arXiv subjects

A. Rod Gover

Publications and source records attributed to A. Rod Gover.

At least 55 records · Page 3Linked to original sources

arXiv2016

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.

hep-th↗

arXiv2016

CR embedded submanifolds of CR manifolds

We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naïve methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

math.DG↗

arXiv2016

Zero and negative eigenvalues of the conformal Laplacian

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

math.DG↗

arXiv2016

C-Projective Compactification; (quasi--)Kaehler Metrics and CR boundaries

For complete complex connections on almost complex manifolds we introduce a natural definition of compactification. This is based on almost c--projective geometry, which is the almost complex analogue of projective differential geometry. The boundary at infinity is a (possibly non-integrable) CR structure. The theory applies to almost Hermitean manifolds which admit a complex metric connection of minimal torsion, which means that they are quasi--Kaehler in the sense of Gray--Hervella; in particular it applies to Kaehler and nearly Kaehler manifolds. Via this canonical connection, we obtain a notion of c-projective compactification for quasi--Kaehler metrics of any signature. We describe an asymptotic form for metrics that is necessary and sufficient for c--projective compactness. This metric form provides local examples and, in particular, shows that the usual complete Kaehler metrics associated to smoothly bounded, strictly pseudoconvex domains in C^n are c--projectively compact. For a smooth manifold with boundary and a complete quasi-Kaehler metric $g$ on the interior, we show that if its almost c--projective structure extends smoothly to the boundary then so does its scalar curvature. We prove that $g$ is almost c--projectively compact if and only if this scalar curvature is non-zero on an open dense set of the boundary, in which case it is, along the boundary, locally constant and hence nowhere zero there. Finally we describe the asymptotics of the curvature, showing, in particular, that the canonical connection satisfies an asymptotic Einstein condition. Key to much of the development is a certain real tractor calculus for almost c--projective geometry, and this is developed in the article.

math.DG↗

arXiv2015

Scalar Curvature and Projective Compactness

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.

math.DG↗

arXiv2015

Singular Yamabe Problem Willmore Energies

We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit hypersurface formulae for both the energy functional and the obstruction.

math.DG↗

arXiv2015

An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity

The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as primary motivation the search for conformally invariant tensors and diffrerential operators. The final four lectures apply the conformal tractor calculus to the study of conformally compactified geometries, motivated by the conformal treatment of infinity in general relativity.

math.DG↗

arXiv2015

Projectively related metrics, Weyl nullity, and metric projectively invariant equations

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries which is defined on a rank $(n+1)$-bundle. We show this connection is linked to the metrisability equations that govern the existence of metrics compatible with the structure. The fundamental 2-tensor also leads to a new class of invariant linear differential operators that are canonically associated to these geometries; included is a third equation studied by Gallot et al. We apply the results to study the metrisability equation, in the nullity setting described. We obtain strong local and global results on the nature of solutions and also on the nature of the geometries admitting such solutions, obtaining classification results in some cases. We show that closed Sasakian and Kähler manifold do not admit nontrivial solutions. We also prove that, on a closed manifold, two nontrivially projectively equivalent metrics cannot have the same tracefree Ricci tensor. We show that on a closed manifold a metric having a nontrivial solution of the metrisablity equation cannot have two-dimensional nullity space at every point. In these statements the meaning of trivial solution is dependent on the context. There is a function $B$ naturally appearing if a metric projective structure has nullity. We analyse in detail the case when this is not a constant, and describe all nontrivially projectively equivalent Riemannian metrics on closed manifolds with nonconstant $B$.

math.DG↗

arXiv2015

Conformal Holonomy Equals Ambient Holonomy

This paper studies the relation between two notions of holonomy on a conformal manifold. The first is the conformal holonomy, defined to be the holonomy of the normal tractor connection. The second is the holonomy of the Fefferman-Graham ambient metric of the conformal manifold. It is shown that the infinitesimal conformal holonomy and the infinitesimal ambient holonomy always agree up to the order that the ambient metric is defined.

math.DG↗

arXiv2015

Projective Compactness and Conformal Boundaries

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $α$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on $M$. We study the relation between interior geometry and the possibilities for compactification, and then develop the tools that describe the induced geometry on the boundary. We prove that a pseudo--Riemannian metric on $M$ which is projectively compact of order two admits a certain asymptotic form. This form was known to be sufficient for projective compactness, so the result establishes that it provides an equivalent characterization. From a projectively compact connection on $M$, one obtains a projective structure on $\overline{M}$, which induces a conformal class of (possibly degenerate) bundle metrics on the tangent bundle to the hypersurface $\partial M$. Using the asymptotic form, we prove that in the case of metrics, which are projectively compact of order two, this boundary structure is always non--degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non--degenerate boundary geometry gives rise to a (conformal) standard tractor bundle endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.

