A fourth order curvature flow on a CR 3-manifold

Abstract

Abstract. Let (M 3 , J, θ 0 ) be a closed pseudohermitian 3-manifold. Suppose the associated torsion vanishes and the associated Q-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian 3-manifold, we study the change of the contact form according to a certain version of normalized Q-curvature flow. This is a fourth order evolution equation. We prove that the solution exists for all time and converges smoothly to a contact form of zero Q-curvature. We also consider other background conditions and obtain a priori bounds up to high orders for the solution.

On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold

“…We observe that for a standard pseduhermitian (2n + 1)-sphere (S 2n+1 , J , θ) with the induced natural CR structure from C n+1 and the standard contact form θ , one can show that [1,4] …”

“…1 The commutation relation (3.11) of Greenleaf has something wrong (see [15]). Then the coefficient of torsion term in the Bochner formula (4.2) of Greenleaf should be − n 2 .…”

“…Let (M, J, θ) be a closed pseudohermitian 3-manifold with zero torsion. Then the corresponding CR Paneitz operator is essentially positive [1,2].…”

The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold

“…Note that the positivity of the CR Paneitz operator P 0 is of great importance in studying the Q-curvature flow on a CR 3-manifold (see [1]). …”

“…Nonetheless, we also give examples in Section 4 to show that we can use Theorem 1.2 here to obtain the lower bound of the first positive eigenvalue of the sublaplacian. But [ In Section 4, we also point out that if the torsion of M is zero, then the corresponding CR Paneitz operator must be positive (for the detail, see [1]). In this case, Theorem 1.1 is identical to [16,Theorem 1.2] and we get the following Corollary:…”