# On the CR analogue of Frankel conjecture and a smooth representative of the first Kohn-Rossi cohomology group

## TL;DR

This paper proves a CR analogue of the Frankel conjecture for certain spherical, strictly pseudoconvex CR manifolds, establishing conditions under which the first Kohn-Rossi cohomology group has specific properties.

## Contribution

It provides a criterion for pseudo-Einstein contact forms and confirms the CR Frankel conjecture in particular geometric settings, extending previous complex geometric results.

## Key findings

- CR Frankel conjecture holds for spherical, strictly pseudoconvex CR manifolds with nonnegative pseudohermitian curvature.
- The conjecture is valid when the first Kohn-Rossi cohomology group vanishes.
- A criterion for pseudo-Einstein contact forms is established.

## Abstract

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we obtain the CR Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold with the vanishing first Kohn-Rossi cohomology group. In particular, this conjecture holds in a spherical boundary of the Stein manifold.