent years in order to understand its singularity formation and its asymptotics. We will take a look at some of the main points of this theory, including how the flow decomposes a map into minimal immersions. The main focus of the talk will be the recent work joint with Kohout and Rupflin that addresses the question of whether the flow enjoys unique limits or not. We will see when it does and when it does not.