Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature

Abstract: In this paper, we show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal. We also prove that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature on each connected component. This gives partial affirmative answers to Chen's conjecture, to the generalized Chen's conjecture in hyperbolic spaces, and to the Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.

• The paper proves that every biharmonic Wintgen ideal submanifold in a space form with nonpositive sectional curvature is minimal, without completeness, properness, or integrability assumptions.
• Using the Choi–Lu normal form, Chen-submanifold properties, and biharmonic equations, the analysis establishes that positive-curvature examples have constant mean-curvature magnitude on each connected component.
• The result gives conditional progress on Chen’s, generalized Chen’s, and Balmuş–Montaldo–Oniciuc conjectures across arbitrary dimensions and codimensions, while showing that minimality is not generally possible in spheres.

Overview

This paper by Shun Maeta studies biharmonic submanifolds that satisfy the Wintgen ideal condition in space forms. The main result, Theorem 1 of the paper (2606.25791), states that if $M^m$ is a Wintgen ideal submanifold of dimension $m \geq 2$ and codimension $p \geq 1$ in a Riemannian manifold $N^{m+p}(\rho)$ of constant sectional curvature $\rho$, then biharmonicity implies:

1. Minimality when $\rho \leq 0$: every biharmonic Wintgen ideal submanifold in a space form of nonpositive constant sectional curvature is minimal.
2. Constant mean curvature when $\rho > 0$: $|{\bf H}|$ is constant on each connected component.

These conclusions give partial affirmative answers to three well-known conjectures: Chen's conjecture (biharmonic submanifolds of Euclidean space are minimal), the generalized Chen's conjecture (the same for nonpositively curved ambient spaces), and the Balmuş–Montaldo–Oniciuc conjecture (proper biharmonic submanifolds of spheres have constant mean curvature). The restriction to Wintgen ideal submanifolds is essential: the generalized Chen's conjecture is false in general, as shown by Ou and Tang's counterexamples, and the spherical conclusion cannot be strengthened to minimality because small hyperspheres in spheres are proper biharmonic and totally umbilical.

Motivation and context

The paper sits at the intersection of two research programs. The first concerns the Wintgen inequality. For surfaces in $\mathbb{E}^3$, the classical inequality $K \leq |{\bf H}|^2$ holds with equality exactly at umbilical points. In higher codimension, the DDVV inequality refines this to

$m \geq 2$0

where $m \geq 2$1 is the normalized scalar curvature and $m \geq 2$2 the normalized normal scalar curvature. Submanifolds attaining equality are called Wintgen ideal. For codimension two surfaces, equality is equivalent to circularity of the curvature ellipse (superconformality), connecting the class to Bryant's superminimal surfaces in $m \geq 2$3, which are precisely minimal Wintgen ideal surfaces. For $m \geq 2$4, equality yields the Choi–Lu normal form for the shape operators, which is the key structural tool. When $m \geq 2$5, equality forces total umbilicity.

The second program consists of the Chen-type rigidity conjectures. Prior progress has concentrated heavily on hypersurfaces (Hasanis–Vlachos, Defever, Dimitrić, Fu–Hong–Zhan) or on higher-codimensional results requiring additional hypotheses: properness of the immersion (Akutagawa–Maeta, Maeta), square-integrability of $m \geq 2$6 (Nakauchi–Urakawa), $m \geq 2$7 and volume-growth assumptions (Luo), or parallelism of the normalized mean curvature vector (Balmuş–Montaldo–Oniciuc). The present result requires no completeness, no properness, no integrability condition, and no parallelism assumption—only the pointwise Wintgen ideal condition—making it a genuinely higher-codimensional contribution that holds for arbitrary immersed submanifolds.

Structure of the proof

The proof exploits the fact that every Wintgen ideal submanifold in a space form is a Chen submanifold, i.e., its allied mean curvature vector vanishes identically. This forces the vector $m \geq 2$8 to lie in $m \geq 2$9. Combined with an algebraic identity derived from the Choi–Lu normal form,

$p \geq 1$0

where $p \geq 1$1 is the mean curvature component in the distinguished normal 2-plane, a dichotomy follows at any non-umbilical point with $p \geq 1$2: either $p \geq 1$3 or $p \geq 1$4. Each alternative is then eliminated (for $p \geq 1$5) or shown to force local constancy of $p \geq 1$6 (for $p \geq 1$7).

