[Paper Review] On biharmonic submanifolds in non-positively curved manifolds and $\varepsilon$-superbiharmonic submanifolds

Yong Luo|arXiv (Cornell University)|Jun 25, 2013

Geometric Analysis and Curvature Flows3 citations

TL;DR

This paper establishes sufficient conditions under which biharmonic submanifolds in non-positively curved Riemannian manifolds must be minimal, by imposing an $L^p$-norm integral condition on the mean curvature vector. It further extends these results to $varepsilon$-superbiharmonic submanifolds, generalizing prior work and providing new criteria for minimality in curved ambient spaces.

ABSTRACT

In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y-L. Ou and L. Tang in \cite{Ou-Ta}. However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that: (a) Every complete biharmonic submanifolds (resp. hypersurfaces) $(M, g)$ in a Riemnnian manifold $(N, h)$ with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some $p\in (0, +\infty)$, $\int_{M}|\vec{H}|^{p}du_g 0$ must be minimal. We also consider $\varepsilon$-superbiharmonic submanifolds defined recently in \cite{Wh} by G. Wheeler and prove similar results for $\varepsilon$-superbiharmonic submanifolds, which generalize the results in \cite{Wh}.

Motivation & Objective

• To investigate the validity of Chen's generalized conjecture on biharmonic submanifolds in non-positively curved manifolds.
• To identify sufficient integral conditions that guarantee biharmonic submanifolds are minimal, despite known counterexamples.
• To extend the analysis to $\varepsilon$-superbiharmonic submanifolds and generalize recent results in this class.
• To provide a refined criterion for minimality using $L^p$-norms of the mean curvature vector.

Proposed method

• The analysis relies on integral estimates involving the $L^p$-norm of the mean curvature vector $\vec{H}$ over the submanifold $M$.
• It applies Bochner-type formulas and curvature assumptions on the ambient manifold $N$ to derive vanishing results for $\vec{H}$.
• The proof uses the non-positivity of sectional curvature (or Ricci curvature for hypersurfaces) to control the growth of $\vec{H}$.
• It introduces and analyzes the class of $\varepsilon$-superbiharmonic submanifolds, defined via a perturbation of the biharmonic equation.
• The method involves integrating differential identities over $M$ and applying the divergence theorem under completeness assumptions.
• The key technical step is showing that the $L^p$-integrability of $|\vec{H}|$ forces $\vec{H} \equiv 0$ under the curvature and completeness conditions.

Experimental results

Research questions

• RQ1 Under what integral conditions on $|\vec{H}|$ do biharmonic submanifolds in non-positively curved manifolds become minimal?
• RQ2 Can the generalized Chen's conjecture be recovered under $L^p$-norm conditions on the mean curvature vector?
• RQ3 How do $\varepsilon$-superbiharmonic submanifolds behave in non-positively curved ambient spaces?
• RQ4 What role does the completeness of the submanifold play in ensuring minimality?
• RQ5 To what extent do the results generalize previous theorems on biharmonic submanifolds?

Key findings

• Every complete biharmonic submanifold $M$ in a Riemannian manifold $N$ with non-positive sectional curvature is minimal if $\int_M |\vec{H}|^p \, du_g < \infty$ for some $p \in (0, +\infty)$.
• For hypersurfaces, minimality holds if $N$ has non-positive Ricci curvature and $\int_M |\vec{H}|^p \, du_g < \infty$ for some $p \in (0, +\infty)$.
• The $L^p$-integrability of $|\vec{H}|$ implies $\vec{H} \equiv 0$ under the given curvature and completeness assumptions.
• The results extend to $\varepsilon$-superbiharmonic submanifolds, generalizing findings from [Wh].
• The paper provides a new sufficient condition for minimality that avoids the need for pointwise bounds on $\vec{H}$.
• The completeness of $M$ is essential in the argument, as it allows the application of divergence-type theorems to conclude $\vec{H} \equiv 0$.