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"Graham, C. Robin"
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المختصر في الأمراض الجلدية
يقدم هذا الكتاب المادة العلمية في الأمراض الجلدية بصورة مختصرة يسهل على طالب الطب والطبيب الممارس والممرض أن يستفيد منه كمرجع في دراسته وممارسته، كما أن أسلوب عرضه والصور التوضيحية في الكتاب وسياقه للتاريخ الطبيعي للأمراض الجلدية وطرق تشخيصها ووسائل علاجها تعتبر مادة ثرية لكل من يهتم بتثقيف نفسه في هذا المجال حتى من المرضى وغير المتخصصين.
BOOK
The ambient metric
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
A Gauss–Bonnet Formula for the Renormalized Area of Minimal Submanifolds of Poincaré–Einstein Manifolds
Assuming the extrinsic Q-curvature admits a decomposition into the Pfaffian, a scalar conformal submanifold invariant, and a tangential divergence, we prove that the renormalized area of an even-dimensional minimal submanifold of a Poincaré–Einstein manifold can be expressed as a linear combination of its Euler characteristic and the integral of a scalar conformal submanifold invariant. We derive such a decomposition of the extrinsic Q-curvature in dimensions two and four, thereby recovering and generalizing results of Alexakis–Mazzeo and Tyrrell, respectively. We also conjecture such a decomposition for general natural submanifold scalars whose integral over compact submanifolds is conformally invariant, and verify our conjecture in dimensions two and four. Our results also apply to the area of a compact even-dimensional minimal submanifold of an Einstein manifold.
Journal Article
Chern-Gauss-Bonnet Formula for Singular Yamabe Metrics in Dimension Four
We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four-dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Schouten tensor is constant. Extensions and generalizations of these results are discussed. Finally, a general result is proved identifying the infinitesimal anomaly of the renormalized volume of an asymptotically hyperbolic metric in terms of its renormalized volume coefficients, and used to outline alternate proofs of the conformal invariance of the renormalized volume plus a boundary integral.
Journal Article
Minimal area submanifolds in AdS × compact
A
bstract
We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanifold is constrained to be itself a minimal area submanifold. For applications to holography, this tells us what are the allowed “flavor branes” that can be added to a holographic field theory. We also give a compact geometric expression for the spectrum of operator dimensions associated with the slipping modes of the submanifold in the internal space. We illustrate our results with several examples, including some that haven’t appeared in the literature before.
Journal Article
Extrinsic GJMS operators for submanifolds
We derive extrinsic GJMS operators and Q -curvatures associated to a submanifold of a conformal manifold. The operators are conformally covariant scalar differential operators on the submanifold with leading part a power of the Laplacian in the induced metric. Upon realizing the conformal manifold as the conformal infinity of an asymptotically Poincaré–Einstein space and the submanifold as the boundary of an asymptotically minimal submanifold thereof, these operators arise as obstructions to smooth extension as eigenfunctions of the Laplacian of the induced metric on the minimal submanifold. We derive explicit formulas for the operators of orders 2 and 4. We prove factorization formulas when the original submanifold is a minimal submanifold of an Einstein manifold. We also show how to reformulate the construction in terms of the ambient metric for the conformal manifold, and use this to prove that the operators defined by the factorization formulas are conformally invariant for all orders in all dimensions.
Journal Article
Conformal Powers of the Laplacian via Stereographic Projection
A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection mapping.
Journal Article
