Renormalized Volume

We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk geometry, improved discussion of hypersurfaces with boundar

Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a new hypersurface conformal invariant which generalises to higher dimensions the important Willmore invariant of embedded surfaces. We call this the obstruction density. For even dimensional hypersurfaces it is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand machinery. The solution to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers determine an explicit holographic formula for the obstruction density.Comment: 37 pages, LaTeX, abridged version, functionals and explicit invariants from previous version treated in greater detail in another postin

Ballistic missile defence: how soon, how significant, and what should Australia's policy be?

Summary: The issue of ballistic missile defence (BMD) was a controversial one when US President Reagan first advocated a strategic-level system in the early 1980s. It remains so today. What’s Australia’s interest? We live a long way away from most current ballistic missile arsenals. But the ADF frequently deploys within range of ballistic missile systems, especially in Northeast Asia or the Middle East, and those systems might proliferate more widely in the future. The paper considers the two questions we need to decide. The first is the priority for enhancing the ADF’s own BMD capabilities. The second is whether it makes sense for us to participate in a cooperative arrangement with the US or other partners

Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of [31] to a conformally invariant, Poincare--Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulae for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of [9]. This complements the results of Aubry and Guillarmou [3] where associated conformal harmonic spaces parametrise smooth solutions.Comment: 85 pages, LaTeX, typos corrected, references added, to appear in Memoirs of the AM

Diehard Rebels: The Confederate Culture of Invincibility

Southern Superiority During the past several decades, numerous studies have attempted to trace the failure of Confederate nationalism, document a supposed lack of will in the Confederacy, and study desertion and dissent in southern armies and on the home front. Jason Phillips\u27s Diehard...

Wade Hampton: Confederate Warrior to Southern Redeemer

Interview with Rod Andrew, Jr. Interviewed by Christopher Childers Civil War Book Review (CWBR): Wade Hampton\u27s public life spans such a momentous time of American history in which the nature of the American republic changed drastically. What stayed constant in Ha...

The Rebel Yell: A Cultural History

“The Ugliest Sound that Any Mortal Ever Heard: The Rebel Yell in American History and Culture The famous, high-pitched “rebel yell began its career as a distinctive war cry and a means of communication for Confederate soldiers, and soon became a marker of white southern identity. By th...

Halls of Honor: College Men in the Old South

Principle and folly College men balanced mischief with Southern nobility Robert F. Pace, Professor of History at McMurry University in Abilene, Texas, has produced a delightful book that enriches our understanding of male student life in antebellum southern colleges. Pace pursues...