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Gerhard Knieper

Publications and source records attributed to Gerhard Knieper.

At least 19 recordsLinked to original sources
arXiv2025

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

In this article we study geodesic flows on closed Riemannian manifolds without conjugate points and divergence property of geodesic rays. If the fundamental group is Gromov hyperbolic and residually finite we prove, under appropriate assumptions on the expansive set, that the geodesic flow has a unique measure of maximal entropy. This generalizes corresponding results of Climenhaga, Knieper and War proved under the stronger assumption of the existence of a background metric of negative sectional curvature.

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arXiv2025

Horospherical mean curvature functions and D'Atri spaces

We consider simply connected Riemannian manifolds without conjugate points for which the horospherical mean curvature function is continuous, reversible and invariant under the geodesic flow. We show under mild additional curvature tensor conditions that rank one manifolds in this family are automatically asymptotically harmonic. In particular, compact rank one manifolds of this kind must be locally symmetric spaces of negative curvature. Moreover, we show under the same conditions that rank one D'Atri spaces without conjugate points are harmonic. An earlier result of this type was proved by Druetta for certain homogeneous D'Atri spaces.

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arXiv2025

Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature

In his seminal work on Teichm\"uller spaces (\cite{Th98}), Thurston introduced the maximal stretch for a pair of hyperbolic metrics on a closed surface of genus $\geq 2$ and showed that the logarithm of this quantity induces an asymmetric metric in the Teichm\"uller space. He also showed that the subset of the surface on which the maximal stretch is attained is a geodesic lamination. In this paper, we define the maximal stretch analogously for closed manifolds equipped with Riemannian metrics of variable negative curvature and investigate the structure of the related Mather set on the unit tangent bundle. In contrast to the Teichm\"uller space, the Mather set may not be lifts of geodesic laminations in this broader setting. However, in our paper, we will discuss similar features shared by the Mather set with geodesic laminations. We also connect the study of the Mather set with the theory of best Lipschitz maps.

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arXiv2024

Surfaces of section for geodesic flows of closed surfaces

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $Σ$ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partialΣ$ as limit sets of the orbits of the geodesic flow that do not return to $Σ$. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.

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arXiv2022

Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a $C^2$ open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are $C^2$ stably ergodic if and only if they are Anosov.

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arXiv2021

Closed geodesics on surfaces without conjugate points

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.

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arXiv2021

Geodesic stretch, pressure metric and marked length spectrum rigidity

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in \cite{Guillarmou-Lefeuvre-18} and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, that reduces to the Weil-Peterson metric in the case of Teichmüller space and is related to the works of \cite{McMullen,Bridgeman-Canary-Labourie-Sambarino-15}.

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arXiv2020

Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.

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arXiv2019

The Fourier transform on harmonic manifolds of purely exponential volume growth

Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $ρ= h/2$. Fixing a basepoint $o \in X$, for $ξ\in \partial X$, denote by $B_ξ$ the Busemann function at $ξ$ such that $B_ξ(o) = 0$. then for $λ\in \C$ the function $e^{(iλ- ρ)B_ξ}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(λ^2 + ρ^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(λ, ξ) := \int_X f(x) e^{(-iλ- ρ)B_ξ(x)} dvol(x)$$ for all $λ\in \C, ξ\in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(λ, ξ) e^{(iλ- ρ)B_ξ(x)} dλ_o(ξ) |c(λ)|^{-2} dλ$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $λ_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.

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arXiv2019

Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $α\in [0,π/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $α= 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the Ricci curvature of these hypersurfaces and show that all hypersurfaces for $α\in (0,\fracπ{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.

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arXiv2014

Harmonic functions on rank one asymptotically harmonic manifolds

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds $X$ with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon-Nykodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification $\bar{X}$.

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arXiv2014

Noncompact asymptotically harmonic manifolds

In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. We prove the following equivalences for asymptotically harmonic manifolds $X$ under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded: (a) $X$ has rank one; (b) $X$ has Anosov geodesic flow; (c) $X$ is Gromov hyperbolic; (d) $X$ has purely exponential volume growth with volume entropy equals $h$. This generalizes earlier results by G. Knieper for noncompact harmonic manifolds and by A. Zimmer for asymptotically harmonic manifolds admitting compact quotients.

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arXiv2013

Topological entropy of minimal geodesics and volume growth on surfaces

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of (M, g) generalizing work of Freire and Mane.

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arXiv2013

Noncompact harmonic manifolds

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. The classification of all noncompact harmonic spaces is still a very difficult open problem. In this paper we provide a survey on recent results on noncompact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.

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arXiv2011

Minimal geodesic foliation on T^2 in case of vanishing topological entropy

On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all $r \in \mathbb{R} \cup \{\infty\}$ the universal cover $\Br^2$ is foliated by minimal geodesics of rotation number $r$. For irrational $r \in \mathbb{R}$ all geodesics are minimal, for rational $r \in \mathbb{R} \cup \{\infty\}$ all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.

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arXiv2010

Characterization of geodesic flows on T^2 with and without positive topological entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering $\Br^2$. Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on $T^2$.

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