Summer School Curvature & Applications, Chiemsee, 31.8. - 5.9.2025
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| Venue
| Program
Summer School 2025 – Schedule
| Breakfast |
08:00 – 09:00 |
| Lunch |
12:00 – 13:30 (also on Friday) |
| Dinner |
18:00 – 19:30 |
| Discussion Time |
13:30 – 15:00 and evenings |
| Excursion Wednesday (after lunch) |
Lecture Titles and Abstracts
S. Bartels
Approximation of elastic rods
We address the numerical approximation of inextensible elastic rods that are described by framed curves. It turns out that a noncanonical treatment of the arclength constraint is required to obtain optimal convergence rates. Further aspects of the mini course are energy decreasing time-stepping schemes, stable discretizations of twist contributions and the practical realization of self-avoidance.
E. Mäder-Baumdicker
Parabolic monotonicity formulas in geometric flows and their applications
A. Malchiodi
Perturbative methods in geometric analysis
We will describe a versatile finite-dimensional reduction method for a
class of geometric problems that exploits variational structure and
allows to find existence and multiplicity of solutions.
We will focus on applications to the study of isoperimetric sets with
small volume, construction of constant mean curvature surfaces,
and well as Sachk-Uhlenbeck's approximation the harmonic energy.
F. Rupp
Varifolds and Curvature Energies
Varifolds are a central concept in Geometric Measure Theory that provide a flexible framework for describing surfaces and their curvature in a weak sense. We will give a gentle introduction to varifolds with an emphasis on applications in variational models for elastic membranes and discuss recent progress in regularity theory.
B. Schmidt
Curved elastic and inelastic plates
The topic of the lectures is the effective description of thin elastic structures that are subject to bending. We begin by reviewing some classical results on the derivation of suitable plate theories from three-dimensional non-linear elasticity. We then report on ongoing advances in deriving effective theories for thin sheets consisting of multiple layers with mismatching equilibria. Finally, we describe some recent advances which allow to model brittle materials that may develop folds and cracks.
Poster session
Prof. Ismail Merabet (University of Kasdi Merbah-Ouargla, Algeria)
Numerical approximation of nonlinearly elastic rod models
In this work, we intend to propose two mathematical models of a class of bodies that are thin and slender at the same time, a feature that allows us to have recourse to a one-dimensional (1D) theory. The moderately large deflection of thin elastic plates or shells, i.e., bodies which are intrinsically thin, can be well described by the Föppl–von Kármán (FvK) model [1, 2, 3, 4]. The 1D models are obtained by dimensional reduction from thin nonlinearly elastic shallow shell models. The present work aims to derive enhanced models for the classical ones, such as Timoshenko’s or Euler’s models, and to present numerical simulations.
Alexander West (Universität Bonn)
Energy quantization for constrained Willmore surfaces
I present recent joint work with C. Scharrer. We establish an energy quantization for constrained Willmore surfaces, where the constraints are given by area, volume, and total mean curvature, assuming that the underlying conformal structure remains bounded. We introduce a balancing term which quantifies the potential loss of energy in the neck regions. Furthermore, we show strong compactness of constrained Willmore surfaces under some energy threshold. Finally, a consequence of our results is the energy quantization of axisymmetric spherical surfaces without additional assumptions.
Dr. Vanessa Hüsken (University Duisburg-Essen)
Geometric non-linearities in PDEs for Cosserat elasticity – how they affect the regularity of solutions
Since the beginning of the 1900s, linearized Cosserat elasticity is well known and often used in the engineering community for modelling micropolar elastic solids. But from a mathematician’s perspective a geometrically non-linear version of the model is interesting. Existence of solutions for the latter has been known for about 20 years, while regularity questions were investigated only during the last couple of years. We introduce the Cosserat bulk model and give an overview of recently developed different regularity results, for Cosserat energy minimizers as well as for critical points. Classical regularity theory for (p-) harmonic maps into manifolds is an essential tool in deriving those results. At the same time, the geometric nature of the model’s non-linearities in some cases allows not only regular, but also quite singular solutions to exist. Finally, those singular solutions do not exist, when we go away from the (3d-) Cosserat bulk model towards a Cosserat model for shells, i.e. very thin materials, whose elastic behaviour is governed by their (2d-) midsurface.
Roee Leder (The Hebrew Univ. of Jerusalem)
Cohomology for Linearized Boundary-Value Problems on General Riemannian Structures
I shall present a framework for casting the solvability and uniqueness conditions of linearized geometric boundary-value problems in cohomological terms. The theory is designed to be applicable without assumptions on the underlying Riemannian structure and provides tools to study the emergent cohomology explicitly. To achieve this generality, Hodge theory is extended to sequences of Douglas--Nirenberg systems that interact via Green's formulae, overdetermined ellipticity, and a condition I call the order-reduction property, replacing the classical requirement that the sequence form a cochain complex. This property typically arises from linearized constraints and gauge equivariance, as demonstrated by several examples, including the linearized Einstein equations with sources, where the cohomology encodes geometric and topological data.
Nicolas Alexander Hasse (Universität Mannheim)
LIMITING CASES OF THE sinh-GORDON EQUATION AND THE SOUL CONJECTURE
The sinh-Gordon equation is a non-linear elliptic partial differential equation
∆u + sinh(u) = 0
whose solutions form an integrable system corresponding to tori of constant mean curvature in R3. In this talk we will examine limits of spectral curves corresponding to doubly-periodic solutions of the sinh-Gordon equation of genus two. We are interested in limits where one of the main curvatures of the surface explodes. Hence we examine spectral curves whose defining polynomials have a pair of roots that go to infinity and zero. The spectral data consists of a polynomial a of degree four and two meromorphic differential forms defined by polynomials bi of degree three which are uniquely defined by their periods. We will use the information from the periods and the principal values at zero to define a limit of these differential forms and examine it. These limits correspond to non-conformal harmonic maps which also form an integrable system. Such limits are of interested in the study of the so called soul conjecture, which states that for every closed curve in R3 there exists a sequence of cmc tori that slings around this curve in the limit.
Dr. Wei Wei, scholarship holder of the AvH foundation (Nanjing University)
k-Yamabe problem and its related Sobolev inequalities
Inspired by the Sobolev inequalities related with $sigma_2$-curvature and scalar curvature, we provide the Sobolev inequalities connecting with $Q$ curvature and the scalar curvature $R$. And we introduce the following Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$:
Find a conformal metric $g$ in a given conformal class $[g_0]$ with
\[
Q_g/ R_g=const.
\]
We prove that on a closed $n$-dimensional Riemannian manifold $(M,g_{0})$ with semi-positive $Q$-curvature and non-negative scalar curvature, the above Yamabe problem is solvable. This is a joint work with Prof. Y. X. Ge and G. F. Wang.
Mingwei Zhang, CSC scholarship (Univ. of Freiburg)
Stability of spinorial Sobolev inequalities on sphere
We consider the sharp spinorial Sobolev inequality on S^n. From the variation point of view, this is a spinorial analogy of Yamabe problem. It is well known that the optimal Sobolev constant is the so-called Bär-Hijazi-Lott invariant which, as the Yamabe invariant, attains its maximum at round sphere. We prove on S^n that the Sobolev quotient being close to the optimal constant implies that spinor being close to an optimizer. Compared to the function case, the difficulty arises from the more complicated algebraic properties of Dirac operator. This is a joint work with Prof. Guofang Wang.
