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

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 $δ$-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 $δ$-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]).

• The paper establishes sharp pointwise Chen inequalities that bound leaf-wise first δ-invariants using mean curvature, the warping function, its Laplacian, and the ambient holomorphic curvature.
• Its sign-dependent estimates remain valid in complex projective, Euclidean, and hyperbolic space forms, while distinguishing leaf-wise invariants from intrinsic factor invariants through Bishop–O’Neill formulas.
• Equality forces mixed total geodesy and minimality, while the minimal case yields necessary curvature conditions that partially address Chern’s problem but do not prove sufficiency or global existence.

Overview

This paper establishes the first Chen inequality for CR-warped product submanifolds $M^n = N_T \times_f N_\perp$ isometrically immersed in a complex space form $\tilde M^{2m}(c)$ of constant holomorphic sectional curvature $c$. The result controls an intrinsic quantity — the leaf-wise $\delta$-invariant of either factor — by the extrinsic squared norm of the mean curvature vector $\|\vec H\|^2$, thereby answering a problem posed by B.-Y. Chen concerning relationships between intrinsic and extrinsic submanifold invariants. As a geometric application, setting $\vec H = 0$ yields necessary conditions for minimality, giving a partial answer to S.-S. Chern's classical problem on minimal immersions of warped products. The bound is stated to be sharp and uniform in the sign of $c$.

Geometric setup and the leaf-wise invariant

The ambient space is a Kähler manifold of constant holomorphic sectional curvature $c$, whose sectional curvature on any 2-plane satisfies $\min(c/4, c) \le \tilde K(\pi) \le \max(c/4, c)$, with holomorphic 2-planes realizing curvature $c$ and totally real 2-planes realizing $\tilde M^{2m}(c)$0. A CR-submanifold carries an orthogonal decomposition $\tilde M^{2m}(c)$1 into a $\tilde M^{2m}(c)$2-invariant (holomorphic) distribution and an anti-invariant (totally real) distribution; a CR-warped product is $\tilde M^{2m}(c)$3 with $\tilde M^{2m}(c)$4 holomorphic ($\tilde M^{2m}(c)$5 even) and $\tilde M^{2m}(c)$6 totally real.

A key conceptual contribution is the careful distinction between two quantities that are often conflated. The leaf-wise first Chen invariant of a factor at $\tilde M^{2m}(c)$7 is

$\tilde M^{2m}(c)$8

computed with respect to the curvatures of the immersed manifold $\tilde M^{2m}(c)$9, not of the abstract factor. For the holomorphic factor this coincides exactly with the intrinsic Chen invariant $c$0, since leaves $c$1 are totally geodesic in $c$2. For the totally real factor, the Bishop–O'Neill formula gives

$c$3

so the intrinsic form of the inequality acquires both a scaling by $c$4 and a gradient correction involving $c$5. This distinction matters: the warped product structure itself contributes curvature through the term $c$6 (the fundamental warped-product identity), which appears explicitly in all bounds.

The main inequality

The central theorem states, for each point $c$7:

1. If $c$8:

$c$9

2. If $\delta$0:

$\delta$1

Here $\delta$2, so the bound takes different forms depending on the sign of $\delta$3: for $\delta$4 the correction is $\delta$5, while for $\delta$6 it is $\delta$7, which is sharper and necessary because the infimum defining $\delta$8 may then be realized on a holomorphic 2-plane. This sign-dependent treatment is what makes the inequality uniform across complex projective space ($\delta$9), flat space ($\|\vec H\|^2$0), and the complex hyperbolic case ($\|\vec H\|^2$1).

The proof follows the standard Chen program: the Gauss equation yields the fundamental identity $\|\vec H\|^2$2, where the ambient partial scalar curvature computation uses the fact that $\|\vec H\|^2$3 precisely on the $\|\vec H\|^2$4 holomorphic pairs within $\|\vec H\|^2$5. Chen's algebraic lemma is then applied to the diagonal second-fundamental-form coefficients of each distribution separately, producing a lower bound on $\|\vec H\|^2$6 that feeds back into the Gauss equation for an arbitrary 2-plane. Two computational lemmas reorganize the quadratic terms over the leaf/fiber splitting, and the resulting non-negative remainder $\|\vec H\|^2$7 is discarded to obtain the final bound.

Equality case and its consequences

The equality analysis is the strongest part of the paper's conclusions. Equality holds if and only if the minimizing 2-plane attains $\|\vec H\|^2$8 and the shape operators take canonical block forms: a traceless $\|\vec H\|^2$9 block on the minimizing plane, vanishing diagonal traces on each distribution, and vanishing off-diagonal blocks coupling $\vec H = 0$0 and $\vec H = 0$1. Consequently, every equality case forces the CR-warped product to be mixed totally geodesic, simultaneously $\vec H = 0$2-minimal and $\vec H = 0$3-minimal, and hence minimal in $\vec H = 0$4 at the point. In other words, equality in the inequality can only occur for minimal submanifolds — a rigidity statement analogous in spirit to the equality classification in Chen's original 1993 theorem for real space forms.

Application to Chern's problem

Setting $\vec H = 0$5 produces explicit necessary conditions for a minimal immersion:

Case	Necessary condition
Holomorphic factor, $\vec H = 0$6	$\vec H = 0$7
Holomorphic factor, $\vec H = 0$8	$\vec H = 0$9
Totally real factor	$c$0

For the flat Euclidean case these collapse to the clean statements $c$1 and, intrinsically, $c$2. Thus any CR-warped product admitting a minimal immersion into Euclidean space must satisfy these pointwise curvature restrictions — a concrete instance of Chern's request for further necessary conditions on the metric of a minimally immersible submanifold. These are necessary conditions only; the paper does not claim they are sufficient, and no existence results are provided.

Limitations and open questions

Several qualifications apply. First, the inequality is proved pointwise; the paper does not establish global or integrated versions, nor does it address whether the equality conditions can be realized globally by nontrivial examples beyond the local algebraic characterization. Second, the sharpness claim rests on the equality characterization rather than on exhibited families of extremal submanifolds. Third, the framework is restricted to complex space forms; the authors explicitly pose as open problems the extension to locally conformal Kähler space forms, both for the inequality itself and for the corresponding answer to Chern's problem with the same sign-dependent treatment of $c$3. Finally, the answer to Chern's problem is explicitly partial: necessity of the curvature bounds is established, but sufficiency remains unaddressed.

Conclusion

The paper extends Chen's first inequality to CR-warped product submanifolds of complex space forms, introducing the leaf-wise $c$4-invariant as the natural controlled quantity and clarifying its relation to the genuine intrinsic Chen invariants via the Bishop–O'Neill formulas. The resulting bounds are sharp, uniform in the sign of the holomorphic sectional curvature, and rigid at equality, forcing mixed total geodesy and minimality. The corollaries give explicit necessary curvature conditions for minimal immersions, constituting a partial solution to Chern's problem in this class, with extensions to locally conformal Kähler ambients left as open problems.