## Geometric Analysis Seminar

### Organisers

### Talks

The seminar usually takes place on Thursdays, 3-5pm in building 03, room 214 of the Institute of Analysis and Numerics, Otto-von-Guericke University Magdeburg.

#### Spring–Summer 2024

**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.

#### Winter 2022/23

**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.

#### Spring–Summer 2022

**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.

#### Winter 2021/22

**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**:

#### Spring–Summer 2021

**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.