Talks

Magalie Bénéfice (Nancy)

Couplings of Brownian motions in subRiemannian manifolds.

The coupling method on different kind of manifolds is an interesting tool leading to numerous results in analysis, probability and geometry. As an example, couplings of Brownian motions can provide inequalities just like Harnack parabolic inequalities or gradient inequalities as well as results on the harmonic functions or some lower bounds of the spectral gap.
Such results can be obtained by constructing couplings such that the probability that the Brownian motions meet at a certain time is "small enough". When this meeting time is almost surely finite, the coupling is called "successful".
In the case of subRiemannian manifolds, this coupling method can be particularly useful as it can deal with some of the over mentioned problems without the intervention of some geometric or analytic objects that are difficult to define.

The aim of this talk is to present some of these couplings and their applications in Riemannian manifolds and to see how they can be extended to some subRiemannian structures like the Heisenberg group, $SU(2)$ or $SL(2,\mathbb{R})$.

Florian Johne (Freiburg-im-Breisgau)

Topology and geometry of metrics of positive intermediate curvature.

In this talk we explain a non-existence result for metrics of positive
m-intermediate curvature (a notion of curvature reducing to positive
Ricci curvature for m = 1, and positive scalar curvature for m = n-1)
on closed orientable manifolds with topology $N^n = M^{n-m} x
\mathbb{T}^m$ for $n \leq 7$.
Our proof uses a slicing constructed by minimization of weighted areas,
the associated stability inequality, and estimates on the gradients of
the weights and the second fundamental form of the slices. This is
joint work with Simon Brendle and Sven Hirsch.

Eva Kopfer (Bonn)

Density-constrained optimal transport.

We consider the problem of dynamic optimal transport with a density constraint. We derive variational limits in terms of Γ-convergence for singular phenomena.

Leonie Langer (Ulm)

Heterogeneity and Incompressibility in the Evolution of Elastic Wires.

(PDF of the abstract)

Elastic wires are mathematical curves composed of matter. They are used to model approximately one-dimensional elastic objects like plant stems, polymers, marine cables or hair.
The elastic energy of a sufficiently smooth regular curve $\gamma\colon\mathbb{S}^1\to\RR^2$ describing an elastic wire is defined as
\begin{align*}
\mathcal{E}(\gamma)=\frac12\int_\gamma\vert\kappa\vert^2\mathrm{d} s.
\end{align*}
Here, $\kappa=\partial_s^2\gamma$ is the curvature of $\gamma$, $\partial_s=\vert\partial_x\gamma\vert^{-1}\partial_x$, and $\mathrm{d} s=\vert\partial_x\gamma\vert\mathrm{d} x$ is the arclength element.
In the last decades, several authors have studied the $L^2$-gradient flow of the elastic energy in different variants. We briefly summarize their results and then focus on two new variants.

First, we consider elastic wires with a heterogeneity described by a density function. We define a generalization of the elastic energy, which depends on material parameters, captures the interplay between curvature and density effects and resembles the Canham--Helfrich functional.
Describing the closed planar curve by its inclination angle, the $L^2$-gradient flow of this energy is a nonlocal coupled parabolic system of second order. We shortly discuss local well-posedness, global existence and convergence. Then, we show that the (non)preservation of quantities such as convexity as well as the asymptotic behavior of the system depend delicately on the choice of material parameters.

Second, we study the evolution of elastic wires under the assumption of incompressiblity and derive a gradient flow of the elastic energy which preserves the enclosed area of the evolving planar curves. Contrary to an earlier approach using Lagrange multipliers, we give priority to the locality of the evolution equation, accepting it being of sixth order. We prove a global existence result and, by penalizing the length, we show convergence to an area constrained critical point of the elastic energy.

Jianyu Ma (Toulouse)

Displacement functional and absolute continuity of Wasserstein barycenters.

Metric barycenters are used to average probability measures on metric spaces where no affine structure is available in general. On the (Wasserstein) space of probability measures over a given metric space, (Wasserstein) barycenter is a direct generalization of the celebrated McCann interpolation which corresponds to the case of two probability measures. In the talk, we consider Wassertein barycenters on possibly non-compact Riemannian manifolds and explain how to prove the absolute continuity of such a probability measure with respect to Riemannian volume. We shall discuss the displacement functional inequality used in my paper Arxiv 2310.1382 to prove this theorem under natural assumptions including Ricci curvature bounds. While being connected to curvature our approach is different from the widely used displacement convexity.

Dorian Martino (Zürich)

Classification of branched Willmore spheres.

In 1984, Byrant proved that smooth Willmore spheres are all conformal transformations of minimal surfaces. However, the case of branched Willmore spheres was left open. In the 2000s, the interest in analytical questions surrounding Willmore surfaces, such as compactness questions or the Willmore flow, lead to important contributions in the study of branched Willmore surfaces. Two attempts of Lamm-Nguyen in 2015 and Michelat-Rivière 2019 allowed to extend Bryant’s classification to some branched Willmore spheres. In this talk, I will fully answer the question: every branched Willmore sphere is a conformal transformation of a minimal surface. After introducing Willmore surfaces and the conformal Gauss map, I will present some key idea of the proof.

Jona Seidel (Darmstadt)

The soul of Alexandrov spaces.

The Soul Theorem asserts that every complete noncompact Riemannian manifold M with nonnegative curvature contains a closed, totally convex submanifold, a soul, whose normal bundle is diffeomorphic to M.
Alexandrov spaces are metric spaces where curvature bounds are introduced by triangle comparison and serve as a generalization of complete Riemannian manifolds. Perelman proved that the construction of the soul generalizes to Alexandrov spaces. The Soul Theorem, however, does not.
Still, the soul serves as a key tool to study noncompact nonnegatively curved Alexandrov spaces. One such an example is a Theorem for Riemannian manifolds by Sarafutdinov which, as we show, generalizes to Alexandrov spaces. It gives a lower bound on the injectivity radius in terms of an upper curvature bound and the soul.
We first give an overview of the soul construction and, second, outline the generalization of Sarafutdinov's theorem to Alexandrov spaces.

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