[Paper Review] Biharmonic submanifolds in manifolds with bounded curvature

Shun Maeta|arXiv (Cornell University)|May 23, 2014

Geometric Analysis and Curvature Flows3 references3 citations

TL;DR

This paper establishes conditions under which complete biharmonic submanifolds in Riemannian manifolds with sectional curvature bounded above by a non-negative constant $ c $ must have constant mean curvature $ \sqrt{c} $. Using integral and Ricci curvature assumptions, it proves that if the mean curvature is bounded below by $ \sqrt{c} $, then it must be exactly $ \sqrt{c} $, providing partial confirmation of the BMO conjecture for biharmonic submanifolds in spheres.

ABSTRACT

We consider a complete biharmonic submanifold $ϕ:(M,g) ightarrow (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt c$. If (i) $\int_M (|{\bf H}|^2-c)^{p}dv_g

Motivation & Objective

• To provide partial affirmative answers to the BMO conjecture, which posits that all biharmonic submanifolds in spheres have constant mean curvature.
• To extend the conjecture from local to global differential geometry by focusing on complete submanifolds.
• To investigate conditions under which biharmonic submanifolds in manifolds with bounded curvature must have constant mean curvature.
• To establish sufficient geometric and analytic conditions—specifically integral decay or Ricci curvature lower bounds—under which the mean curvature is forced to be exactly $ \sqrt{c} $.

Proposed method

• Analyzes the biharmonic equation for isometric immersions $ \phi: (M,g) \to (N,h) $, where $ N $ has sectional curvature bounded above by $ c \geq 0 $.
• Applies a generalized maximum principle to the function $ |\mathbf{H}|^2 - c $, using a cut-off function $ \lambda $ on a geodesic ball.
• Imposes the condition $ \int_M (|\mathbf{H}|^2 - c)^p \, dv_g < \infty $ for some $ 0 < p < \infty $ to control the decay of the mean curvature deviation.
• Uses the Ricci curvature lower bound on $ M $ as a geometric constraint to force constancy of $ \mathbf{H} $, via spectral and comparison arguments.
• Applies the biconservative condition in space forms to derive a PDE involving $ H $ and the shape operator, leading to constancy of $ H $ under the inequality $ \lambda_i > -\frac{1}{2}mH $.
• Employs the biminimal equation $ \Delta H - H|A|^2 + cmH - \lambda H = 0 $ to analyze critical points and derive umbilical structure under constant $ H $.

Experimental results

Research questions

• RQ1 Under what conditions does a complete biharmonic submanifold in a manifold with sectional curvature ≤ c have constant mean curvature $ \sqrt{c} $?
• RQ2 Can the BMO conjecture be extended from local to global settings by assuming completeness and curvature bounds?
• RQ3 Does the integrability condition $ \int_M (|\mathbf{H}|^2 - c)^p \, dv_g < \infty $ for some $ p > 0 $ imply constancy of the mean curvature?
• RQ4 How do Ricci curvature lower bounds on the submanifold $ M $ influence the constancy of the mean curvature in biharmonic immersions?
• RQ5 What role does the biconservative condition play in forcing constant mean curvature for hypersurfaces in space forms?

Key findings

• If $ \int_M (|\mathbf{H}|^2 - c)^p \, dv_g < \infty $ for some $ 0 < p < \infty $, and the mean curvature satisfies $ |\mathbf{H}| \geq \sqrt{c} $, then $ |\mathbf{H}| = \sqrt{c} $.
• If the Ricci curvature of $ M $ is bounded from below and $ |\mathbf{H}| \geq \sqrt{c} $, then $ |\mathbf{H}| = \sqrt{c} $.
• For compact biharmonic submanifolds with $ |\mathbf{H}| \geq \sqrt{c} $, the mean curvature is necessarily $ \sqrt{c} $, without requiring integral or Ricci conditions.
• In space forms with constant curvature $ c $, any biconservative hypersurface satisfying $ \lambda_i > -\frac{1}{2}mH $ for all $ i $ has constant mean curvature.
• Under the same biconservative condition, if $ H \neq 0 $, then $ c > 0 $ and the hypersurface is totally umbilical with $ |A|^2 = cm $.
• The biminimal equation with constant $ H $ implies $ |A|^2 = cm - \lambda $, and if $ H \neq 0 $, then $ cm - \lambda > 0 $, leading to a totally umbilical structure.