scholarly journals Partial Marked Length Spectrum Rigidity Of Negatively Curved Surfaces

2020 ◽  
Author(s):  
Noelle Sawyer
Keyword(s):  
Length Spectrum ◽  
Curved Surfaces ◽  
Negatively Curved ◽  
Marked Length Spectrum

scholarly journals Marked length rigidity for Fuchsian buildings

2018 ◽  
Vol 39 (12) ◽  
pp. 3262-3291
Author(s):  
DAVID CONSTANTINE ◽  
JEAN-FRANÇOIS LAFONT
Keyword(s):  
Automorphism Group ◽  
Riemannian Metrics ◽  
Length Spectrum ◽  
Negatively Curved ◽  
Spectrum Function ◽  
Marked Length Spectrum

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

Ergodic theory and rigidity on the symmetric space of non-compact type

2001 ◽  
Vol 21 (1) ◽  
pp. 93-114 ◽  
Author(s):  
INKANG KIM
Keyword(s):  
Ergodic Theory ◽  
Symmetric Spaces ◽  
Isometry Group ◽  
Length Spectrum ◽  
Compact Type ◽  
Negatively Curved ◽  
Rank One ◽  
Convex Cocompact ◽  
Marked Length Spectrum ◽  
Locally Symmetric Manifolds

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

Rigidity for non-compact surfaces of finite area and certain Kähler manifolds

1995 ◽  
Vol 15 (3) ◽  
pp. 475-516 ◽  
Author(s):  
Jianguo Cao
Keyword(s):  
Length Spectrum ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Finite Area ◽  
Marked Length Spectrum ◽  
Compact Surfaces

AbstractWe first consider the rigidity of the marked length spectrum for non-compact surfaces of finite area.

scholarly journals Approximate solutions of cohomological equations associated with some Anosov flows

1990 ◽  
Vol 10 (2) ◽  
pp. 367-379 ◽  
Author(s):  
Svetlana Katok
Keyword(s):  
Approximate Solutions ◽  
Geodesic Flows ◽  
Curved Surfaces ◽  
Anosov Flow ◽  
3 Dimensional ◽  
Negatively Curved ◽  
Anosov Flows ◽  
Higher Differentiability ◽  
Special Case ◽  
Cohomological Equations

AbstractThe Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

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