Christian Bär
Wintersemester 2011/2012
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Donnerstag, 14:15-15:45
Haus 8, Raum 0.53
| Datum | Vortrag | Referent | Inhalt |
| 20.10.11 | Curvature estimates for cylindrically bounded submanifolds | Pacelli Bessa | {tex} We give sectional curvature estimates for complete properly immersed submanifolds $\varphi : M^{m} \to N^{n-l}}\times \mathbf^{R}^k$, $n+k \leq 2m -1$. {/tex} |
| 27.10.11 | Dirac-harmonic maps and indes theory | Bernd Ammann | {tex} \textit{joint work with Nicolas Ginoux, Regensburg}\\ We prove existence results for Dirac-harmonic maps using index theoretical tools. Let $M$ and $N$ be closed Riemannian manifolds, and assume that $M$ is equipped with a spin structure. The fermionic energy functional $E(f,\phi):=\int_M |df|^2 + \langle D^{f^*TN}\psi,\psi\rangle$ is defined for maps $f:M\to N$ and a twisted spinor $\psi\in\Gamma(\Sigma M\otimes f^*TN)$. The operator $D^{f^*TN}$ denotes the associated twisted Dirac operator. The fermionic energy functional is the super-manifold analogue of the classical energy of a map $f:M\to N$. Stationary points of the fermionic energy functional are called Dirac-harmonic maps. The case ${\rm dim}\; M=2$ is of particular interest. On the one hand, the fermionic energy is invariant under conformal changes of $M$, and as a consequence sphere-bubble phenomena may appear which make analytic statements harder or more interesting. On the other hand this dimension is a mathematical model for strings in a riemannian space-time, usually called non-linear sigma-models in physics. We will use the fact that the kernel of $D^{f^*TN}$ defines an index ${\rm ind}D^{f^*TN}\in KO_2(point) \cong \mathbb{Z}_2$. We use this index to prove the existence of non-trivial Dirac-harmonic maps. Our proofs also rely on known existence results for harmonic maps in the classical sense (e.g. by Sacks-Uhlenbeck, Parker, Eells-Sampson). For example we will see: if $N$ is not spin, but if the universal covering of $N$ is spin, then there is a non-constant harmonic map $f:S^2\to N$ for which there is a $4$-dimensional family of Dirac-harmonic maps of the form $(f,\psi)$. We obtain several similar conclusions under other geometric conditions. {/tex} |
| 03.11.11 | Properties of the Dirac spectrum on lens spaces | Sebastian Boldt | The spectrum of the (classical) Dirac-Operator on the round sphere is very well known. On spherical space forms the eigenvalues are the same, but their multiplicities are in general smaller. Encoding these into complex power series defines holomorphic functions on the interior of the unit disc which extend to meromorphic funtions on the whole complex plane. The spectrum of the Dirac-Operator determines these generating functions and vice versa. We will investigate the properties of the generating functions on lens spaces, certain spherical space forms with cyclic fundamental group, showing that the order of the fundamental group of a lens space as well as the isometry class of a homogeneous lens space are spectral invariants. We will have a closer look at three dimensional lens spaces, specifying a number theoretic system of equations which holds for isospectral lens spaces that stems from the residues of the generating functions at their poles. By solving this system of equations in the case that the order of the fundamental group is a prime number, we prove that three dimensional isospectral lens spaces with the order of their fundamental group a prime number are isometric. |
| 10.11.11 | Poisson algebraic property | Joakim Arnlind | We show that the differential geometry of almost Kähler submanifolds can be formulated in terms of the Poisson structure induced by the inverse of the Kähler form. More precisely, the submanifold relations, such as Gauss' and Weingarten's equations, (as well as many other objects) can be expressed as Poisson brackets of the embedding coordinates. It is then natural to ask the following question: Are there abstract Poisson algebras for which such equations hold? We answer this question by introducing Kähler-Poisson algebras and show that an affine connection can be defined, fulfilling all the desired symmetries and relations, e.g. the Bianchi identities. Furthermore, as an illustration of the new concepts we derive algebraic versions of some well known theorems in differential geometry. In particular, we prove (in the purely algebraic setting) that Schur's lemma holds and that a lower bound on the Ricci curvature induces a bound on the eigenvalues of the Laplace operator. |
