Adjoints of elliptic cone operators

Gil, Juan B.; Mendoza, Gerardo A.

doi:10.1353/ajm.2003.0012

Cited by 70 publications

(130 citation statements)

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“…We refer to [26] for a discussion of this general theory, and to [6] or [13] for its application in the conic setting.…”

Section: Domains Of Closed Extensions Of Conic Operatorsmentioning

confidence: 99%

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Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

Mazzeo¹,

Montcouquiol²

2011

J. Differential Geom.

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The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less than 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli space of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff [7] when the singular set has no vertices, and by Weiß [32] when the cone angles are less than π. We prove here an infinitesimal rigidity result valid for cone angles less than 2π, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author [22], see also the concurrent work by Weiß [33].

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“…We refer to [26] for a discussion of this general theory, and to [6] or [13] for its application in the conic setting.…”

Section: Domains Of Closed Extensions Of Conic Operatorsmentioning

confidence: 99%

“…It is proved in [6] (in the scalar case, but the techniques readily apply to the vector-valued case), see also [13], that if u ∈ D max , i.e. u ∈ L 2 and Lu ∈ L 2 , then using Proposition 6,…”

Section: Now Definementioning

confidence: 99%

Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

Mazzeo¹,

Montcouquiol²

2011

J. Differential Geom.

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“…Theorem 1.1 is proved in Section 6. We start with a formula for the boundary sesquilinear form [ , ] A taken from Gil-Mendoza [4]; see Theorem 6.1. The proof of Theorem 1.1 then consists in evaluating this formula, where the latter basically means to "take the residue" of the formula from Theorem 5.1.…”

Section: Description Of the Contentmentioning

confidence: 99%

Green’s formulas for cone differential operators

Witt

2007

Trans. Amer. Math. Soc.

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Abstract. Green's formulas for elliptic cone differential operators are established. This is achieved by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint; thereby utilizing the concept of a discrete asymptotic type. From this description, the singular coefficients replacing the boundary traces in classical Green's formulas are deduced.1. Introduction 1.1. The main result. Let X be a compact C ∞ -manifold with non-empty boundary, ∂X. On the interior X• := X \ ∂X, we consider differential operators A which on U \ ∂X for some collar neighborhood U ∼ = [0, 1) × Y of ∂X, with coordinates (t, y) and Y being diffeomorphic to ∂X, take the formSuch differential operators A are called cone-degenerate, or of Fuchs type, and this is written as A ∈ Diff µ cone (X). They arise, e.g., when polar coordinates are introduced near a conical point.Throughout, we shall fix some reference weight δ ∈ R. This means that we will be working in the weighted L 2 -space H 0,δ (X) as the basic function space; see (1.12) and also Appendix B. Let A * ∈ Diff µ cone (X) be the formal adjoint to A, i.e,where ( , ) denotes the scalar product in H 0,δ (X). Then it is customary to consider the maximal and minimal domains of A,

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“…Because this is not central to our discussion, we leave details to the reader (and refer to [10] for a thorough discussion of the conic case).…”

Section: The Normal Operator Of D Amentioning

confidence: 99%

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Hunsicker¹,

Mazzeo²

2005

Internat. Math. Res. Notices

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Let (X, g) be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of X, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai [5]. Because the metric g is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of L 2 harmonic forms for any complete edge metric on the regular part of X.

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Adjoints of elliptic cone operators

Abstract: Abstract. We study the adjointness problem for the closed extensions of a general b-elliptic operatorThe case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator.

Cited by 70 publications

References 12 publications

Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

Green’s formulas for cone differential operators

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