An Optimal Gap Theorem in a Complete Strictly Pseudoconvex CR Manifold

TL;DR

This paper establishes an optimal gap theorem for complete strictly pseudoconvex CR manifolds, showing that under certain curvature decay conditions, the manifold must be flat, using advanced heat equation techniques.

Contribution

It introduces a novel application of the Li-Yau-Hamilton inequality and heat equation monotonicity to derive a sharp curvature decay condition implying flatness.

Findings

Manifolds with scalar curvature decaying faster than o(r^{-2}) are flat.

The method combines heat equation inequalities with geometric analysis.

Provides a new criterion for flatness in CR geometry.

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 pseudocovex CR 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 centered at some point o decays as $o(r^{-2})$, then the manifold is flat.