# Programme

All talks take place in building 4 West, room 1.7 (the Wolfson Lecture Theatre). Refreshments will be served in the 4 West Atrium, and lunch in the Wessex Restaurant (Wessex House, level 2). Click here for maps.

## Tuesday, 5 June

### 15:20 – 16:05   Frédéric Robert (Henri Poincaré University, Nancy)

##### The Lin-Ni conjecture: the influence of the curvature
###### Abstract.
Lin and Ni conjectured that the Neumann system $\begin{cases} -\Delta u + \epsilon u = u^{\frac{n + 2}{n - 2}} & \mbox{in } \Omega \subset \subset \mathbb{R}^n \\ u > 0 & \mbox{in } \Omega \\ \partial_\nu u = 0 & \mbox{on } \partial \Omega \end{cases}$ admits only the constant solution for small $\epsilon > 0$. This conjecture has stimulated and generated intensive contributions in the past decades. In particular, it has been known for long that the conjecture is not valid in small dimensions (except 3). Moreover, recent examples of Wei show that it is not valid when the mean curvature is negative and for infinite energies. We prove here that the conjecture is valid in the positive mean curvature case and with finite energy when $n = 3$ or $n \ge 7$. This is joint work with O. Druet and J. Wei.

### 16:50 –17:35   Tobias Lamm (Karlsruhe Institute of Technology)

##### Quantitative rigidity results for conformal immersions
###### Abstract.
By a classical result of Codazzi every closed, totally umbilic surface in $\mathbb{R}^n$ is a round sphere. De Lellis and Müller proved a rigidity statement corresponding to this result. More precisely, they showed that for every closed surface in $\mathbb{R}^3$, whose traceless second fundamental form is "small" in $L^2$, there exists a conformal parametrization whose distance to a standard parametrization of a round sphere is small in $W^{2,2}$.
In a recent joint work with H. Nguyen (Warwick) we were able to extend this result to arbitrary codimensions. Moreover, we obtained related rigidity results for inversions of the catenoid and Enneper's minimal surface.

## Wednesday, 6 June

### 9:30 –10:15   Tristan Rivière (ETH Zürich)

##### Energy identities for linear and non-linear critical PDE's
###### Abstract.
We will first present a quantization result – also called ''energy identity'' – for the angular part of the energy of solutions to second order elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. It is optimal in the sense that one can construct sequences of solutions to such systems whose radial part of the energy is not quantized. We will derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudo-holomorphic curves on degenerating Riemann surfaces. We will finally explain how these results extend to linear and non-linear higher order systems. All these results have been obtained in collaboration with Paul Laurain.

### 15:20 –16:05   Melanie Rupflin (Albert Einstein Institute, Potsdam)

##### Uniqueness for the polyharmonic map flow
###### Abstract.
We consider the question of uniqueness of weak solutions for the extrinsic polyharmonic map heat flows in their critical dimension. We first extend the results of Freire and of Lamm-Rivière on the uniqueness of weak solutions of the harmonic resp. biharmonic map flow with non-increasing energy to flows of arbitrary order and then prove that (under a weak additional assumption) any possible loss of uniqueness must be due to reverse bubbling.

### 16:50 – 17:35   Anna Dall'Acqua (University of Magdeburg)

##### Unstable Willmore surfaces of revolution
###### Abstract.
In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. We demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions. This is a joint work with K. Deckelnick and G. Wheeler.

## Thursday, 7 June

### 9:30 – 10:15   Klaus Deckelnick (University of Magdeburg)

##### Error analysis for the approximation of the Willmore flow of graphs by $C^1$-finite elements
###### Abstract.
The evolution of two-dimensional graphs under Willmore flow gives rise to a highly nonlinear initial-boundary value problem for the height function of the graph. We introduce a semidiscrete numerical scheme which uses $C^1$ finite elements for the approximation in space and derive optimal error estimates.

