An Optimal Gap Theorem in a Complete Strictly Pseudoconvex CR $$(2n+1)$$ ( 2 n + 1 ) -Manifold

Abstract

In this paper, by applying a linear trace Li–Yau–Hamilton inequality for a positive (1, 1)-form solution of the CR Hodge–Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudoconvex CR \((2n+1)\)-manifold with nonnegative pseudohermitian bisectional curvature and vanishing torsion. We prove that if the average of the Tanaka–Webster scalar curvature over a ball of radius r centered at some point o decays as \(o\left( r^{-2}\right) \), then the manifold is flat.