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"Conformal geometry"
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Asymptotic Counting in Conformal Dynamical Systems
In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic
subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the
former.
We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic
orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features
of the distribution of their weights.
These results have direct applications to a wide variety of examples, including the case of
Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions,
Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known
results and proves new results.
Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius
operators and Tauberian theorems as used in classical problems of prime number theory.
eBook
Conformal symmetry breaking differential operators on differential forms
We study conformal symmetry breaking differential operators which map differential forms on
eBook
The power of geometric algebra computing
\"The Power of Geometric Algebra Computing for Engineering and Quantum Computing is based on GAALOPWeb, a new user-friendly, web-based tool for the generation of optimized code for different programming languages as well as for the visualization of Geometric Algebra algorithms for a wide range of engineering applications. This book includes applications from the fields of computer graphics, robotics and quantum computing and will help students, engineers and researchers interested in really computing with Geometric Algebra\"-- Provided by publisher.
Book
On the stability of free boundary minimal submanifolds in conformal domains
Given an n -dimensional Riemannian manifold with non-negative sectional curvature and convex boundary, that is conformal to a Euclidean convex bounded domain, we show that it does not contain any compact stable free boundary minimal submanifold of dimension 2 k n-2 , provided that either the boundary is strictly convex with respect to any of the two metrics or the sectional curvature is strictly positive.
Journal Article
Geometric pressure for multimodal maps of the interval
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting
of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric
Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized
multimodal maps, that is smooth maps
eBook
The Ambient Metric (AM-178)
This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric inn+2dimensions that encodes a conformal class of metrics inndimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric inn+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics.
The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.
eBook
Holography of Higher Codimension Submanifolds: Riemannian and Conformal
We provide a natural generalization to submanifolds of the holographic method used to extract higher-order local invariants of both Riemannian and conformal embeddings, some of which depend on a choice of parallelization of the normal bundle. Qualitatively new behavior is observed in the higher-codimension case, giving rise to new invariants that obstruct the order-by-order construction of unit defining maps. In the conformal setting, a novel invariant (that vanishes in codimension 1) is realized as the leading transverse-order term appearing in a holographically-constructed Willmore invariant. Using these same tools, we also investigate the formal solutions to extension problems off of an embedded submanifold.
Journal Article
The Dirichlet-to-Neumann map for Poincaré–Einstein fillings
We study the non-linear Dirichlet-to-Neumann map for the Poincaré–Einstein filling problem. For even-dimensional manifolds, the range of this non-local map is described in terms of a rank-two “Dirichlet-to-Neumann tensor” along the boundary determined by the Poincaré–Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincaré–Einstein metrics. We construct natural “Dirichlet-to-Neumann hypersurface invariants” that are conformally invariant and recover all Dirichlet-to-Neumann tensors. We give an explicit formula for these hypersurface invariants and use a new vanishing result for odd order T -curvatures to show that they are the unique, natural conformal hypersurface invariant of transverse order equalling the boundary dimension. We also construct such conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré–Einstein fillings for odd-dimensional manifolds with conformally flat boundary.
Journal Article