Georg Frenck

• PostDoc Position at University of Augsburg

We show that in every dimension $n \geq 8$, there exists a smooth closed manifold $M^n$ which does not admit a smooth positive scalar curvature ("psc") metric, but $M$ admits an $\mathrm{L}^\infty$-metric which is smooth and has psc outside a singular set of codimension $\geq 8$. This provides counterexamples to a conjecture of Schoen. In fact, the...

In this article we study the space of positive scalar curvature metrics on totally nonspin manifolds with spin boundary. We prove that for such manifolds of certain dimensions, those spaces are not connected and have nontrivial fundamental group. Furthermore we show that a well-known propagation technique for detection results on spaces of positive...

In this article we study the space of positive scalar curvature metrics on totally nonspin manifolds with spin boundary. We prove that for such manifolds of certain dimensions, those spaces are not connected and have nontrivial fundamental group. Furthermore we show that a well-known propagation technique for detection results on spaces of positive...

We present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M . This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high d...

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{ams...

Given a manifold $M$, we completely determine which rational $\kappa$-classes are non-trivial for (fiber homotopy trivial) $M$-bundles over the $k$-sphere, provided that the dimension of M is large compared to $k$. We furthermore study the vector space of these spherical $\kappa$-classes and give an upper and a lower bound on its dimension. The pro...

We construct and study an H H -space multiplication on R + ( M ) {\mathcal {R}}^+(M) for manifolds M M which are nullcobordant in their own tangential 2 2 -type. This is applied to give a rigidity criterion for the action of the diffeomorphism group on R + ( M ) {\mathcal {R}}^+(M) via pullback. We also compare this to other known multiplicative st...

This is an expository article without any claim of originality. We give a complete and self-contained account of the Gromov–Lawson–Chernysh surgery theorem for positive scalar curvature metrics.

We prove the existence of elements of infinite order in the homotopy groups of the spaces $\mathcal{R}_{Ric>0}(M)$ and $\mathcal{R}_{sec>0}(M)$ of positive Ricci and positive sectional curvature, provided that $M$ is high-dimensional and Spin, admits such a metric and has a non-vanishing rational Pontryagin class.

We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing $\hat{A}$-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all ${{\operatorname{Spin}}}$ manifolds of dimension 6 or at least 10, which admit such...

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

We construct smooth bundles with base and fiber products of two spheres whose total spaces have non-vanishing $\hat{A}$-genus. We then use these bundles to locate non-trivial rational homotopy groups of the space of metrics of positive Ricci curvature for all manifolds of dimension $6$ or at least $10$ which admit such a metric and are a connected...

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

We construct and study an $H$-space multiplication on $\mathcal R^+(M)$ for manifolds $M$ which are nullcobordant in their own tangential $2$-type. This is applied to give a rigidity criterion for the action of the diffeomorphism group on $\mathcal R^+(M)$ via pullback. We also compare this to other known multiplicative structures on $\mathcal R^+(...

We present a rigidity theorem for the action of the mapping class group $\pi_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is f...

In this article, we give a complete and self--contained account of Chernysh's strengthening of the Gromov--Lawson surgery theorem for metrics of positive scalar curvature. No claim of originality is made.