arXiv subjects

Karen Butt

Publications and source records attributed to Karen Butt.

6 recordsLinked to original sources

arXiv2025

Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric

In 2004, Manning showed that the topological entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly decreasing along the normalized Ricci flow, and he asked if an analogous result holds in higher dimensions for metrics in a neighborhood of a hyperbolic metric. In this paper, we affirmatively answer this question. Namely, we show that the topological entropy of the geodesic flow of a closed Riemannian manifold that carries a hyperbolic metric is indeed strictly decreasing along the normalized Ricci flow starting from a metric of variable negative sectional curvature sufficiently close to the hyperbolic metric.

math.DG↗

arXiv2025

Marked Poincar\'e rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3

Let $M$ be a compact manifold without boundary equipped with a Riemannian metric $g$ of negative curvature. In this paper, we introduce the marked Poincar\'e determinant (MPD), a homothety invariant of $g$ depending on differentiable periodic data of its geodesic flow. The MPD associates to each free homotopy class of closed curves in $M$ a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. We prove a local MPD rigidity result in dimension 3: if $g$ is sufficiently close to a hyperbolic metric $g_0$ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of $g_0$ is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.

math.DG↗

arXiv2025

Quantitative marked length spectrum rigidity for surfaces

We consider a closed negatively curved surface $(M, g)$ with marked length spectrum sufficiently close (multiplicatively) to that of a hyperbolic metric $g_0$ on $M$. We show there is a smooth diffeomorphism $F:M \to M$ with derivative bounds close to 1, depending on the ratio of the two marked length spectrum functions. This is a two-dimensional version of our main result in [But25b].

math.DG↗

arXiv2025

Monotonicity of the Liouville entropy along the Ricci flow on surfaces

We show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is eventually strictly increasing along the normalized Ricci flow (NRF). More precisely, we obtain a new expression for the derivative of the Liouville entropy along an arbitrary conformal deformation in dimension 2, and we prove it is positive in the direction of the NRF for 1/6-pinched metrics. This partially answers a question of Manning from 2004. In addition, we show that the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is strictly increasing along the NRF starting from any metric of non-constant negative curvature.

math.DS↗

arXiv2022

Approximate control of the marked length spectrum by short geodesics

The marked length spectrum (MLS) of a closed negatively curved manifold $(M, g)$ is known to determine the metric $g$ under various circumstances. We show that in these cases, (approximate) values of the MLS on a sufficiently large finite set approximately determine the metric. Our approach is to recover the hypotheses of our main theorems in arXiv:2203.12128, namely multiplicative closeness of the MLS functions on the entire set of closed geodesics of $M$. We use mainly dynamical tools and arguments, but take great care to show the constants involved depend only on concrete geometric information about the given Riemannian metrics, such as the dimension, sectional curvature bounds, and injectivity radii.

math.DG↗

arXiv2022

Quantitative marked length spectrum rigidity

We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenst\"adt, we show the volumes of $M$ and $N$ are approximately equal. We then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenst\"adt and Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh.

math.DG↗