Resolvent of the Laplacian on strictly pseudoconvex domains

Epstein, Charles L.; Melrose, Richard; Mendoza, Gerardo A.

doi:10.1007/bf02392446

Cited by 118 publications

(164 citation statements)

References 16 publications

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“…Fractional CR covariant operators of order 2γ, γ ∈ R, may be defined from scattering theory on a Kähler-Einstein manifold X [21,45,40,33], they are pseudodifferential operators whose principal symbol agrees with the pure fractional powers of the CR sub-Laplacian (−∆ b ) γ on the boundary M = ∂X . In the particular case of the Heisenberg group H n , they are simply the intertwining operators on the CR sphere calculated in [8,51,9] using classical representation theory tools.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…In particular, we consider the Θ-structures introduced by Epstein-Melrose-Mendoza [21]. We will only give brief description and we refer the interested reader to the detailed description in [40].…”

Section: 2mentioning

confidence: 99%

“…Let X be a m-dimensional complex manifold with compact, strictly pseudoconvex boundary ∂X = M. Let g + be a Kähler metric on X such that there exists a globally defined approximate solution ϕ of the Monge-Ampère equation that makes g + an approximate Kähler-Einstein metric in the sense of (2.17) and X = {ϕ < 0}. In particular, the metric g + belongs to the class of Θ-metrics considered by [21].…”

Section: Scattering Theory and The Conformal Fractional Sub-laplacianmentioning

confidence: 99%

“…Note that S(s) is a meromorphic family of pseudodifferential operators in the Θ-calculus of [21] of order 2(2s − m), and self-adjoint when s is real.…”

Section: Scattering Theory and The Conformal Fractional Sub-laplacianmentioning

confidence: 99%

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An extension problem for the CR fractional Laplacian

Frank

González

Monticelli

et al. 2015

Advances in Mathematics

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Scattering Theory and The Conformal Fractional Sub-laplacianmentioning

confidence: 99%

“…Note that S(s) is a meromorphic family of pseudodifferential operators in the Θ-calculus of [21] of order 2(2s − m), and self-adjoint when s is real.…”

Section: Scattering Theory and The Conformal Fractional Sub-laplacianmentioning

confidence: 99%

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An extension problem for the CR fractional Laplacian

Frank

González

Monticelli

et al. 2015

Advances in Mathematics

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“…Simultaneously, the infinity of such manifolds, that is the spherical CauchyRiemannian manifolds furnish the simplest examples of manifolds with contact structures. Second, such manifolds provide simplest examples of negatively curved manifolds not having constant sectional curvature, and already obtained results show surprising differences between geometry and topology of such manifolds and corresponding properties of (real hyperbolic) manifolds with constant negative curvature, see [BS,BuM,EMM,Go1,GM,Min,Yu1]. Third, such manifolds occupy a remarkable place among rank-one symmetric spaces in the sense of their deformations: they enjoy the flexibility of low dimensional real hyperbolic manifolds (see [Th,A1,A2] and §7) as well as the rigidity of quaternionic/octionic hyperbolic manifolds and higher-rank locally symmetric spaces [MG1,Co2,P].…”

Section: Introductionmentioning

confidence: 99%

Geometry and topology of complex hyperbolic and Cauchy-Riemannian manifolds

Apanasov¹

1997

Russ. Math. Surv.

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ABSTRACT. We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.

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Short‐time existence for the network flow

Lira

Mazzeo

Pluda

et al. 2023

Comm Pure Appl Math

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This paper contains a new proof of the short‐time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard‐Reitich, Mantegazza‐Novaga‐Tortorelli and eventually, in the most general case of irregular networks by Ilmanen‐Neves‐Schulze. Although the present paper proves a result similar to the one in Ilmanen et al., the method here provides substantially more detailed information about how an irregular network “resolves” into a regular one. Either approach relies on the existence of self‐similar expanding solutions found in Mazzeo and Saez. As a precursor to the main theorem, we also prove an unexpected regularity result for the mixed Cauchy‐Dirichlet boundary problem for the linear heat equation on a manifold with boundary.

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Resolvent of the Laplacian on strictly pseudoconvex domains

Cited by 118 publications

References 16 publications

An extension problem for the CR fractional Laplacian

An extension problem for the CR fractional Laplacian

Geometry and topology of complex hyperbolic and Cauchy-Riemannian manifolds

Short‐time existence for the network flow

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