The sharp upper bounds for the first positive eigenvalue of the Kohn–Laplacian on compact strictly pseudoconvex hypersurfaces

Li, Song-Ying; Lin, Guijuan; Son, Duong Ngoc

doi:10.1007/s00209-017-1922-z

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…(1.13) Remark 1.6. When j k = δ jk , the Kronecker symbol, it was proved by Li, Lin, and the author [7] that…”

Section: Tor(z Z)mentioning

confidence: 99%

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

Son

2022

Colloq. Math.

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We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in C 2 from 2012. In fact, the main result proves a stronger positivity property, namely the 1 2 -positivity in the sense of Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a Kähler manifold of nonnegative holomorphic sectional curvature. Our approach is rather simple and uses a version of the Gauss equation for semi-isometric CR immersions of pseudohermitian manifolds into Kähler manifolds.

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“…(1.13) Remark 1.6. When j k = δ jk , the Kronecker symbol, it was proved by Li, Lin, and the author [7] that…”

Section: Tor(z Z)mentioning

confidence: 99%

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

Son

2022

Colloq. Math.

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“…For example, in [3] a Lichnerowicz-type estimate for the first positive eigenvalue in terms of the Webster scalar curvature (or Ricci curvature) was established, while the characterization of the equality case in this estimate was treated in [6] and [2], for the higher-dimensional and the three-dimensional cases respectively. In [5], several upper bounds for the first positive eigenvalue of the Kohn Laplacian on the boundaries of certain domains in C 2 were obtained; the bounds are extrinsic, depending on the realization of the CR manifolds as embedded compact real hypersurfaces in C 2 . All of the aforementioned bounds are sharp, with equality holding if (M, θ) is isomorphic to the standard CR sphere.…”

Section: Introductionmentioning

confidence: 99%

An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaces

Dall’Ara¹,

Son²

2021

Preprint

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A real hypersurface in C 2 is said to be Reinhardt if it is invariant under the standard T 2 -action on C 2 . Its CR geometry can be described in terms of the curvature function of its "generating curve", i.e., the logarithmic image of the hypersurface in the plane R 2 . We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the T 2 -action being free). Our bound is expressed in terms of the L 2 -norm of the curvature function of the generating curve and is attained if and only if the curve is a circle.

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“…The Bergman kernels of the ellipsoids were studied by Hirachi, [13]. The ellipsoids have also been studied in pseudo-Hermitian CR geometry, see [11,12]. The final result of the paper characterizes the ellipsoids which satisfy (1.4).…”

Section: Introductionmentioning

confidence: 99%

Rigidity theorem of the Bergman kernel by analytic capacity

Treuer¹

2021

Preprint

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In [7], Dong and I proved that the domains D ⊂ C of finite volume whose on-diagonal Bergman kernels K(•, •) satisfy K(z 0 , z 0 ) = V olume(D) −1 are disks minus closed polar sets. We utilized the solution of the Suita conjecture, a deep theorem of several complex variables. In this note, I present a significantly more elementary proof of this theorem that does not use several complex variables. As a corollary, a new lower bound for the on-diagonal Bergman kernel is given. Finally, I show that the only real ellipsoid in Webster normal form which satisfies K(0, 0) = V olume(D) −1 is the unit ball.

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The sharp upper bounds for the first positive eigenvalue of the Kohn–Laplacian on compact strictly pseudoconvex hypersurfaces

Cited by 6 publications

References 26 publications

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaces

Rigidity theorem of the Bergman kernel by analytic capacity

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