This website is meant to inform about the seminar series entitled "KIT Geometric Analysis Seminar" held (typically in-person) at the Karlsruher Institut für Technologie every Tuesday at 14:00 German time.

If you are interested, you can subscribe to the mailing list of the seminar by sending an e-mail to Gianmichele Di Matteo. In this way, you will receive the title, the abstract and the MS Teams link of the upcoming talk on a weekly basis. To clarify, it will always be possible to attend the seminars online, even when these will be held in person!

Upcoming talks

 Past talks (2022)

• 25/01/2022: Gianmichele Di Matteo (Karlsruher Institut für Technologie)

Title: CMC and isoperimetric small double bubbles in compact manifolds

Abstract: In this talk, we will show how CMC and isoperimetric sets enclosing and separating two different small volumes in a manifold, arise as small perturbations of Euclidean standard double bubbles. These latter were shown to be isoperimetric minimizers given two fixed volumes by Hutchings, Morgan, Ritoré and Ros. Due to the singular nature of the model minimizer, the perturbations considered contain a non-trivial tangential component. The proof is based on elements from oscillation theory for degenerate elliptic operators, Hodge theory for overdetermined non-elliptic problems and perturbation theory. 

• 01/02/2022: Mario Schulz (University of Münster)

Title: Noncompact self-shrinkers for mean curvature flow (joint work with Reto Buzano and Huy The Nguyen) 

Abstract: In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. We employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. 

No seminars from 02/02/2022 to 07/03/2022!

08/03/2022: Alessandro Pigati (Courant Institute of Mathematical Sciences)

Title: Nondegenerate minimal submanifolds as energy concentration sets 

Abstract: Various energies of physical significance have been shown to effectively approximate the area functional. These energies are defined on the set of functions on a given ambient manifold, and for critical points they tend to concentrate towards a (possibly singular) minimal submanifold. 

In this talk we answer the converse problem: we show that any nondegenerate minimal submanifold arises in this way. The strategy is entirely variational and generalizes a recent work for geodesics (by Colinet, Jerrard, and Sternberg), by extending two key g.m.t. results to arbitrary dimension. (Joint work with Guido De Philippis)

No seminars from 09/03/2022 to 28/03/2022!

29/03/2022: Alessandra Pluda (University of Pisa)

Title: Resolution of singularities of the network flow 

Abstract: The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature (at any point and time) and can be interpreted as the gradient flow of the length. We consider the same flow for networks (finite unions of sufficiently smooth curves whose end points meet at junctions). Because of the variational nature of the problem, one expects that for almost all the times the evolving network will possess only triple junctions where the unit tangent vectors forms angles of 120 degrees (regular junctions). However, even if the initial network has only regular junctions, this property is not preserved by the flow and junctions of four or more curves may appear during the evolution. The aim of this talk is first to describe the process of singularity formation and then to explain the resolution of such singularities and how to continue the flow in a classical PDE framework. This is a research in collaboration with Jorge Lira (Universidade Federal do Ceará), Rafe Mazzeo (Stanford University) and  Mariel Saez (P. Universidad Catolica de Chile)

05/04/2022: Shengwen Wang (University of Warwick)

Title: A Brakke type regularity for the Allen-Cahn flow 

Abstract:  We will talk about an analogue of the Brakke's local regularity theorem for the $\epsilon$ parabolic Allen-Cahn equation. In particular, we show uniform $C_{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon$ tend to 0 under the almost unit-density assumption. This can be viewed as a diffused version of the Brakke regularity for the limit mean curvature flow. This talk is based on joint work with Huy Nguyen.

12/04/2022: Mattia Fogagnolo (Scuola Normale Superiore di Pisa) 

Title: New integral estimates in substatic manifolds and the Alexandrov Theorem 

Abstract: The classical Alexandrov Theorem in the Euclidean space asserts that any bounded set with a smooth boundary of constant mean curvature is a ball. This result can be more quantitatively expressed  by showing that an integral deficit from being of constant mean curvature dominates suitable analytic quantities that vanish exactly when the domain is a ball. In this talk, we provide generalizations of this in the context of substatic manifolds with boundary, that constitute a vast generalization of the family of manifolds with nonnegative Ricci curvature, and that are of particular importance in General Relativity. Our approach is based on the discovery of a vector field with nonnegative divergence involving the solution to a torsion-like boundary value problem introduced by Li-Xia in a related earlier work.

