Seminar on Differential Geometry and Analysis

Marius Müller (Universität Augsburg):
The(?) biharmonic Alt-Caffarelli problem

The classical Alt–Caffarelli problem concerns the minimization of a functional consisting of two competing terms: the Dirichlet energy and the Lebesgue measure of the positivity set. Minimizers must balance out these competing interests -- low variation against adhesion to the zero level on a large set.

In this talk, we investigate what happens when the Dirichlet energy is replaced by the biharmonic energy. The resulting loss of the maximum principle gives rise to several new and surprising phenomena, which we will explore.

I will present results on the regularity and geometry of minimizers and discuss their optimality. Moreover, we will study regularity and optimal shape of the zero level sets. Finally, we will take a closer look at the blow-up behavior of minimizers near their zero level sets.

Many of these results are from joint work with Hans-Christoph Grunau.

Julian Scheuer (Goethe Universität Frankfurt):
Survey on fully nonlinear curvature flows and applications

Abstract:

Abstract: Curvature flows have been one of the key subjects in geometric analysis for almost half a century, with striking applications in other fields of geometry. They all share the property of being partial differential equations. The famous mean curvature flow is a quasi-linear example, but also fully nonlinear flows have allowed applications for example in convex geometry, in particular for isoperimetric type inequalities.

In this talk we survey the historical motivation to study these flows and give a few very recent highlights and developments, especially in the context of spherical geometry.

Felix Schulze (University of Warwick):
PIC1 pinched manifolds are flat or compact

Abstract: Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, was resolved recently using Ricci flow. In this talk I will explain a direct analogue of that result in all dimensions. One aspect of the work is a new lifting technique that allows us to handle manifolds that are collapsed at infinity; until now this could only be achieved in 3D via work of Lott. The new theorem builds on earlier work of Deruelle, Simon and the speaker and separately of Lee-Topping. Joint work with Alix Deruelle, Man Chun Lee, Miles Simon and Peter Topping.

Madaleine Jotz (Universität Würzburg):
Deformation theory of ideals in Lie algebras

Abstract: The smooth deformations of an ideal 𝔦 in a Lie algebra 𝔤 differentiate to cohomology classes in the cohomology of 𝔤 with values in its adjoint representation on Hom(𝔦, 𝔤/𝔦). This cohomology associated with the ideal 𝔦 in 𝔤 is compared with other Lie algebra cohomologies defined by 𝔦, such as the cohomology defined by 𝔦 as a Lie subalgebra of 𝔤 in (Richardson 1969), and the cohomology defined by the Lie algebra epimorphism 𝔤 → 𝔤/𝔦.

After a choice of complement of the ideal 𝔦 in the Lie algebra 𝔤, its deformation complex is enriched to the differential graded Lie algebra that controls its deformations, in the sense that its Maurer-Cartan elements are in one-to-one correspondence with the (small) deformations of the ideal. Furthermore, the L∞-algebra that simultaneously controls the deformations of 𝔦 and of the ambient Lie bracket is identified. Under appropriate assumptions on the low degrees of the deformation cohomology of a given Lie ideal, the (topological) rigidity and  stability of ideals are  described, as well as obstructions to deformations of ideals of Lie algebras.

This is a joint work with Ilias Ermeidis.

Anna Dall'Acqua (Universität Ulm):
Unwinding an elastic wire

Abstract: How does a thin elastic rod choose its shape? A natural way to model this is to assign an energy that measures how much the rod bends, and to look for shapes that minimize this energy. We begin by exploring these optimal configurations. We then ask what happens if the rod is allowed to move: how does its shape evolve if it continuously tries to reduce its bending? We discuss general ideas that ensure this evolution is well-defined and eventually settles into an equilibrium.
Finally, we show how this one-dimensional picture sheds light on the behavior of certain surfaces, revealing an unexpected connection between the motion of curves and geometric evolution in higher dimensions.

Filippo Gazzola (Politecnico di Milano):
Daily mathematics in fluid-structure interactions

Abstract: Daily life is full of phenomena related to fluid-structure interactions that we often fail to recognize.
Mathematics, the natural sciences, physics, and engineering are not always able to explain what we observe.
Sometimes, different theories lead to conflicting conclusions.
The purpose of my talk is to uncover these interactions, to connect some seemingly different phenomena, and to raise unanswered questions.
I will also discuss some personal contributions.
No specific knowledge is required, just a strong sense of observation, coupled with curiosity and a thirst for knowledge. 

Guido Sweers (Universität zu Köln):
Bending clamped plates

Abstract: Pulling a string or pushing a membrane fixed at the boundary results in the string/membrane to move in the same direction. More than 100 years ago one expected to same to hold for a plate fixed at it boundary. It turned out to be not that obvious or even true in general. The first models lead to boundary value problems with second order (elliptic) differential equations, while the plate leads us to a fourth order elliptic problem. For the general audience I hope to explain the concepts, some known results and two open problems including clues what to expect. 

