 |
Universität Augsburg
|
|
|
Universität Augsburg
|
|
Professor Dr. Bernhard Hanke
Universität Augsburg
spricht am
Montag, 12. Oktober 2026
um
16:00 Uhr
im
Raum 2004 (L1)
über das Thema:
| Abstract: |
|
Let $M$ be a closed, connected smooth manifold of dimension at least $2$. In the first part of the talk, we will discuss results on the homotopy type of the space of Riemannian metrics of positive scalar curvature on $M$, developed over the past several years by various authors. Combining methods from geometric analysis and geometric topology, in particular surgery and bordism theory, one finds that, whenever this space is nonempty, its homotopy type can be remarkably rich. In the second part of the talk, I will introduce a spectral generalization of positive scalar curvature. More precisely, for a positive real parameter $\gamma$, we say that a Riemannian metric $g$ has positive spectral scalar curvature if the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{scal}_g$ is strictly positive. I will report on recent joint work with Gioacchino Antonelli and Georg Frenck, in which we determine the homotopy type of the space of metrics with positive spectral scalar curvature. Depending on $\gamma$, this space is either homotopy equivalent to the space of metrics of positive scalar curvature or contractible. Furthermore, the threshold value of $\gamma$ at which this ''phase transition'' occurs is precisely the value appearing in the classical conformal Laplacian. In particular, our result provides a homotopy-theoretic strengthening of a conjecture of Gromov from his Four Lectures on Scalar Curvature, in the maximal possible range of the coefficient $\gamma$. |
| Hierzu ergeht herzliche Einladung. |
Kaffee, Tee und Gebäck eine halbe Stunde vor Vortragsbeginn im Raum 2006 (L1).