$$C_0$$-positivity and a classification of closed three-dimensional CR torsion solitons

Abstract

A closed CR 3-manifold is said to have C 0 -positive pseudohermitian curvature if (W + C 0 T or)(X, X) > 0 for any 0 = X ∈ T 1,0 (M ). We discover an obstruction for a closed CR 3-manifold to possess C 0 -positive pseudohermitian curvature. We classify closed three-dimensional CR Yamabe solitons according to C 0 -positivity and C 0 -negativity whenever C 0 = 1 and the potential function lies in the kernel of Paneitz operator. Moreover, we show that any closed threedimensional CR torsion soliton must be the standard Sasakian space form. At last, we discuss the persistence of C 0 -positivity along the CR torsion flow starting from a pseudo-Einstein contact form. ∂J ∂t = 2A J,θ , ∂θ ∂t = −2W θ.

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

“…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].…”

“…In fact, there are CR manifolds diffeomorphic to the 3-torus T 3 which admit pseudohermitian structures of positive Tanaka-Webster scalar curvature (see Remark 3.4), while tori of dimension at least 3 admit no positive scalar curvature Riemannian metric [13]. Recently, Cao, Chang, and Chen [2] considered a condition stronger than the positivity of the Tanaka-Webster scalar curvature, the C 0 -positivity, which involves both curvature and torsion. For compact CR 3-manifolds they proved that the C 0 -positivity with C 0 ≥ 1/2 implies the positivity of an adapted Riemannian metric.…”

“…In particular, it does not imply the positivity of the scalar curvature of any adapted Webster metric. In [2], Cao, Chang, and Chen introduced the following notion exhibiting the importance of the torsion: A closed CR 3-manifold is said to have C 0 -positive pseudohermitian curvature if there exists a pseudohermitian structure θ with curvature and torsion satisfying…”

“…Observe that the inequality is strict since M is supposed to be strictly pseudohermitian C-convex. When n = 1, it follows that (M, θ) is 1 2 -positive in the sense of [2] and hence the corollary follows from [2, Theorem 1.1].…”

“…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 .…”