[Paper Review] Biharmonic hypersurfaces with constant scalar curvature in space forms

Yu Fu, Min-Chun Hong|arXiv (Cornell University)|Jun 10, 2016

Geometric Analysis and Curvature Flows23 references5 citations

TL;DR

This paper proves that biharmonic hypersurfaces with constant scalar curvature in space forms $ℝ^{n+1}(c)$ have constant mean curvature if $c > 0$, and are minimal if $c \leq 0$, provided they have at most six distinct principal curvatures. The result partially confirms Chen’s conjecture and its generalized version, establishing the nonexistence of proper biharmonic hypersurfaces with constant scalar curvature in $ℝ^{n+1}$ or $ℂ^{n+1}$ for $n < 7$.

ABSTRACT

Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c&gt;0$ and $M^n$ is minimal if $c\leq0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^{n+1}$ or hyperbolic space $\mathbb H^{n+1}$ for $n&lt;7$.

Motivation & Objective

• Address the classification of biharmonic hypersurfaces with constant scalar curvature in space forms, motivated by Chen’s conjecture and its generalized version.
• Investigate whether such hypersurfaces must be minimal or have constant mean curvature under geometric constraints.
• Provide a partial confirmation of Chen’s conjecture and the generalized Chen’s conjecture in the case of constant scalar curvature hypersurfaces.
• Extend previous results on biharmonic hypersurfaces with few principal curvatures to the case of constant scalar curvature.
• Establish the nonexistence of proper biharmonic hypersurfaces with constant scalar curvature in Euclidean and hyperbolic spaces for dimensions $n < 7$.

Proposed method

• Use the biharmonic equation for hypersurfaces in space forms, specifically the condition $\Delta \vec{H} + \text{trace } R^N(d\phi, \vec{H})d\phi = 0$, to analyze the mean curvature vector field.
• Apply the assumption of constant scalar curvature and at most six distinct principal curvatures to reduce the complexity of the system of equations.
• Employ a case-by-case analysis based on the non-vanishing structure constants $\omega_{ij}^k$ of the connection forms associated with principal curvature distributions.
• Derive polynomial equations in the principal curvatures by differentiating the biharmonic condition and using the Codazzi equations.
• Use symmetry and algebraic constraints from the curvature tensor and structure equations to show that principal curvatures must be constant, leading to contradiction unless mean curvature is constant.
• Utilize differential identities and curvature relations in space forms to eliminate inconsistent configurations, particularly in cases involving zero principal curvatures or equal curvatures.

Experimental results

Research questions

• RQ1 Under what conditions do biharmonic hypersurfaces with constant scalar curvature in space forms $\mathbb{M}^{n+1}(c)$ have constant mean curvature?
• RQ2 Does the generalized Chen’s conjecture hold for biharmonic hypersurfaces with constant scalar curvature in space forms with $c \leq 0$?
• RQ3 Can proper biharmonic hypersurfaces with constant scalar curvature exist in Euclidean space $\mathbb{E}^{n+1}$ or hyperbolic space $\mathbb{H}^{n+1}$ for $n < 7$?
• RQ4 How does the number of distinct principal curvatures (up to six) affect the integrability and rigidity of biharmonic hypersurfaces with constant scalar curvature?
• RQ5 Are there algebraic obstructions to the existence of such hypersurfaces when the structure constants $\omega_{ij}^k$ are non-zero for multiple triplets of principal curvatures?

Key findings

• Biharmonic hypersurfaces with constant scalar curvature in $\mathbb{M}^{n+1}(c)$ have constant mean curvature if $c > 0$, provided they have at most six distinct principal curvatures.
• For $c \leq 0$, such hypersurfaces are minimal, confirming a stronger rigidity condition in non-positive curvature spaces.
• The result confirms a partial case of Chen’s conjecture: there are no proper biharmonic hypersurfaces with constant scalar curvature in $\mathbb{E}^{n+1}$ or $\mathbb{H}^{n+1}$ for $n < 7$.
• Under the assumption of at most six distinct principal curvatures, the system of biharmonic equations forces the principal curvatures to be constant unless the mean curvature is constant.
• The analysis shows that configurations with multiple non-vanishing $\omega_{ij}^k$ lead to algebraic contradictions unless the mean curvature is constant.
• Specific curvature relations such as $\omega_{pp}^1 \omega_{vv}^1 = -\lambda_p\lambda_v - c$ are used to derive polynomial equations in $\lambda_1$, forcing it to be constant and thus implying constant mean curvature.