scholarly journals Boundary rigidity of negatively-curved asymptotically hyperbolic surfaces

2020 ◽  
Vol 95 (1) ◽  
pp. 129-166
Author(s):  
Thibault Lefeuvre
Keyword(s):  
Hyperbolic Surfaces ◽  
Negatively Curved ◽  
Boundary Rigidity ◽  
Asymptotically Hyperbolic

scholarly journals X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds

2020 ◽  
Vol 69 (7) ◽  
pp. 2857-2919 ◽  
Author(s):  
C. Robin Graham ◽  
Colin Guillarmou ◽  
Plamen Stefanov ◽  
Gunther Uhlmann
Keyword(s):  
Hyperbolic Manifolds ◽  
X Ray ◽  
Ray Transform ◽  
Asymptotically Hyperbolic Manifolds ◽  
Boundary Rigidity ◽  
Asymptotically Hyperbolic

scholarly journals Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

2017 ◽  
Vol 2017 (724) ◽  
Author(s):  
Hiroshi Isozaki ◽  
Yaroslav Kurylev ◽  
Matti Lassas
Keyword(s):  
Inverse Problem ◽  
Riemann Surface ◽  
Inverse Scattering ◽  
Scattering Matrix ◽  
Hyperbolic Surfaces ◽  
Conical Singularities ◽  
Compact Surfaces ◽  
Asymptotically Hyperbolic ◽  
Generalized Scattering Matrix

AbstractWe consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface

scholarly journals Marked boundary rigidity for surfaces

2016 ◽  
Vol 38 (4) ◽  
pp. 1459-1478 ◽  
Author(s):  
COLIN GUILLARMOU ◽  
MARCO MAZZUCCHELLI
Keyword(s):  
Riemannian Metrics ◽  
Compact Surface ◽  
Conjugate Points ◽  
Geodesic Flows ◽  
Convex Boundary ◽  
Strictly Convex ◽  
Negatively Curved ◽  
Boundary Rigidity ◽  
Boundary Distance

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

THE RENORMALIZED VOLUME AND UNIFORMIZATION OF CONFORMAL STRUCTURES

2016 ◽  
Vol 17 (4) ◽  
pp. 853-912 ◽  
Author(s):  
Colin Guillarmou ◽  
Sergiu Moroianu ◽  
Jean-Marc Schlenker
Keyword(s):  
Einstein Metrics ◽  
Conformal Class ◽  
Conformal Boundary ◽  
Negatively Curved ◽  
Non Linear Equation ◽  
Higher Dimensional ◽  
Dimensional Version ◽  
Cotangent Space ◽  
Conformal Structures ◽  
Asymptotically Hyperbolic

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$, we denote it by $\text{Vol}_{R}(M,g;h_{0})$. We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$, satisfied in particular by Einstein metrics. When $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$-Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces

2014 ◽  
Vol 2014 (697) ◽  
pp. 1-14 ◽  
Author(s):  
Sai-Kee Yeung
Keyword(s):  
Number Field ◽  
Rational Points ◽  
Two Dimensional ◽  
Hyperbolic Surfaces ◽  
Integral Points ◽  
Negatively Curved ◽  
Ball Quotients ◽  
Complex Ball ◽  
Compact Surfaces ◽  
Dimensional Ball

AbstractThe first goal of this paper is to study the question of finiteness of integral points on a cofinite non-compact complex two-dimensional ball quotient defined over a number field. Along the process we will also consider some negatively curved compact surfaces. Using some fundamental results of Faltings, the question is to reduce to a conjecture of Borel about existence of virtual holomorphic one-forms on cofinite non-cocompact complex ball quotients. The second goal of this paper is to study the conjecture on such non-compact surfaces.

Double-Helix Supramolecular Nanofibers Assembled from Negatively Curved Nanographenes

2021 ◽  
Author(s):  
Kenta Kato ◽  
Kiyofumi Takaba ◽  
Saori Maki-Yonekura ◽  
Nobuhiko Mitoma ◽  
Yusuke Nakanishi ◽  
...  
Keyword(s):  
Double Helix ◽  
Negatively Curved
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