arXiv subjects

Frieder Jäckel

Publications and source records attributed to Frieder Jäckel.

6 recordsLinked to original sources

arXiv2026

Constant potentials do not minimize the fundamental gap on convex domains in negatively curved Hadamard manifolds

We show that for every negatively curved Hadamard manifold $X$ and every $D > 0$ there exists a convex domain $Ω\subseteq X$ with diameter $D$ and a convex potential $V$ on $Ω$ such that the fundamental gap of the operator $-Δ+V$ is strictly smaller than the fundamental gap of $-Δ$. This shows that the second part of the fundamental gap conjecture is wrong in every negatively curved manifold. This is significantly harder than in the previously known case of hyperbolic space because, due to the lack of symmetry, one has to study a true PDE, and not just an ODE.

math.AP↗

arXiv2025

Effective stability of negatively curved Einstein metrics in dimensions at least $4$

We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in a suitable sense, then it admits a genuine Einstein metric of negative sectional curvature. Importantly, the pinching constant measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.

math.DG↗

arXiv2025

Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space

We show that for every $n \geq 2$ and $D > 0$ there exist a convex domain $Ω\subseteq \mathbb H^n$ with diameter $D$ and a convex potential $V$ on $Ω$ such that the fundamental gap of the operator $-Δ+V$ is strictly smaller than the fundamental gap of $-Δ$. In comparison to previous work, this result requires more refined control of the eigenfunctions.

math.AP↗

arXiv2025

Negatively curved Einstein metrics on Gromov-Thurston manifolds

For every $n\geq 4$ we construct infinitely many mutually not homotopic closed manifolds of dimension $n$ which admit a negatively curved Einstein metric but no locally symmetric metric.

math.DG↗

arXiv2023

Improved decay rate in a stability theorem for hyperbolic metrics

Recently, Ursula Hamenstädt and the author proved a stability result for finite volume hyperbolic metrics in dimension three that does not assume any upper volume bounds, but that requires an exponentially fine control of the metric in the thin part of the manifold. We use a bootstrap argument to extend the result allowing for a weaker exponential control of the metric. This is achieved by formulating an abstract axiomatic framework.

math.DG↗

arXiv2022

Stability of Einstein metrics and effective hyperbolization in large Hempel distance

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $C^{2,α}$-topology. In dimension $3$ the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of $3$-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many cusps and tubes.

math.DG↗