Organizers: Miles Simon and Florian Litzinger

Institute of Analysis and Numerics, Otto-von-Guericke University Magdeburg, Germany.

The seminars are held on Thursday, 15:00-17:00, G02-20.

**Thursday,
April 25, 2024**

Hannover-Magdeburg seminar (in Magdeburg)

**Thursday,
May 16, 2024, 15:00-17:00**

**Rudolf Zeidler** (University of Münster)

**Title**:
Rigidity of spin fill-ins with non-negative scalar curvature

**Abstract**:
Given a closed Riemannian spin manifold $\Sigma$ with certain properties, I will explain various rigidity results for compact spin manifolds $M$ that have $\Sigma$ as boundary and satisfy lower bounds on scalar- and mean curvature. This includes rigidity in terms of the hyperspherical radius of $\Sigma$ as conjectured by M. Gromov. The talk will be based on joint work with S. Cecchini (Texas A&M University) and Sven Hirsch (IAS Princeton).

**Thursday,
May 23, 2024, 15:00-17:00**

**Elena
Mäder-Baumdicker** (Technical University of Darmstadt)

**Title**:
A monotonicity formula for the bi-heat equation

**Abstract**:
For geometric evolution equations of second order, (parabolic) monotonicity formulas have been proven to be very powerful tools. One example is the monotonicity formula for the harmonic map heat flow which lead to regularity results and global existence results for small initial energy in higher dimension (Struwe 1988). Another example is the Mean Curvature Flow, where many results rely on Huisken's famous monotonicity formula. In this talk, I will explain that also fourth order equations satisfy a certain kind of monotonicity formula in some cases. One of the challenges in these computations consists in getting some control of the Euclidean bi-heat kernel in certain regimes. This talk is based on work with Casey Kelleher and Nils Neumann.

**Thursday,
June 6, 2024, 15:00-17:00**

**Mario Schulz** (University of Münster)

**Title**:
Free boundary minimal surfaces

**Abstract**:
Free boundary minimal surfaces naturally appear in various contexts, including partitioning problems for convex bodies, capillarity problems for fluids, and extremal metrics for Steklov eigenvalues on manifolds with boundary. Constructing embedded free boundary minimal surfaces is challenging, especially in ambient manifolds like the Euclidean unit ball, which only allow unstable solutions. Min-max theory offers a promising avenue for existence results, albeit with the added complexity of requiring control over the topology of the resulting surfaces. This presentation will offer an overview of recent results and applications.

**Thursday,
July 4, 2024**

Hannover-Magdeburg seminar (in Hannover)

**Thursday,
October 27, 2022, 15:00-17:00**

**Maxwell
Stolarski** (University of Warwick)

**Title**:
Closed
Ricci Flows with Singularities Modeled on Asymptotically Conical
Shrinkers

**Abstract**:
Shrinking
Ricci solitons are Ricci flow solutions that self-similarly shrink
under the flow. Their significance comes from the fact that
finite-time Ricci flow singularities are typically modeled on
gradient shrinking Ricci solitons. Here, we shall address a certain
converse question, namely, "Given a complete, noncompact
gradient shrinking Ricci soliton, does there exist a Ricci flow on a
closed manifold that forms a finite-time singularity modeled on the
given soliton?" We'll discuss work that shows the answer is yes
when the soliton is asymptotically conical. No symmetry or Kahler
assumption is required, and so the proof involves an analysis of the
Ricci flow as a nonlinear degenerate parabolic PDE system in its full
complexity. We'll also discuss applications to the (non-)uniqueness
of weak Ricci flows through singularities.

**Thursday,
November 10,
2022, 15:00-17:00**

**Florentin
Münch **(Max Planck Institute
for Mathematics, Leipzig)

**Title**:
Discrete Ricci curvature, betweenness centrality and rigidity

**Abstract**:
We give an introduction about discrete Ricci curvature and present
various discrete versions of the discrete Bonnet Myers diameter
bound. We will focus on one version upper bounding the average
distance in terms of the weighted average of the Ricci curvature,
where the weight is the betweenness centrality. We will show that
this bound is attained precisely for Cartesian products of cocktail
party graphs, Johnson graphs, demi-cubes, the Gosset graph, and the
Schläfli graph.

