Results 1  10
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36
The WeilPetersson metric and volumes of 3dimensional hyperbolic convex cores
 J. Amer. Math. Soc
, 2001
"... Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of ..."
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Cited by 44 (6 self)
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Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of hyperbolic 3manifolds
Fredholm operators and Einstein metrics on conformally compact manifolds
"... Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, t ..."
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Cited by 42 (2 self)
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Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for selfadjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.
Complete Manifolds With Positive Spectrum, II
, 2003
"... In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for ndimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of ..."
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Cited by 27 (11 self)
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In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for ndimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of a theorem of Cheng.
the spectrum of an asymptotically hyperbolic Einstein manifold
 Comm. Anal. Geom
, 1995
"... Abstract. This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its “ideal boundary” at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an (n + 1)dimensional manifold is the ra ..."
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Cited by 24 (3 self)
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Abstract. This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its “ideal boundary” at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an (n + 1)dimensional manifold is the ray [n 2 /4, ∞), with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the continuous spectrum. The main result of this paper is that, if the Yamabe invariant of the conformal structure on the boundary is nonnegative, then there are no such eigenvalues. This generalizes results of R. Schoen, S.T. Yau, and D. Sullivan for the case of hyperbolic manifolds. 1.
Hausdorff dimension and conformal dynamics III: Computation of dimension
 Amer. J. Math
"... This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Di ..."
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Cited by 21 (5 self)
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This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials fc(z) = z 2 + c, c ∈ [−1, 1/2]; and (c) the family of rational maps ft(z) = z/t+ 1/z, t ∈ (0, 1]. We also calculate H. dim(Λ) ≈ 1.305688 for the Apollonian gasket, and H. dim(J(f)) ≈ 1.3934 for Douady’s rabbit, where f(z) = z 2 + c
Comparison theorem for Kähler manifolds and positivity of spectrum
 J. DIFFERENTIAL GEOM
, 2005
"... The first part of this paper is devoted to proving a comparison theorem for Kähler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are CP m, Cm,andCH m. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from ..."
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Cited by 16 (8 self)
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The first part of this paper is devoted to proving a comparison theorem for Kähler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are CP m, Cm,andCH m. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m2 for a complete, mdimensional, Kähler manifold with holomorphic bisectional curvature bounded from below by −1. The second part of the paper is to show that if this upper bound is achieved and when m = 2, then it must have at most four ends.
Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups, preprint Geometry
 Topology Monographs, Volume 1 (1998) James W Anderson
"... This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H 3 /Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3 /Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topolog ..."
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Cited by 12 (0 self)
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This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H 3 /Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3 /Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H. dim(Λ) < 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higherdimensional manifolds. Applications
Conformal and Harmonic Measures on Laminations Associated with Rational Maps
, 2002
"... The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination A and the associated hyperbolic 3lamination H endowed with an action of ..."
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Cited by 12 (4 self)
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The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination A and the associated hyperbolic 3lamination H endowed with an action of a discrete group of isomorphisms.
Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Preprint available at http://www.math.psu.edu/sarig
"... Abstract. We describe the ergodic invariant Radon measures for the horocycle flow on general (infinite) regular covers of finite volume hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the covering surface. ..."
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Cited by 9 (1 self)
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Abstract. We describe the ergodic invariant Radon measures for the horocycle flow on general (infinite) regular covers of finite volume hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the covering surface. 1.
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 7 (4 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.