On the CR–Obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in $\mathbb{C}^{n+1}$

Li, Song-Ying; Tran, MyAn

doi:10.1090/s0002-9947-2011-05396-1

Cited by 5 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Corollary 1.4 generalizes [4, Theorem 1.2] and [9] about the positivity of the Tanaka-Webster scalar curvature on real ellipsoids. Recall that an ellipsoid E in C 2 ∼ = R 4 is a compact real hypersurface given by the equation…”

Section: Tor(z Z)mentioning

confidence: 53%

“…We mention here, for example, the Lichnerowicz-type estimate for the first positive eigenvalue of the sub-Laplacian (see e.g. [9] and the references therein) and the classification of the closed CR torsion solitons [2].…”

Section: Tor(z Z)mentioning

confidence: 99%

“…Explicit computations of the (Tanaka-Webster) scalar, Ricci, as well as the full curvature tensor have been done only for a few examples, see e.g., [9,3,14]. Proposition 4.3 below gives another one: We use the Gauss equations in Proposition 2.2 to compute the holomorphic sectional curvature of a Brieskorn manifold.…”

Section: Proofs Of Theorem 12 and Its Corollariesmentioning

confidence: 99%

“…Chanillo, Chiu, and Yang [4] proved, among other interesting results, that the scalar curvature R is positive on a real ellipsoid of C 2 [4, Theorem 1.2]. In general dimension, a positive lower bound for the Tanaka-Webster Ricci tensor on a real ellipsoid can be deduced from Li-Tran [9] which studies a lower estimate for the eigenvalue of the sub-Laplacian [9,Theorem 1.1]. In [4, p. 81], Chanillo, Chiu, and Yang posed an interesting question whether the scalar curvature R must be positive on strictly convex domains in C 2 .…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Son

2022

Colloq. Math.

View full text Add to dashboard Cite

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.

show abstract

Section: Tor(z Z)mentioning

confidence: 53%

Section: Tor(z Z)mentioning

confidence: 99%

Section: Proofs Of Theorem 12 and Its Corollariesmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

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

Son

2022

Colloq. Math.

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

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

Lin

Son

2017

Math. Z.

Self Cite

View full text Add to dashboard Cite

show abstract

On the CR–Obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in $\mathbb{C}^{n+1}$

Abstract: Abstract. This paper studies the CR-version of Obata theorem on a pseudoHermitian CR-manifold (M, θ). The main result of the paper is proving that CR-Obata theorem holds on real ellipsoid E(A) with contact form θ = 1 2ij − 1 with A j ∈ (−1, 1).

Cited by 5 publications

References 24 publications

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

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

Rigidity theorem of the Bergman kernel by analytic capacity

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

Contact Info

Product

Resources

About