**Thursday,
December 15, 2022, 15:00-17:00**

**Chaona
Zhu** (University
of Rome Tor Vergata)

**Title**:
Compactness of Stationary Dirac-Harmonic maps

**Thursday,
January 12, 2023, 15:00-17:00**

**Christian
Ketterer** (University of Freiburg)

**Title**:
Rigidity and stability results for mean convex subsets in RCD spaces

**Abstract**:
I present splitting theorems
for mean convex subsets in RCD spaces. This extends results for
Riemannian manifolds with boundary by Kasue, Croke and Kleiner to a
non-smooth setting. A corollary is a Frankel-type theorem. I also
show that the notion of mean curvature bounded from below for the
boundary of an open subset is stable w.r.t. to uniform convergence of
the corresponding boundary distance function.

**Thursday,
April 28, 2022, 15:00-17:00**

**Christian
Rose** (University of
Potsdam)

**Title**:
Geometric properties of manifolds with Kato-bounded Ricci
curvature

**Abstract**:
In
my last talk I gave an overview on results based on Kato-bounds on
the Ricci curvature. I will recall the main motivation and
definitions and discuss some geometric consequences of almost
everywhere positive Ricci curvature in Kato sense, such as Betti
number bounds and compactness.

**Thursday,
May 12, 2022, 15:00-17:00**

**Joshua
Daniels-Holgate** (University of Warwick)

**Title**:
Approximation of mean curvature flow with generic
singularities by smooth flows with surgery

**Abstract**:
We construct smooth flows with surgery that approximate weak
mean curvature flows with only spherical and neck-pinch
singularities. This is achieved by combining the recent work of
Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White,
establishing canonical neighbourhoods of such singularities, with
suitable barriers to flows with surgery. A limiting argument is then
used to control these approximating flows. We demonstrate an
application of this surgery flow by improving the entropy bound on
the low-entropy Schoenflies conjecture.

**Thursday,
June 2 , 2022, 15:00-17:00**

**Matthias
Erbar** (Bielefeld
University)

**Title**:
Synthetic (super) Ricci flows

**Abstract**:
In the talk I would like to
review several approaches to define notions of (super) Ricci flows
for time-dependent families of metric measure spaces. These are based
on ideas from optimal transport of measures and take their starting
point in the successful theory of synthetic Ricci curvature (lower
bounds) for metric measure spaces.

**Thursday,
June 9, 2022, 15:00-17:00**

**Boris
Vertman** (Carl von Ossietzky University of Oldenburg)

**Title**:
Prescribed mean curvature flow in Lorentzian space times

**Abstract**:
We report on our recent work about the prescribed mean
curvature flow on non-compact space-like Cauchy hypersurfaces in
generalized Robertson-Walker space times. We prove existence and
convergence of the flow under additional conditions. This is joint
work with Giuseppe Gentile and generalizes the previous results by
Ecker, Huisken Gerhard and others.

**Thursday,
November 4, 2021, 15:00-17:00**

**Jiawei
Liu** (Otto-von-Guericke-University
Magdeburg)

**Title**:
Ricci flow starting from an embedded closed convex surface in R^3

**Abstract**:
I will talk about the existence and uniqueness of Ricci flow that
admits an embedded closed convex surface in R^3 as metric initial
condition. The main point is a family of smooth Ricci flows starting
from smooth convex surfaces whose metrics converge uniformly to the
metric of the initial surface in intrinsic sense.

**Thursday,
November 25, 2021, 15:00-17:00**

**Gianmichele
Di Matteo** (Karlsruhe
Institute of Technology)

**Title**:
A Local Singularity Analysis for the Ricci flow

**Abstract**:
In
this talk, I will describe a refined local singularity analysis for
the Ricci flow developed jointly with R. Buzano. The key idea is to
investigate blow-up rates of the curvature tensor locally, near a
singular point. Then I will show applications of this theory to Ricci
flows with scalar curvature bounded up to the singular time.

