Oberseminar Differentialgeometrie

24.04.2025 (O)

Dr. Türkü Özlüm Çelik (Max Planck Institute of Molecular Cell Biology and Genetics, Dresden)

Title: 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.

Title: 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).

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Prof. Dr. Shubham Dwivedi (Hamburg)

21.11.2024 (O)

Prof. Dr. Sebastian Goette (Freiburg)

Title: Leibenson's equation on Riemannian manifolds

Abstract: We consider on an arbitrary Riemannian manifold the Leibenson equation u_t=Δ_pu^q, where Δ_p is the p-Laplace-Beltrami operator. This equation comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium. Our main result is that if p>1 and q>1/(p-1)  then bounded solutions of
this equation have finite propagation speed. In Rⁿ this result has been long known due to explicit Barenblatt solutions. The proof on manifolds is based on a certain mean value inequality for subsolutions. This work is joint with Philipp Sürig.

Prof. Dr. Guofang Wang

(Universität Freiburg)

Title: Capillary hypersurfaces: Geometric inequalities, flows and related problems

Abstract: In this talk, we first introduce geometric quantities for capillary hypersurfaces, and then establish isoperimetric type inequalities, Alexandrov-Fenchel inequalities, by studying corresponding geometric flows. We will also talk about a related Minkowski type problem and other related problems. The talk is based on a joint work with Chao Xia and other collaborators.

18.04.2024 (O)

Prof. Dr. Karl-Theodor Sturm

(Universität Bonn)

Title: Metric Measure Spaces and Synthetic Ricci Bounds

Abstract: Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. The talk will provide a brief introduction to the concept of uniform lower Ricci bounds as introduced by Lott–Villani and myself, and illustrate some of its geometric, analytic, and probabilistic consequences.
Finally, I will present the more general concept of metric measure spaces (X,d,m) with distribution-valued lower Ricci bounds

• for which we prove the equivalence with sharp gradient estimates,
•  the class of which will be preserved under time changes with arbitrary Lipschitz functions, and
•  which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets of X.

Dr. Pak-Yeung Chan

(UC San Diego/Warwick University)

Title: Gap theorem for nonnegatively curved manifolds

Abstract: In this seminar, we shall discuss some recent results on the gap theorem of nonnegatively curved manifolds with small curvature in an average integral sense, which can be viewed as a Riemannian analogue of the optimal gap result by Ni on Kaehler manifolds. In dimension 3, we also establish a gap theorem for Ricci nonnegative manifolds with pointwise quadratic curvature decay and fast average integral curvature decay. This talk is based on some joint works with Man-Chun Lee.

Dr. Huy 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.

Dr. Riccardo Tione

(MPI Leipzig)

Title: On the Lawson-Osserman conjecture

Abstract: In 1977, H.B. Lawson and R. Osserman conjectured that Lipschitz maps which are critical with respect to outer variations of the area functional are also critical with respect to domain variations. In this talk I will present a solution to this conjecture in the planar case. This result was obtained in collaboration with J. Hirsch and C. Mooney.

Prof. Dr. Christian Bär

(Universität Potsdam)

Title: A holographic index theorem and applications to scalar curvature geometry

Abstract: In the first part of the talk I will discuss a "holographic" index theorem for compact manifolds with boundary. It relates the index of a boundary value problem to the index of an operator on the boundary. It will then be applied to scalar curvature geometry. We will show a ridigidity theorem for scalar curvature on certain warped product spaces. This implies, in particular, Llarull's theorem for the punctured sphere, as conjectured by Gromov.
This is based on joint work with Simon Brendle, Bernhard Hanke, and Yipeng Wang.

Prof. Dr. Luciano Mari

(University of Milan)

Prof. Dr. Boris Vertman

(Universität Oldenburg)

Title: Symmetric Instantons

Abstract: With Spencer Whitehead, we developed a systematic framework to study instantons on R^4 that are invariant under groups of isometries. In this presentation, I will describe this framework and some results obtained using it.