The talk is based on a joint work with A. Pinamonti (Trento). 

19/04/2022: Tristan Ozuch-Meersseman (Massachusetts Institute of Technology) 

Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows 

Abstract: With A. Deruelle, we define a Perelman like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities. 

26/04/2022: Michał Miśkiewicz (University of Warsaw)

Title: Perspectives on the p-harmonic map flow 

Abstract: The p-harmonic map flow is a flow of maps u : M → N between two given manifolds, and it is designed to decrease the Dirichlet p-energy (i.e. the integral of the gradient raised to the p-th power) in time. It is useful in studying p-harmonic maps, which are simply its stationary solutions. Since the pioneering work of Eells-Sampson ('64) and Chen-Struwe ('89), much is known about the special case p=2, but the general case is still full of open questions. In addition to difficulties related to the p-Laplace operator, a major obstacle seems to be the lack of a local energy monotonicity formula (due to Struwe in the case p=2). I will describe a new, alternative formulation of the p-harmonic map flow, based on the homogeneous (a.k.a. game-theoretic) p-Laplace operator. This homogeneous formulation indeed implies a Struwe-type monotonicity formula, but it comes with new fundamental challenges. 

The talk is based on joint work with Aaron Naber and Erik Hupp. 

03/05/2022: Giada Franz (ETH Zürich)

Title: Equivariant min-max theory to construct free boundary minimal surfaces in the unit ball 

Abstract: A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e., the unit sphere). A very natural question is whether there are FBMS in the unit ball of any given topological type. 

In this talk, we will present the construction of a family of FBMS with connected boundary and arbitrary genus, via an equivariant version of Almgren-Pitts min-max theory à la Simon-Smith. We will see how this method allows us to control the topology of the resulting surface and also to obtain information on its index.

10/05/2022: Florian Litzinger (Otto-von-Guericke-University Magdeburg)

Title: Singularities of high codimension curve shortening flow 

Abstract: We investigate the formation of singularities of curve shortening flow in an Euclidean background of any dimension. In particular, for type-II singularities, we prove the existence of a sequence of space-time points along which the curvature tends to infinity such that a rescaling of the solution along it converges to the Grim Reaper solution, paralleling Altschuler's work in the case of space curves. Furthermore, we demonstrate that the curve shortening flow of initial curves with an entropy bound converges to a round point in finite time. 

17/05/2022: Lothar Schiemanowski (Leibniz Universität Hannover)

Title: Special holonomy and topology of asymptotically conical manifolds in dimensions 6 and 7 

Abstract: We derive topological properties of asymptotically conical manifolds with special holonomy using analytical techniques. We then discuss applications of these results. 

24/05/2022: Fabian Rupp (University of Vienna)

Title: Li-Yau inequalites for the Helfrich functional 

Abstract: In order to study the shape of red blood cells, Canham and Helfrich suggested a variational model involving a curvature dependent energy functional. In the simplest case, this reduces to the well-known Willmore functional and the correspoding minimization problem has been extensively studied throughout the last decade. An essential tool in this analysis is an inequality of Li and Yau which implies that any immersed surface with energy strictly below 8π must not intersect itself. In this talk, we will prove an extension of this inequality to the Helfrich energy. One crucial consequence is that smoothly embedded minimizers in the Canham-Helfrich model exist if the energy is not too large. This is joint work with Christian Scharrer (Bonn). 

07/06/2022: Konstantinos Zemas (Universität Münster)

Title: Geometric rigidity in variable domains and applications to SDRI models

Abstract: Quantitative rigidity results, besides their inherent geometric interest, have played a prominent role in the mathematical study of models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for functionals within the framework of nonlinear elasticity. In this talk I will present a generalization of this result to the setting of variable domains, where the geometry of the domain comes into play in terms of a suitable surface energy of its boundary.  As a first application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of Γ-convergence. This is joint work with Manuel Friedrich and Leonard Kreutz. 