### 10:45 – 11:30   Robert Nürnberg (Imperial College London)

##### Parametric approximation of elastic flow for curves and curve networks
###### Abstract.
We introduce a parametric finite element approximation of the Willmore/elastic flow of a closed curve in $\mathbb{R}^d$, which has good mesh properties. We show that in the continuous-in-time semidiscrete case it satisfies a stability bound and an equidistribution mesh property. We extend this approach to a single open curve that satisfies various boundary conditions and to curve networks.

### 11:35 –12:20   Chun-Chi Lin (National Taiwan Normal University, Taipei)

##### The $L^2$-flow of elastic curves with or without boundary
###### Abstract.
The long-time existence of smooth solutions for the $L^2$-flow of elastic closed curves, as a fourth-order parabolic equation, has been studied in the past by many researchers (for examples, by Wen in 1995, by Polden in 1995 and by Dziuk, E. Kuwert, R. Schätzle in 2002). Here, the elastic energy of curves corresponds to the Euler-Bernoulli model of elastic rods in the literature.
In this talk, we will first discuss applications of the results in the literature to related cases, e.g., Kirchhoff elastic rods, elastic knots. Then, we demonstrate our investigation on the cases of non-closed curves with either clamped boundary conditions or certain dynamic boundary conditions. The $L^2$-flow of non-closed elastic curves is not only a natural generalization of the case of closed curves but also motivated from applications, e.g., in nonlinear spline interpolation or path-planning problem. We will show how to derive the long-time existence of smooth solutions for the fourth-order parabolic equation when the initial curves are smooth and the boundary conditions are properly posed.
This talk is mainly based on the joint works with H. Schwetlick.

### 16:50 – 17:35   Udo Hertrich-Jeromin (University of Bath)

##### Conformal coordinates
###### Abstract.
We shall discuss criteria for and consequences of the existence of conformal coordinates for a hypersurface in Euclidean space.

## Friday, 8 June

### 9:30 –10:15   Andrea Malchiodi (SISSA, Trieste)

##### Non-uniqueness results for critical metrics of regularized determinants
###### Abstract.
On compact four-dimensional manifolds, Branson and Oersted proved an extension of Polyakov's formula for the regularized determinants of conformally covariant operators (such as the conformal Laplacian and the Paneitz operator). We study critical metrics for some regularized determinants on $S^4$, and prove non uniqueness results. This is a joint work with M. Gursky.

### 10:45 –11:30   Bin Zhou (Peking University)

##### The Bernstein theorem for a class of fourth order equations
###### Abstract.
In this talk, I will talk about a class of fourth order equations of Monge-Ampere type. In particular, Abreu's equation arising from scalar curvature equation on toric Kahler manifolds is included. It is an extension of the affine maximal surface equation, but new difficulty comes from the lack of affine invariance. I will prove the 2-dimensional Bernstein theorem for these equations, i.e., the entire solution to these equations must be quadratic polynomials. This extends the affine Bernstein theorem, which was first proved by Trudinger-Wang. The main ingredients are interior estimates and the proof of strict convexity for the solutions.

### 11:35 –12:20   Andreas Gastel (University of Duisburg-Essen)

##### Remarks on higher order gauge invariant functionals
###### Abstract.
(Joint work with Christoph Scheven, Erlangen.) We study variational integrals of higher order which are invariant under gauge transformations. For these we discuss questions that are by now classical for the Yang-Mills functional, like the role of Uhlenbeck gauges, and existence and regularity of minimizers.

### 14:30 – 15:15   Paweł Strzelecki (University of Warsaw)

##### Global curvatures of compact sets and a characterization of manifolds of class $W^{2,p}$
###### Abstract.
We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $S$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $S$ satisfying a mild general condition relating the size of holes in $S$ to the flatness of $S$ measured in terms of beta numbers) is in fact an embedded manifold of class $W^{2,p}$, $p>m=\dim S$. The results are based on a careful analysis of Morrey estimates for integral curvature-like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $S$ or (b) the size of spheres tangent to $S$ at one point and passing through another point of $S$. Appropriately defined maximal functions of such integrands turn out to be in $L^p$ if and only if the local graph representations of $S$ have second order derivatives in $L^p$ and $S$ is an embedded submanifold. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $S$ is a round sphere.
This is joint work with S. Kolasinski and H. von der Mosel.