**Thursday,
December 9, 2021, 15:00-17:00, Zoom Meeting**

**Olaf Müller**
(Humboldt University of
Berlin)

**Title**:
Compactness and Finiteness Theorems in Riemannian and
Lorentzian Geometry

**Abstract**:
This
talk would circle around two results of mine, a smooth Cheeger-Gromov
compactness theorem for manifolds with boundary and the (to my
knowledge) first known Lorentzian finiteness theorem.

**Thursday,
December 16, 2021, 16:00-17:00**

**Florian
Litzinger** (Otto-von-Guericke-University
Magdeburg)

**Title**:
Regularity theory and singularity analysis
for some geometric PDE

**Abstract**:
In
this talk, I will outline two sets of problems I have been working on
recently. First, I will introduce the regularity theory for
two-dimensional Pfaffian systems and the related fundamental theorem
of surface theory. In particular, we will see how to obtain the
optimal regularity result. Second, we will consider singularities of
curve shortening flow in arbitrary codimension.

**Thursday,
January 13, 2022, 15:00-17:00, Zoom Meeting**

**Christian
Rose** (University of
Bremen)

**Title**:
Compact manifolds with Kato-bounded Ricci curvature

**Abstract**:
It is a classical fact that all compact Riemannian manifolds
with prescribed uniform lower bound on the Ricci curvature and upper
diameter bound possess similar geometric and spectral estimates. In
the last decades there was an increasing interest in relaxing the
uniform lower Ricci curvature bounds to integral bounds as they are
more stable with respect to perturbations of the metric. Even more
general than the usual L^p-curvature restrictions is the so-called
Kato condition on the negative part of the Ricci curvature. It proved
to be a powerful assumption to generalize many of the existing
estimates on eigenvalues, heat kernel, and Betti number estimates,
which I will discuss in my talk. In parts joint work with Gilles
Carron and Guofang Wei.

**Thursday,
January 20, 2022, 15:00-17:00**

**Boris
Vertman** (Carl von Ossietzky University of Oldenburg)

**Due
to the pandemic, this talk has been moved to the Summer Seminars!**

**Tuesday,
February 8, 2022, 15:00-17:00, Zoom Meeting**

**Marius
Müller** (Albert-Ludwigs-University Freiburg)

**Title**:
An obstacle problem for the p-elastic energy

**Abstract**:

**Thursday,
April 8, 15:00-16:00, Zoom Meeting**

**Alix
Deruelle **(Institut de Mathématiques de Jussieu-Paris Rive
Gauche)

**Title**:
A relative entropy for expanders of the Ricci flow

**Abstract**:
Expanding self-similar solutions of the Ricci flow are solutions
which evolves by scaling and diffeomorphisms only. Such solutions are
also called expanding gradient Ricci solitons. These "canonical"
metrics are potential candidates for smoothing out isolated
singularities instantaneously. These heuristics apply to the
Kähler-Ricci flow too. In this talk, we ask the question of
uniqueness of such self-similar solutions coming out of a given
metric cone over a smooth link. As a first step, we make sense of a
suitable Lyapunov functional also called relative entropy in this
setting.

**Thursday,
April 15, 15:00-16:00, Zoom Meeting**

**Man-Chun
Lee** (Northwestern University and University of Warwick)

**Title**:
d_p convergence and epsilon-regularity theorems for entropy and
scalar curvature lower bound

**Abstract**:
In this talk, we consider Riemannian manifolds with almost
non-negative scalar curvature and Perelman entropy. We establish an
epsilon-regularity theorem showing that such a space must be close to
Euclidean space in a suitable sense. We will illustrate examples
showing that the result is false with respect to the Gromov-Hausdorff
and Intrinsic Flat distances, and more generally the metric space
structure is not controlled under entropy and scalar lower bounds. We
will introduce the notion of the d_p distance between (in particular)
Riemannian manifolds, which measures the distance between W^{1,p}
Sobolev spaces, and it is with respect to this distance that the
epsilon regularity theorem holds. This is joint work with A. Naber
and R. Neumayer.

