First Chen Inequality for CR-Warped Product Submanifolds of a Complex Space Form and Applications

TL;DR

This paper establishes a new Chen inequality for CR-warped product submanifolds in complex space forms, linking intrinsic and extrinsic invariants, and explores conditions for minimality and open problems.

Contribution

It proves the first Chen inequality for CR-warped product submanifolds, relating intrinsic invariants to mean curvature, and addresses minimality conditions in complex space forms.

Findings

The inequality is sharp and uniform across the sign of holomorphic sectional curvature.

Derived necessary conditions for CR-warped products to be minimal in complex space forms.

Addressed open problems related to Chen inequalities and minimal submanifolds.

Abstract

In this paper, the first Chen inequality is proved for CR-warped product submanifolds in complex space forms. This inequality involves intrinsic invariants (a leaf-wise $\delta$-invariant and the sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer to Problem [1]. We carefully distinguish the leaf-wise $\delta$-invariant of a factor (used in the bound) from the intrinsic Chen invariant of the same factor, the two being related, on the totally real factor, by the Bishop--O'Neill formula. The bound is sharp and is uniform in the sign of the holomorphic sectional curvature $c$. As a geometric application, we derive necessary conditions for the immersed CR-warped product submanifold to be minimal in a complex space form, providing a partial answer to a well-known problem proposed by S.S. Chern (Problem [2]). For further research directions, we address a couple of open problems (Problem [3]} and Problem [4]).