An Optimal Gap Theorem in a Complete Strictly Pseudoconvex CR (2n+1)(2n+1) Manifold
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)(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) $or 2, then the manifold is flat.