21/09/2022: Andrea Malchiodi (Scuola Normale Superiore di Pisa)

Title: Some existence and regularity results for the Born-Infeld model 

Abstract: We consider a degenerate elliptic equation that describes spacelike hypersurfaces in Minkowski's spacetime having prescribed Lorentzian mean curvature, as well as the electric potential in Born-Infeld theory.

The problem is variational, and minimizers can be found under rather general assumptions. We provide conditions to guarantee that these weakly solve the required Euler-Lagrange equation, and provide counterexamples as well. This is joint work with Jaeyoung Byeon Norihisa Ikoma and Luciano Mari. 

05/10/2022: Louis Yudowitz (Queen Mary University of London) 

Title: Bubble Tree Convergence of Ricci Shrinkers 

Abstract: Introduced by Richard Hamilton in 1982, Ricci flow has been used to solve a variety of problems in geometry and topology.  A vital part of such proofs is a good understanding of finite time singularities. While we have such an understanding in dimensions 2 and 3, singularity models in higher dimensions are still relatively mysterious. This is partially due to the existence of singularity models which are singular themselves. In this talk, we will prove bubble tree convergence of certain shrinking singularity models. This allows us to analyze the case when the singular set consists of isolated points. As a consequence, we will recover any topology lost due to the formation of the singular points, as well as prove a qualitative classification result. This is based on a joint work with Reto Buzano.  

09/11/2022: Elena Mäder-Baumdicker (Technische Universitat Darmstadt) 

Title: Progress in the theory of the volume preserving Mean curvature flow 

Abstract: The volume preserving mean curvature flow (VPMCF) deforms a hypersurface along the (negative) gradient of the area functional while keeping the enclosed volume fixed. Due to the global constraint the VPMCF behaves quite differently compared to the mean curvature flow. I will point our differences and report on recent work with Ben Lambert about the VPMCF in the closed, Euclidean setting. We were able to obtain estimates of non-local quantities that allow us to apply classical parabolic blowup procedures. Furthermore, we found a property that is preserved along the VPMCF under reasonable assumptions.

16/11/2022: Ernst Kuwert (Albert-Ludwigs-Universität Freiburg)  

Title: Curvature varifolds with orthogonal boundary

Abstract: NOT AVAILABLE

30/11/2022: Renan Assimos (Leibniz Universitat Hannover)   

Title: When discrete geometry meets a geometric flow. 

Abstract:  We use ideas from discrete geometry to build new degree zero harmonic maps from surfaces of genus greater than one into two dimensional spheres. Those surfaces are then used as counter examples to a conjecture of Emery about harmonic map images and closed geodesics. 

09/05/2023: Armin Schikorra (University of Pittsburgh )   

Title: div-curl estimates and harmonic maps: local and nonlocal  

Abstract:  I will present definitions and applications of a notion of fractional div-curl structures. I will talk about their role in the

theory of fractional harmonic maps, such as regularity theory and conservation laws.

12/12/2023: Artemis Vogiatzi (Queen Mary University of London)   

Title: Singularities of High Codimension Mean Curvature Flow of Pinched Submanifolds 

Abstract:  In this talk, by assuming a quadratic curvature pinching condition, we show that the submanifold evolving by mean curvature flow becomes approximately codimension one, in high curvature regions. This fundamental codimension estimate along with a cylindrical type estimate, at a singularity, allows us to establish the existence of a rescaling, which converges to a smooth codimension-one limiting flow in Euclidean space, which is weakly convex and either moves by translation or is a self-shrinker. This is possible using special pointwise gradient estimates for the second fundamental form.

24/06/2024: Huy The Nguyen (Queen Mary University of London)   

Title: Mean Curvature Flow in the Sphere  

Abstract:  In this talk, I will discuss some recent results analysing singularities of the mean curvature flow in the sphere both in the hypersurface case and the high codimension case.