scholarly journals The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

2021 ◽  
pp. 1
Author(s):  
Francesco Lin ◽  
Michael Lipnowski
Keyword(s):  
Length Spectrum ◽  
Hyperbolic Three Manifolds

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

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 $X$.

scholarly journals Local Mixing and Invariant Measures for Horospherical Subgroups on Abelian Covers

2018 ◽  
Vol 2019 (19) ◽  
pp. 6036-6088
Author(s):  
Hee Oh ◽  
Wenyu Pan
Keyword(s):  
Lie Group ◽  
Invariant Measures ◽  
Dense Subgroup ◽  
Abelian Cover ◽  
Rank One ◽  
Hyperbolic Three Manifolds ◽  
Abelian Covers ◽  
Prime Geodesic Theorem ◽  
Convex Cocompact ◽  
Frame Flow

Abstract Abelian covers of hyperbolic three-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic three-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as well as to other counting and equidistribution problems. Our results are proved for any abelian cover of a homogeneous space Γ0∖G where G is a rank one simple Lie group and Γ0 < G is a convex cocompact Zariski dense subgroup.

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.

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