Calabi-Yau theorem and Hodge-Laplacian heat equation ina closed strictly pseudoconvex CR manifold

Abstract

In this paper, we address the Calabi-Lee conjecture for pseudo-Einstein contact structure via the CR Poincaré-Lelong equation. Then we confirm the Calabi-Yau Theorem via Hodge-Laplacian heat flow in a closed strictly pseudoconvex CR $(2n + 1)$-manifold $(M , \theta)$ for $n \geq 2$. With its applications, we affirm a partial answer of the CR Frankel conjecture in a closed spherical strictly pseudoconvex CR $(2n + 1)$-manifold.