In negatively curved 3-manifolds, the marked p-energy spectrum and the modified energy entropy of k-surfaces are rigid: equality or asymptotic behavior forces the ambient curvature to be constant.
A counterexample to marked length spectrum semi-rigidity
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abstract
Given a closed orientable negatively curved Riemannian surface $(M,g)$, we show how to construct a perturbation $(M,g^\prime)$ such that each closed geodesic becomes longer, and yet there is no diffeomorphism $f : (M,g^\prime) \rightarrow (M,g)$ which contracts every tangent vector.
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Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces
In negatively curved 3-manifolds, the marked p-energy spectrum and the modified energy entropy of k-surfaces are rigid: equality or asymptotic behavior forces the ambient curvature to be constant.