Volume renormalization for singular Yamabe metrics

Graham, C. Robin

doi:10.1090/proc/13530

Cited by 38 publications

(54 citation statements)

References 18 publications

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“…Finally we prove that l(x) = 1 5 E 2 (see (6.18)). This is the CR analogue of the result for the singular Yamabe problem ( [7], [8]). In particular, E 2 = 0 is also an obstruction to the smoothness of solutions to the singular CR Yamabe problem.…”

Section: Introduction and Statement Of The Resultssupporting

confidence: 70%

“…We also prove that the coefficient of the first log term in the expansion of a formal solution is given by the variational derivative of E 2 . This is an analogue of the result in singular Yamabe problem ( [8]). We follow Graham's ideas ([8]) in the conformal case.…”

Section: Volume Renormalization and Esupporting

confidence: 68%

“…Motivated by the recent study ( [8]) of the singular Yamabe problem and the associated volume renormalization, we look into the analogous situation in CR geometry. For a CR analogue of the Willmore energy in the surface case, one of us found two CR invariant surface area elements dA 1 , dA 2 in 1995 (see [1]).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Proof. We use an argument which is analogous to [8]. We have We then consider the second term in (6.20), i.e.…”

Section: 5mentioning

confidence: 99%

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Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

Chêng

Yang

Zhang

2018

Advances in Mathematics

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We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization.So they are minimizers for E 1 with zero energy. See Example 2 in Subsection 4.1.Conjecture 1 The shifted Heisenberg spheres are the only closed minimizers for E 1 (with zero energy) in H 1 .On the other hand, usual distance spheres (or Heisenberg spheres) defined by r 4 + 4t 2 = ρ 4 0 (ρ 0 > 0) are critical points of higher energy level for E 1 . See the remark in the end of Example 3 of Subsection 4.1. Another interesting example is the Clifford torus in S 3 . It is a critical point of E 1 with positive energy.Conjecture 2 The Clifford torus is the unique minimizer among all surfaces of torus type for E 1 up to CR automorphisms of S 3 .Critical points of E 2 include vertical planes in H 1 , the surface defined by t = √ 3 2 r 2 in H 1 and surfaces foliated by a linear combination ofe 1 ande 2 (e 1 := ∂ x +y∂ t , e 2 := ∂ y − x∂ t ). We show that E 2 is unbounded from below and above in general. See the remark in the end of Example 3 of Subsection 4.2. 2115-M-001-013-and the National Center for Theoretical Sciences for the constant support. P. Yang would like to thank the NSF of the U.S. for the support of the grant: DMS 1509505. Y. Zhang would like to thank Proessor Alice Chang for her invitation and the Department of Mathematics of Princeton University for its hospitality. He is supported by CSC scholarship 201606345025 and the Fundamental Research Funds for the Central Universities. P. Yang thanks Sean Curry for posing this question. Two CR invariant surface area elementsIn [1], the first author constructed two CR invariant area elements on a nonsingular (noncharacteristic) surface Σ in a strictly pseudoconvex CR 3-manifold (M, ξ, J). Here ξ denotes a contact bundle and J : ξ → ξ is an endomorphism such that J 2 = −Id. We recall ([3]) that a point p ∈ Σ is called singular if its tangent plane T p Σ coincides with the contact plane ξ p at p. We call a surface nonsingular if

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Section: Introduction and Statement Of The Resultssupporting

confidence: 70%

Section: Volume Renormalization and Esupporting

confidence: 68%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Proof. We use an argument which is analogous to [8]. We have We then consider the second term in (6.20), i.e.…”

Section: 5mentioning

confidence: 99%

See 2 more Smart Citations

Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

Chêng

Yang

Zhang

2018

Advances in Mathematics

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“…More recently it was realized that working with general conformally compact manifolds leads naturally to an extension to higher dimensional analogs of the Willmore equation [GW13, GW15,GW16b], and via volume expansions, energy functionals for these [Gra16,GW16a] (see also [GW16b,GGHW15] for an alternative approach). Here we show that there is tremendous gain in treating a new but related problem, namely solid infinite volume regions embedded in manifolds of the same dimension.…”

Section: Introductionmentioning

confidence: 99%

Renormalized volumes with boundary

Gover

Waldron

2019

Commun. Contemp. Math.

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We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré-Einstein structure, this result recovers Branson's Q curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

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Renormalized entanglement entropy and curvature invariants

Taylor

Too

2020

J. High Energ. Phys.

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Renormalized entanglement entropy can be defined using the replica trick for any choice of renormalization scheme; renormalized entanglement entropy in holographic settings is expressed in terms of renormalized areas of extremal surfaces. In this paper we show how holographic renormalized entanglement entropy can be expressed in terms of the Euler invariant of the surface and renormalized curvature invariants. For a spherical entangling region in an odd-dimensional CFT, the renormalized entanglement entropy is proportional to the Euler invariant of the holographic entangling surface, with the coefficient of proportionality capturing the (renormalized) F quantity. Variations of the entanglement entropy can be expressed elegantly in terms of renormalized curvature invariants, facilitating general proofs of the first law of entanglement.

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Volume renormalization for singular Yamabe metrics

Cited by 38 publications

References 18 publications

Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

Renormalized volumes with boundary

Renormalized entanglement entropy and curvature invariants

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