Biharmonic rotational surfaces in the four-dimensional Euclidean space are minimal

Abstract: In this paper, we show that any biharmonic rotational surface in the four-dimensional Euclidean space is minimal. The proof is based on reducing the biharmonic equation to a system of ordinary differential equations for the profile curve and then excluding all possible non-minimal branches. This is a partial affirmative answer to Chen's conjecture.

• The paper demonstrates that any biharmonic rotational surface in E4 is minimal by systematically excluding all non-minimal configurations.
• The methodology leverages ODE reduction, Taylor expansion, and Groebner basis computations to rule out alternative branches.
• The result advances Chen's conjecture in codimension two, offering a robust local, assumption-free proof for rotational surfaces.

Biharmonic Rotational Surfaces in $\mathbb{E}^4$ Are Minimal

Context and Motivation

The paper addresses Chen's conjecture, a longstanding open problem in differential geometry concerning biharmonic submanifolds in Euclidean spaces. Specifically, Bang-Yen Chen formulated the conjecture that any biharmonic submanifold in Euclidean space is minimal, i.e., its mean curvature vector vanishes. Significant progress has been achieved for hypersurfaces in dimensions three through six, but the codimension-two scenario (surfaces in $\mathbb{E}^4$) remains less understood, especially without global immersion assumptions.

Problem Setting and Approach

This work focuses on rotational surfaces in $\mathbb{E}^4$, defined by

$$X(s, \theta) = (x(s), y(s), r(s) \cos\theta, r(s) \sin\theta),$$

with $r(s) > 0$ and $s$ being the arc-length parameter of the profile curve in $\mathbb{E}^3$. The biharmonic equation (the vanishing of $\Delta H$ for the mean curvature vector $H$) is reduced to a system of ODEs for functions parametrizing the profile curve.

The approach is to systematically exclude all possible non-minimal biharmonic branches—without assuming properness or completeness of the immersion—by leveraging compatibility conditions derived from the ODE system and by algebraically analyzing polynomial identities arising in the process.

Main Results

Theorem

Any biharmonic rotational surface in the four-dimensional Euclidean space is minimal.

This result is achieved without imposing additional restrictions (such as parallel normalized mean curvature vector or constant scalar curvature) on the rotational surfaces. The proof includes:

• Writing the biharmonic equation in terms of three functions $a$, $\mathbb{E}^4$0, $\mathbb{E}^4$1 of a new variable $\mathbb{E}^4$2 linked to the arc-length parameter.
• Systematically analyzing all possible cases (seven in total) for potential non-minimal configurations, and excluding each via contradiction—typically by reduction to incompatible algebraic conditions or polynomial identities.
• In the generic case (case VII), a Groebner basis calculation yields an obstruction, ruling out all non-minimal biharmonic rotational surfaces.

Strong Numerical and Logical Results

The proof relies on quantitative elimination techniques, including the use of Taylor expansion and Groebner basis computation in case VII, to show that all non-minimal scenarios necessarily lead to contradictions. The result covers all possible local configurations and thus provides a robust answer within the class of rotational surfaces.

Comparison and Relation to Prior Work

Prior partial results concerning biharmonic surfaces in $\mathbb{E}^4$3 typically required additional assumptions (e.g., properness, convexity, parallel mean curvature vector) and were mainly global in nature. For hypersurfaces, affirmative answers to Chen's conjecture have been obtained for up to six dimensions, and global properness guarantees minimality in all codimensions.

This work is distinctive in its purely local treatment and its avoidance of any geometric constraints on the mean curvature vector, yielding a result that is not subsumed by prior theorems concerning properly immersed submanifolds.

Rotational symmetry, which often emerges in self-similar geometric flows and biconservative theory, is pivotal in the analysis. The paper carefully distinguishes the stronger biharmonic condition from the weaker biconservative condition, emphasizing that non-minimal rotational surfaces exist for biconservative but not for biharmonic cases in $\mathbb{E}^4$4.

Theoretical and Practical Implications

The theorem advances the understanding of Chen’s conjecture in higher codimension, contributing a local non-existence result for non-minimal biharmonic rotational surfaces. Practically, this restricts the types of surface geometries which can satisfy the biharmonic condition without being minimal, with implications for modeling equilibrium surfaces in geometric analysis and for the study of geometric flows.

Theoretically, the result supports the conjectural landscape that minimality is forced by biharmonicity in Euclidean spaces, at least within the class of rotational surfaces. The techniques—especially the reduction to ODEs and algebraic elimination—provide a template for further investigation in higher codimension and more general symmetry classes.

Speculation on Future Developments

Given the effectiveness of local, algebraic approaches in excluding non-minimal biharmonic configurations, it is plausible that analogous methods may be extended to other high-codimension scenarios or to other symmetry classes beyond rotational. The recent flow-theoretic insights into biharmonic map flows could synergize with local ODE reductions to yield further progress.

Moreover, the interplay between the biharmonic and biconservative conditions suggests rich avenues for classifying critical points of higher-order Laplacians on submanifolds. Future research might seek to fully resolve Chen's conjecture locally and globally for arbitrary submanifolds in $\mathbb{E}^4$5 and beyond.

Conclusion

This paper establishes that every biharmonic rotational surface in $\mathbb{E}^4$6 is minimal, providing a local, assumption-free, partial affirmation of Chen's conjecture in codimension two. The result rests on a rigorous exclusion of all possible non-minimal branches via compatibility analysis and algebraic obstruction, marking a significant theoretical advance in biharmonic submanifold geometry (2605.09587).