06.12.2023

Unusual Day and Location: Welfengarten 1 (Hauptgebäude), G005

Dr. Enno Keßler (Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig)

Title: Super J-holomorphic curves

Abstract: J-holomorphic curves or pseudoholomorphic curves are maps from Riemann
surfaces to almost Kähler manifolds satisfying the Cauchy-Riemann equations.
J-holomorphic curves are of great interest because they allow to construct
invariants of symplectic manifolds and those invariants are deeply related to
topological superstring theory. A crucial step towards Gromov–Witten
invariants is the compactification of the moduli space of J-holomorphic curves
via stable maps.

In this talk, I want to report on a supergeometric generalization of J-
holomorphic curves and stable maps where the domain is a super Riemann
surface. Super Riemann surfaces have first appeared in superstring theory as
generalizations of Riemann surfaces with an additional anti-commutative
variable. Super J-holomorphic curves are solutions to a system of partial
differential equations on the underlying Riemann surface coupling the Cauchy-
Riemann equation with a Dirac equation for spinors. I will explain how to
construct moduli spaces of super J-holomorphic curves, their compactification
via super stable maps in genus zero and hint at a possible generalization of
Gromov-Witten invariants.

Title: A gluing construction for complex surfaces with hyperbolic cusps

Abstract: We will describe an example of a degeneration of degree 6 algebraic surfaces in CP^3 with only ordinary triple point singularities on its central fiber. Then we will show how the unique negative Kähler-Einstein metrics on the smooth fibers, which exist by the Aubin-Yau theorem, disintegrate into three distinct geometric pieces on approach to the central fiber: (1) Kobayashi's complete Kähler-Einstein metric on the complement of the triple points, (2) long thin neck regions, and (3) Tian-Yau's complete Ricci-flat Kähler metrics in small neighborhoods of the vanishing cycles. Joint work with Xin Fu and Xumin Jiang.

Title: Translators in Lagrangian mean curvature flow

Abstract: Lagrangian mean curvature flow is potentially a powerful tool in solving problems in symplectic topology.  One of the key challenges is the understanding of formation of singularities, which is conjectured to have links to J-holomorphic curves, stability conditions and the Fukaya category.   Unlike the usual mean curvature flow for hypersurfaces, here one is expected to have to tackle singularities modelled on translating solutions to the flow.   I will describe joint work with Felix Schulze and Gabor Szekelyhidi which allows one to recognize a singularity model in Lagrangian mean curvature flow as a translator - this is the first such result in any form of mean curvature flow beyond curves.

Title: Curvature varifolds with orthogonal boundary.

Abstract: We consider surfaces in a domain with orthogonal boundary condition. The problem to obtain mass bounds in terms of curvature bounds leads to a varifold setting (joint work with Marius Mueller).

Prof. Dr. Oliver Schnürer (Konstanz

Title: Unbounded solutions to mean curvature flow

Abstract:We discuss several aspects of complete noncompact solutions to graphical mean curvature flow in Euclidean space in codimension one:  
1) different variants of stability and instability and 
2) beauty and the beast in mean curvature flow without singularities corresponding to nice and pathological solutions that are graphical over proper subsets of Euclidean space.

Prof. Tobias Lamm

(KIT Karlsruhe)

Dr. Athanasios Chatzikaleas

(Uni Münster)

24.11.2022
(O)

Prof. Umberto Hryniewicz

(RWTH Aachen)

Systolic inequalities in Metric Symplectic Geometry

Abstract: The prototypical question of metric systolic geometry is to give upper bounds on the length of the shortest closed geodesics on a closed Riemannian manifold in terms of its volume. Systolic questions have a natural generalization to conservative dynamical systems, in which closed geodesics are replaced by periodic orbits, length by period and volume by phase space volume. This generalization turns out to be quite fruitful: on the one hand, symplectic methods allow us to solve some long standing questions in metric systolic geometry, on the other hand many interesting new questions arise. These new questions are related to challenging open problems in symplectic and in convex geometry. This talk is based on joint works with Gabriele Benedetti, Barney Bramham, Umberto Hryniewicz and Pedro Salomão.

Phase field models with connectedness constrains

Local and Global Scalar curvature rigidity of Einstein Manifolds

09.05.2019
(M,S)