**Thursday,
May 6, 15:00-16:00, Zoom Meeting**

**Tobias
Marxen** (Carl von Ossietzky University of Oldenburg)

**Title**:
Ricci flow on incomplete manifolds

**Abstract**:
The Ricci flow has become famous
since being the essential tool in the proof of Thurston's
geometrization conjecture and the Poincare conjecture by Perelman.
The idea is that the flow smoothes out irregularities of a given
initial Riemannian metric and ideally converges to a canonical
metric, from which the topological structure of the manifold can be
read off. In order to get the Ricci flow started and to smooth out a
given initial metric, short-time existence is needed. This was
established by Hamilton for closed manifolds. Shi obtained short-time
existence for complete manifolds with bounded curvature. We extend
Shi's result to incomplete manifolds with bounded curvature, thereby
showing that any manifold of bounded curvature can be flown for a
short time. For technical reasons, instead of the Ricci flow we
consider the related Ricci de Turck flow. This is joint work with
Boris Vertman.

**Thursday,
May 20, 15:00-16:00, Zoom Meeting**

**Klaus
Kröncke **(Universität Hamburg)

**Title**:
L^p-stability
and positive scalar curvature rigidity of Ricci-flat ALE manifolds

**Abstract**:
We prove stability of integrable ALE manifolds with a parallel spinor
under Ricci flow, given an initial metric which is close in $L^p\cap
L^{\infty}$, for any $p\in (1,n)$, where n is the dimension of the
manifold. In particular, our result applies to all known examples of
4-dimensional gravitational instantons. The result is obtained by a
fixed point argument, based on novel estimates for the heat kernel of
the Lichnerowicz Laplacian. It allows us to give a precise
description of the convergence behaviour of the Ricci flow. Our decay
rates are strong enough to prove positive scalar curvature rigidity
in $L^p$, for each $p\in [1,\frac{n}{n-2})$, generalizing a result by
Appleton. This is joint work with Oliver Lindblad Petersen.

**Thursday,
May 27, 15:00-16:00, Zoom Meeting**

**Oliver
Lindblad Petersen **(Stanford
University)

**Title**:
Analyticity of quasinormal modes in the Kerr and Kerr-de
Sitter spacetimes

**Abstract**:
Quasinormal modes are fundamental in the study of wave equations on
black hole spacetimes. In this talk, I will explain why the
quasinormal modes in the Kerr and Kerr-de Sitter spacetimes are real
analytic. This requires new analysis of quasinormal modes near
horizons which relies on an important geometric difference between
the horizon Killing vector field and the stationary Killing vector
field. The analyticity then follows from a recent result in the
microlocal analysis of radial points by Galkowski-Zworski. This is
joint work with Andras Vasy.

**Thursday,
June 17, 15:00-16:00, Zoom Meeting**

**Felix
Schulze** (University of Warwick)

**Title**:
Mean curvature flow with generic initial data

**Abstract**:
Mean curvature flow is the gradient flow of the area
functional and constitutes a natural geometric heat equation on the
space of hypersurfaces in an ambient Riemannian manifold. It is
believed, similar to Ricci Flow in the intrinsic setting, to have the
potential to serve as a tool to approach several fundamental
conjectures in geometry. The obstacle for these applications is that
the flow develops singularities, which one in general might not be
able to classify completely. Nevertheless, a well-known conjecture of
Huisken states that a generic mean curvature flow should have only
spherical and cylindrical singularities. As a first step in this
direction Colding-Minicozzi have shown in fundamental work that
spheres and cylinders are the only linearly stable singularity
models. As a second step toward Huisken's conjecture we show that
mean curvature flow of generic initial closed surfaces in R^3 avoids
asymptotically conical and non-spherical compact singularities. The
main technical ingredient is a long-time existence and uniqueness
result for ancient mean curvature flows that lie on one side of
asymptotically conical or compact self-similarly shrinking solutions.
This is joint work with Otis Chodosh, Kyeongsu Choi and Christos
Mantoulidis.

**Thursday,
July 1, 15:00-16:00, Zoom Meeting**

**Peter
Topping** (University of Warwick)

**Title**:
Ricci flow on surfaces with rough initial data

**Abstract**:
There has been much work in recent years on starting the Ricci
flow with initial data that is rougher than a Riemannian metric. A
notable example is when the initial data is a metric space with some
basic regularity. There has also been significant development of
Ricci flow theory on noncompact manifolds. In this talk, I will
survey selected results in these directions and describe some
forthcoming work joint with Hao Yin in two dimensions.