The argument proceeds by open-set case analysis using a smooth adapted Choi–Lu frame on neighborhoods where $p \geq 1$8. In each case, Codazzi equations, the tangential biharmonic equation, and inner products of the normal biharmonic equation with $p \geq 1$9 yield sum-of-squares identities of the form

$N^{m+p}(\rho)$0

which are impossible when $N^{m+p}(\rho)$1 unless $N^{m+p}(\rho)$2, and which pin down $N^{m+p}(\rho)$3 when $N^{m+p}(\rho)$4.

The most delicate part is the degenerate case where $N^{m+p}(\rho)$5 (with $N^{m+p}(\rho)$6 after rotation), in which the linear system for the derivatives of $N^{m+p}(\rho)$7 becomes singular. Here the proof splits by dimension:

• For $N^{m+p}(\rho)$8, Codazzi equations involving tangent directions $N^{m+p}(\rho)$9 ($\rho$0) show that all remaining normal connection coefficients vanish, and combining the Gauss equation with the Ricci equation for the normal pair $\rho$1 produces the algebraic constraint

$\rho$2

which is impossible for $\rho$3 (since $\rho$4) and forces $\rho$5 to be constant for $\rho$6.

• For $\rho$7 (Wintgen ideal surfaces), $\rho$8 and the coefficient $\rho$9 degenerates, so a separate computation is required. Keeping the normal connection coefficients $\rho \leq 0$0, the Ricci equation gives $\rho \leq 0$1, impossible for $\rho \leq 0$2; a comparison between the normal biharmonic equation and the Gauss equation gives $\rho \leq 0$3, impossible for $\rho \leq 0$4.

The global conclusions follow by a standard continuity argument: for $\rho \leq 0$5, $\rho \leq 0$6 is locally constant on the set $\rho \leq 0$7, and since $\rho \leq 0$8 cannot jump from a positive value to zero across the boundary of a connected component of $\rho \leq 0$9, it is constant on each connected component of $\rho > 0$0.

Significance of the results

The theorem extends Chen-type rigidity into higher codimension under a purely pointwise hypothesis. Notably, the proof handles dimensions $\rho > 0$1 and arbitrary codimension $\rho > 0$2 uniformly, with only two points where the surface case $\rho > 0$3 requires separate treatment—the absence of tangent directions beyond the distinguished 2-plane. The result also clarifies the sharpness of the spherical statement: totally umbilical hyperspheres of suitable radius in $\rho > 0$4 are proper biharmonic with $\rho > 0$5, so constant-mean-curvature is the best possible conclusion there.

Within the Wintgen ideal class, the paper effectively resolves the analogues of all three conjectures simultaneously, since the hypotheses cover Euclidean space ($\rho > 0$6), hyperbolic space ($\rho > 0$7), and spheres ($\rho > 0$8) in one framework.

Limitations and open questions

The result is conditional on the Wintgen ideal equality in the DDVV inequality; nothing is claimed for general submanifolds, and extending these methods beyond the Choi–Lu normal form appears nontrivial. The paper also leaves open whether the full Chen conjecture and Balmuş–Montaldo–Oniciuc conjecture hold without the Wintgen ideal restriction, particularly in higher codimension where existing results require properness, integrability, or parallel-normalized-mean-curvature assumptions. A further natural question raised implicitly by the dimension split in the proof is whether the degenerate-case analysis for surfaces ($\rho > 0$9) can be unified with the higher-dimensional argument, or whether genuinely distinct phenomena occur at $|{\bf H}|$0.

Conclusion

The paper establishes that biharmonic Wintgen ideal submanifolds in space forms of nonpositive curvature are minimal, and those in positive curvature have constant mean curvature on each connected component. The proof combines the allied-vector vanishing property of Chen submanifolds with careful local analysis of the Choi–Lu normal form, handling the surface and higher-dimensional cases separately at the critical degenerate step. The results constitute partial affirmative answers to Chen's conjecture, its generalized version in hyperbolic space, and the Balmuş–Montaldo–Oniciuc conjecture in spheres, within a natural higher-codimensional geometric class and without completeness or global hypotheses.