Anna Dall'Acqua (Universität Ulm):
On the free boundary elastic flow

Abstract: We study the length-preserving elastic flow in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We discuss how a suitable non-flatness assumption ensures global existence and subconvergence to critical points. 
This is joint work with Manuel Schlierf.

Kai Zehmisch (Ruhr-Universität Bochum):
Contactomorphic Vertically Convex Domains

Abstract: We consider the standard Darboux space equipped with the radial symmetric contact form. We study co-orientation preserving contactomorphisms between bounded domains up to the boundary. We determine the contactomorphism classes among all strict vertically convex domains over a round ball in the Liouville hyperplane that are radially symmetric about the Reeb axis and whose boundary coincides along a neighbourhood of the common equator. The total invariant is the mean curvature of the bounding sphere at the umbilic points with the same sign.

Marco Pozzetta (Politecnico di Milano):
A sharp spectral splitting theorem

Abstract: The celebrated Cheeger-Gromoll splitting theorem implies that noncompact nonnegatively curved manifolds with multiple ends must be isometric to the product of a compact manifold with a Euclidean factor. In this seminar we will present a generalization of such a statement where nonnegativity of curvature is encoded in the nonnegativity of a suitable Schrodinger operator involving the Ricci curvature of the manifold. We will also discuss the sharpness of the assumptions. The proof of the result is based on a variational argument that exploits suitably constructed minimizers to a prescribed mean curvature problem.
The seminar is intended to be self-contained and accessible also to non-experts in Riemannian geometry. We will heuristically recall the basic concepts and discuss motivations.
The talk is based on a joint work in collaboration with Gioacchino Antonelli (Notre Dame University) and Kai Xu (UC Berkeley).

Anna Balci (Universität Bielefeld):
Nonlinear models with differential forms

Abstract: The talk focuses on the analytical and numerical challenges that arise in nonlinear models with differential forms. In particular, we address the lack of density of smooth forms in the relevant energy spaces, which leads to energy gap type phenomena. We identify structural conditions that prevent such gaps and provide criteria for the existence of minimizers within the natural function spaces. In addition, we establish regularity results and elliptic estimates for solutions to the associated nonlinear systems and develop numerical methods that preserve the underlying geometric structure of the problem, including compatibility with the Hodge decomposition.

Duc Viet Vu (Universität zu Köln):
Kähler-Einstein metrics on quasi-projective manifolds

Abstract: I will present my recent joint work with Quang-Tuan Dang on Kähler-Einstein metrics of negative curvature on quasi-projective manifolds. Under a standard assumption on a divisor D on a compact Kähler manifold X, we proved that  the singular Kähler-Einstein metric on the complement of D constructed by Berman-Guenancia is almost-complete in the sense of Tian-Yau, and this metric is the weak limit of a sequence of conic twisted global Kähler-Einstein metrics on X.

Georg Frenck (Universität Augsburg):
Positive scalar curvature and point singularities 

Abstract: This talk will focus on so-called $L^\infty$ metrics with lower scalar curvature bounds. While some phenomena from the smooth world still occur in this particular singular setting, one observes some important differences. In particular, there are manifolds which admit an $L^\infty$ metric of positive scalar curvature, even though there cannot be a smooth such metric, which negates a conjecture of Schoen. This is joint work with Simone Cecchini and Rudolf Zeidler.

Louis Yudowitz (KTH Stockholm):
Semi-Continuity of the Morse Index for Ricci Shrinkers

Abstract: When studying the compactness theory for solutions to geometric PDEs, one can encounter the formation of singularities. This occurs, for instance, for sequences of harmonic maps, Einstein manifolds, minimal surfaces, and gradient shrinking Ricci solitons. Provided the singular set consists of isolated points, a common strategy to study them is to construct a “bubble tree” through an iterative blow-up procedure. Moreover, supplementing this with analytic, geometric, and topological arguments allows for the recovery of information lost due to the bubbling behavior. In this talk I will discuss bubbling for gradient Ricci shrinkers, with a focus on how it affects the stability of singularity models for Ricci flow. In particular, I will give an overview of how to prove upper and lower semi-continuity of the Morse index for Ricci shrinkers by studying a certain weighted eigenvalue problem. I will also explain how the same techniques can be applied to show a quantitative relation between the stability of an asymptotically conical Ricci shrinker and that of its asymptotic cone. 

Fabian Rupp (Universität Wien):
A new energy method for shortening and straightening complete curves  

Abstract: The curve shortening flow arises as the L^2-gradient flow of the length functional. In this talk, we will discuss a novel energy method that reinterprets “curve shortening” as “tangent aligning”. This interpretation allows us to apply energy-based methods to study the long-time behavior of the curve shortening flow of curves of infinite length. Our method also applies to higher order flows such as the curve diffusion flow, Chen’s flow, and the elastic flow, which involves not only the curve shortening but also the curve straightening effect. This is joint work with T. Miura (Kyoto).