Conformal geometry of embedded manifolds with boundary from universal holographic formulæ

Arias, Cesar; Gover, A. Rod; Waldron, Andrew

doi:10.1016/j.aim.2021.107700

Cited by 9 publications

(5 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is an analogue for normal Killing solutions of the results for almost Einstein scales found in [49,51,33]. Those results for Einstein scales (and their generalisations to so-called ASC scales in [51]) were key in the (earlier mentioned) development of a holographic approach to hypersurfaces via a singular Yamabe problem in [2,8,58,59,61], as well as a boundary calculus of asymptotically hyperbolic manifolds [57]. We believe the results in Theorem 7.2 should provide one of the key insights for the analogous treatment of submanifolds of higher codimension.…”

Section: Main Results and A Technical Overviewsupporting

confidence: 51%

“…As an application we prove that the zero locus of suitable overdetermined PDE solutions are necessarily distinguished submanifolds; see Theorem 7.2. This shows how distinguished submanifolds fit into the curved orbit theory of [20,21] and, along with Proposition 7.3, is a first step toward understanding how to generalise the holography approach of [2,8,58,59,60,62,61] to higher codimensions.…”

Section: Introductionmentioning

confidence: 73%

“…For example it's failure to be parallel along the hypersurface is captured by a tractor second fundamental form L. In particular one obtains, in a simple explicit way, conformal analogues of the Gauss-Codazzi-Ricci theory, see [33] and references therein. Moreover the normal tractor has a remarkable link to other objects in, for example, Poincaré-Einstein manifolds and related structures, that has led to some deep results (e.g., that link the so-called conformal volume anomaly to higher Willmore invariants [63,2,59]).…”

Section: Main Results and A Technical Overviewmentioning

confidence: 99%

“…Toward resolving this, a logical step is to use the conformal Cartan/tractor connection of [24,95,5,22], and for the special case of hypersurfaces, meaning embedded submanifolds of codimension one, an effective approach was initiated in [5]. With a view to various applications, this hypersurface theory was extended in the works [9,67,93,51,98] and this approach has proved to be central in a number of further extensions and applications [2,8,57,58,59,62,61,75]. Rather separately from the general consideration of submanifolds, the distinguished curves in conformal manifolds known as conformal circles or conformal geodesics have been studied classically (see, e.g., [39,85,86,100,101]) and from various modern perspectives recently [44,4,5,96,90,56,35,68,72,14].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

CR Embedded Submanifolds of CR Manifolds

Curry¹,

Gover²

2019

Memoirs of the AMS

View full text Add to dashboard Cite

For conformal geometries of Riemannian signature, we provide a comprehensive and explicit treatment of the core local theory for embedded submanifolds of arbitrary dimension. This is based in the conformal tractor calculus and includes a conformally invariant Gauss formula leading to conformal versions of the Gauss, Codazzi, and Ricci equations. It provides the tools for proliferating submanifold conformal invariants, as well for extending to conformally singular Riemannian manifolds the notions of mean curvature and of minimal and CMC submanifolds.A notion of distinguished submanifold is defined by asking the tractor second fundamental form to vanish. We show that for the case of curves this exactly characterises conformal geodesics, also called conformal circles, while for hypersurfaces it is the totally umbilic condition. So, for other codimensions, this unifying notion interpolates between these extremes, and we prove that in all dimensions this coincides with the submanifold being weakly conformally circular, meaning that ambient conformal circles remain in the submanifold. We prove that submanifolds are conformally circular, meaning submanifold conformal circles coincide with ambient conformal circles, if and only also a second conformal invariant also vanishes.Next we provide a very general theory and construction of quantities that are necessarily conserved along distinguished submanifolds. This first integral theory thus vastly generalises the results available for conformal circles in [56]. We prove that any normal solution to an equation from the class of first BGG equations can yield such a conserved quantity, and we show that it is easy to provide explicit formulae for these.Finally we prove that the property of being distinguished is also captured by a type of moving incidence relation. This second characterisation is used to show that, for suitable solutions of conformal Killing-Yano equations, a certain zero locus of the solution is necessarily a distinguished submanifold.

show abstract

Section: Main Results and A Technical Overviewsupporting

confidence: 51%

Section: Introductionmentioning

confidence: 73%

Section: Main Results and A Technical Overviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

CR Embedded Submanifolds of CR Manifolds

Curry¹,

Gover²

2019

Memoirs of the AMS

View full text Add to dashboard Cite

show abstract

“…Note that these quantities and operators have all appeared previously in the recent literature, derived from parallel approaches to this problem. See (most comprehensively) [3], as well as [8,10,12]. Theorem 5.5.…”

Section: Explicit Computationsmentioning

confidence: 99%

Scattering on singular Yamabe spaces

2022

View full text Add to dashboard Cite

We apply scattering theory on asymptotically hyperbolic manifolds to singular Yamabe metrics, applying the results to the study of the conformal geometry of compact manifolds with boundary. In particular, we define extrinsic versions of the conformally invariant powers of the Laplacian, or GJMS operators, on the boundary of any such manifold, along with associated extrinsic Q-curvatures. We use the existence and uniqueness of a singular Yamabe metric in any conformal class to define also nonlocal extrinsic fractional GJMS operators on the boundary, and draw other global conclusions about the scattering operator, including a Gauss-Bonnet theorem in dimension four.

show abstract

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Juhl

Ørsted

2022

Advances in Mathematics

View full text Add to dashboard Cite

Conformal geometry of embedded manifolds with boundary from universal holographic formulæ

Cited by 9 publications

References 43 publications

CR Embedded Submanifolds of CR Manifolds

CR Embedded Submanifolds of CR Manifolds

Scattering on singular Yamabe spaces

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Contact Info

Product

Resources

About