Essential spectra of tensor product Hilbert complexes and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:math>-Neumann problem on product manifolds

Berger, Franz

doi:10.1016/j.jfa.2016.06.004

Cited by 3 publications

(6 citation statements)

References 20 publications

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“…This finishes the proof of (i) to (iv). Again by [2,Theorem 5.6], N ϕ 0,n is compact if and only if all N ϕ j 0,1 are compact, which is the case if and only if (5.4) holds for all 1 ≤ j ≤ n. It remains to show that this is equivalent to (5.5). By a simple scaling argument, the claim is equivalent tô…”

Section: Decoupled Weightsmentioning

confidence: 98%

“…Here, ϕ j 0,q j denotes the complex Laplacian for the weight ϕ j (of one variable) on C. This can be seen as a special case of [2,Theorem 5.5]. Indeed, the ∂-operator acting on L 2 0,q (C n , e −ϕ ) can be understood geometrically as the ∂ E -operator for the trivial line bundle E := C n × C → C n , and with Hermitian metric (z, v 1 ), (z, v 2 ) := v 1 v 2 e −ϕ(z) on E. If now E j is C × C → C with metric given by ϕ j : C → R, then it is easy to see that under the isomorphism (C n × C → C) ∼ = π * 1 E 1 ⊗ · · · ⊗ π * n E n the trivial line bundle on C n carries the metric given by (5.1), where π j : C n → C is the projection onto the jth factor.…”

Section: Decoupled Weightsmentioning

confidence: 99%

“…holds for all u ∈ dom(∂) ∩ dom(∂ * ϕ ) ∩ ker( ϕ 0,q ) ⊥ , see the arguments in [13,23,2]. We are interested in spectral properties of N ϕ 0,q , in particular the problem of deciding its compactness in terms of easily accessible properties of ϕ.…”

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confidence: 99%

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On some spectral properties of the weighted ∂¯-Neumann operator

Berger¹,

Haslinger²

2019

Kyoto J. Math.

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We study necessary conditions for compactness of the weighted ∂-Neumann operator on the space L 2 (C n , e −ϕ ) for a plurisubharmonic function ϕ. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non-) compactness of the ∂-Neumann operator for decoupled weights, which are of the form ϕ(z) = ϕ1(z1) + · · · + ϕn(zn). More can be said if every ∆ϕj defines a nontrivial doubling measure.2010 Mathematics Subject Classification. Primary 32W05; Secondary 30H20, 35N15, 35P10.

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Section: Decoupled Weightsmentioning

confidence: 98%

Section: Decoupled Weightsmentioning

confidence: 99%

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On some spectral properties of the weighted ∂¯-Neumann operator

Berger¹,

Haslinger²

2019

Kyoto J. Math.

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“…It is this asymmetry that gives us the freedom to choose, given µ, the most appropriate metric, e.g., one that makes h,µ coercive (if it exists). See, e.g., [Str10,Ber16] for proofs of the well-known facts just discussed.…”

Section: )mentioning

confidence: 99%

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Berger

Dall’Ara

Son

2019

Complex Variables and Elliptic Equations

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Given a smooth positive measure µ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on (0, 1)-forms. Thanks to an appropriate version of the Bochner-Kodaira-Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold.Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam.We also show in an appendix that the "twisted basic identities" of, e.g., [MV15] are standard basic identities with respect to conformally Kähler metrics.

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“…6 By this we mean a (co)chain complex of closed and densely defined operators between Hilbert spaces, see [BL92] or also [Ber16].…”

Section: Preliminaries On Complex and Hermitian Geometrymentioning

confidence: 99%

Discreteness of spectrum for the $\overline\partial$-Neumann Laplacian on manifolds of bounded geometry

Berger

2018

Preprint

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For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with ∂-Neumann boundary conditions, which is a selfadjoint operator on the space of square-integrable differential forms with values in the given holomorphic bundle. We argue that some known results on the spectral properties of this operator on pseudoconvex domains in C n continue to hold on Kähler manifolds satisfying certain bounded geometry assumptions. In particular, we will consider the Dolbeault complex for forms with values in a line bundle, where known results from magnetic Schrödinger operator theory can be applied.

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Essential spectra of tensor product Hilbert complexes and the ∂‾-Neumann problem on product manifolds

Cited by 3 publications

References 20 publications

On some spectral properties of the weighted ∂¯-Neumann operator

On some spectral properties of the weighted ∂¯-Neumann operator

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Discreteness of spectrum for the $\overline\partial$-Neumann Laplacian on manifolds of bounded geometry

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