## Tuesday, 5 June

#### 14:00 Registration

### 14:30 – 15:15 **Michael Struwe** (ETH Zürich)

##### Partial regularity of stationary biharmonic maps, revisited

### 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.
#### 16:10 Coffee and tea

### 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.
#### 10:20 Coffee and tea

### 10:45 – 11:30 **David Calderbank** (University of Bath)

##### Extremal Kahler metrics: analysis, geometry, examples

### 11:35 – 12:20 **Jan Metzger** (University of Potsdam)

##### Willmore surfaces subject to an area bound

#### 12:40 Lunch

### 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.
#### 16:10 Coffee and tea

### 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.
#### 10:20 Coffee and tea

### 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.
#### 12:40 Lunch

### 15:20 – 16:05 **Friedemann Schuricht** (TU Dresden)

##### Contact problems for nonlinearly elastic rods

#### 16:10 Coffee and tea

### 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.
#### 10:20 Coffee and tea

### 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.
#### 12:40 Lunch

### 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.
### 15:20 –16:05 **Ernst Kuwert** (University of Freiburg)

##### Minimizing curvature functionals among spheres in 3-manifolds

#### 16:10 Coffee and tea