| 17.11.11 | Surfaces in 3-dimensional Thurston geometries via Spin^c spinors | Roger Nakad | {tex} Simply connected 3-dimensional homogeneous manifolds $\mathbb{E}(\kappa, \tau)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $\mathbb{E}(\kappa, \tau)$. As application, we get an elementary proof of a generalized Lawson correspondence for constant mean curvature surfaces in $\mathbb{E}(\kappa, \tau)$. {/tex} |
| 24.11.11 | The evolution equations for Dirac-harmonic maps from surfaces | Volker Branding | In this talk we will consider Dirac-harmonic maps from surfaces. After introducing the notion of Dirac-harmonic maps we will summarize the current known results for Dirac-harmonic maps from surfaces. The main part of the talk will then focus on their evolution equations. |
| 01.12.11 | Variational stability of Einstein metrics | Klaus Kröncke | We will study the Einstein Hilbert functional on compact Riemannian manifolds. Particularly, we will study the second variation of the functional on its critical points. After introducing some notions of stability and discussing examples, specific attention will be paid on manifolds which admit a Killing spinor. |
| 08.12.11 | The evolution equation for magnetic harmonic maps | Immanuel Asmus | Magnetic harmonic maps are defined as a generalization of harmonic maps. We will summarize the known results for magnetic harmonic maps. In the main part of the talk we focus on their evolution equations. |
| 15.12.11 | Asymptotic behavior of eigenvalues and eigenfunction in the semiclassical limit | Matthias Ludewig | {tex} Let $L$ be a self-adjoint Laplace type operator on a vector bundle $E$ over a compact riemannian manifold $M$. We are interested in the eigenvalues and eigenfunctions of the operator $\hbar^2 L + V$ in the semiclassical limit $\hbar \to 0$, where $V$ is a symmetric endomorphism field of $E$. At least in special cases, we derive asymptotic expansions for both and look at the relations between the semiclassical behavior and topology of $M$. {/tex} |
| 02.02.12 | Higher Symmetries and their classification | Christoph Wockel | We will describe how Lie 2-groups and the associated notion of principal 2-bundles can be used to realise degree three cohomology classes geometrically. One important example will be the canonical 3-class on a compact simple Lie group. The geometric representative has a Lie 2-group structure itself, which is known as the String 2-group. |
| 09.02.12 | Spectral Functions on Teichmueller Space: Analysis and Computations | Alexander Strohmaier | I will briefly discuss two simple analytical estimates for eigenfunctions that allow to use computer algebra to establish eigenvalue inclusions and non-inclusions on hyperbolic surfaces within small intervals. It is then demonstrated how the spectral determinant and other values of the spectral Zeta functions can be computed to high accuracy for a given point in Teichmueller space. The implemented numerical algorithm is fast enough to investigate the behaviour of the spectral functions as functions on Teichmueller space. Other applications such as wave propagation in negative curvature and long time behavior of the geodesic flow is also discussed. If time allows I will demonstrate our free Fortran program that computes eigenvalues on genus 2 hyperbolic surfaces. |
| 08.03.12 | The space of metrics of positive scalar curvature | Thomas Schick | Fix a compact smooth manifold without boundary.
One of the intensively studied questions of global Riemannian geometry is: "is there a metric of positive scalar curvature on M, and if so, how many". A more precise variant of the second part is: what can one say about the space of metrics of positive scalar curvature, if it is not empty. We will show how one can use methods from index theory (of the Dirac operator) to give answers to this question, and will point to very recent constructions of Crowley-S. and Hanke-S.-Steimle of examples where these methods can indeed be applied. (Haus 8, Raum 0.59) |
| 08.03.12 | Mathematical aspects of the supergravity r- and c-maps | Vicente Cortes | I will explain these constructions for mathematicians and present some new results and open problems.
In particular, I will show how to construct explicit examples of complete quaternionic Kähler manifolds from cubic surfaces. (Haus 8, Raum 0.59) |
| 22.03.12 | Torsion, Bochner formulas and Applications | Frank Pfäffle | I will report on a joint project with Christoph Stephan. We consider Dirac operators induced by torsion connections and compute the Chamseddine-Connes spectral action for the corresponding generalized Laplacians. Incorporating chiral projections produces actions used in Loop Quantum Gravity. (Haus 8, Raum 0.53) |
| 22.03.12 | About eta-invariants | Werner Ballmann | I will explain how eta-invariants enter into index formulas concerning certain noncompact manifolds and will present some explicit results. (Haus 8, Raum 0.53) |