math.DG↗

arXiv2014

Tractor calculus, BGG complexes, and the cohomology of Kleinian groups

For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= Γ\backslash \mathbb{H}^{n}$ where $Γ$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential geometric cohomology on $M$ and algebraic invariants of the group $Γ$. In particular for $\mathbb{F}$ an irreducible $SO(n,1)$-module, we show that the group cohomology with coefficients $H^{\bullet}(Γ, \mathbb{F})$ is isomorphic to the cohomology of an appropriate projective BGG complex on $M$. This yields the geometric interpretation that $H^{\bullet}(Γ, \mathbb{F})$ parameterises solutions to certain distinguished natural PDEs of Riemannian geometry, modulo the range of suitable differential coboundary operators. Viewed in another direction, the construction shows one way that non-trivial cohomology can arise in a BGG complex, and sheds considerable light on its geometric meaning. We also use the tools developed to give a new proof that $H^{1} (Γ, S_{0}^{k} \mathbb{R}^{n+1}) \neq 0$ whenever $M$ contains a compact, orientable, totally geodesic hypersurface. All constructions use another result that we establish, namely that the canonical flat connection on a hyperbolic manifold coincides with the tractor connection of projective differential geometry.

math.DG↗

arXiv2014

Generalising the Willmore equation: submanifold conformal invariants from a boundary Yamabe problem

The Willmore energy, alias bending energy or rigid string action, and its variation-the Willmore invariant-are important surface conformal invariants with applications ranging from cell membranes to the entanglement entropy in quantum gravity. In work of Andersson, Chrusciel, and Friedrich, the same invariant arises as the obstruction to smooth boundary asymptotics to the Yamabe problem of finding a metric in a conformal class with constant scalar curvature. We use conformal geometry tools to describe and compute the asymptotics of the Yamabe problem on a conformally compact manifold and thus produce higher order hypersurface conformal invariants that generalise the Willmore invariant. We give a holographic formula for these as well as variational principles for the lowest lying examples.

hep-th↗

arXiv2014

Projective Compactifications and Einstein metrics

For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e.\ geodesic path data) of the given connection. The definition of projective compactness involves a real parameter $α$ called the order of projective compactness. For volume preserving connections, this order is captured by a notion of volume asymptotics that we define. These ideas apply to complete pseudo-Riemannian spaces, via the Levi-Civita connection, and thus provide a notion of compactification alternative to conformal compactification. For each order $α$, we provide an asymptotic form of a metric which is sufficient for projective compactness of the given order, thus also providing many local examples. Distinguished classes of projectively compactified geometries of orders one and two are associated with Ricci-flat connections and non--Ricci--flat Einstein metrics, respectively. Conversely, these geometric conditions are shown to force the indicated order of projective compactness. These special compactifications are shown to correspond to normal solutions of classes of natural linear PDE (so-called first BGG equations), or equivalently holonomy reductions of projective Cartan/tractor connections. This enables the application of tools already available to reveal considerable information about the geometry of the boundary at infinity. Finally, we show that metrics admitting such special compactifications always have an asymptotic form as mentioned above.

math.DG↗

arXiv2014

Holonomy reductions of Cartan geometries and curved orbit decompositions

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third order differential equation that governs the existence of a compatible Einstein metric. In CR-geometry we discuss an invariant equation that governs the existence of a compatible Kähler-Einstein metric.

math.DG↗

arXiv2013

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.

math.DG↗

arXiv2013

Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms $L^\ell_k$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as explicit factored polynomials in second order differential operators. In the case the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the $L^\ell_k$ in terms of the null spaces of mutually commuting second order factors.

math.DG↗

arXiv2012

Detecting Einstein geodesics: Einstein metrics in projective and conformal geometry

Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that the vanishing of these is necessary for the existence of such a metric, and in generic settings the vanishing of these is also sufficient. We also obtain results for the problem of metrisability (without the Einstein condition): We show that the odd Chern type invariants of an affine connection are projective invariants that obstruct the existence of a projectively related Levi-Civita connection. In addition we discuss a concrete link between projective and conformal geometry and the application of this to the projective-Einstein problem.

math.DG↗

arXiv2012

Calculus and invariants on almost complex manifolds, including projective and conformal geometry

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants.

math.DG↗