[Paper Review] Biminimal properly immersed submanifolds in complete Riemannian manifolds of non-positive curvature

Shun Maeta|arXiv (Cornell University)|Aug 2, 2012

Geometric Analysis and Curvature Flows16 references4 citations

TL;DR

This paper proves that any non-negative biminimal properly immersed submanifold in a complete Riemannian manifold with non-positive sectional curvature and polynomially bounded curvature (order < 2) is minimal. Using a generalized maximum principle on an elliptic inequality derived from the biminimal condition, the authors establish that the mean curvature vector vanishes, confirming minimality. The result provides a partial affirmative answer to the global generalized Chen’s conjecture for biharmonic submanifolds.

ABSTRACT

We consider a non-negative biminimal properly immersed submanifold $M$ (that is, a biminimal properly immersed submanifold with $λ\geq0$) in a complete Riemannian manifold $N$ with non-positive sectional curvature. Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\geq-L(1+{ m dist}_N(\cdot, q_0)^2)^{\fracα{2}}$ for some $L&gt;0,$ $2&gt;α\geq 0$ and $q_0\in N$. Then, we prove that $M$ is minimal. As a corollary, we give that any biharmonic properly immersed submanifold in a hyperbolic space is minimal. These results give affirmative partial answers to the global version of generalized Chen's conjecture.

Motivation & Objective

• To investigate the minimality of biminimal submanifolds in complete Riemannian manifolds with non-positive curvature.
• To extend the global version of Chen’s conjecture to biminimal submanifolds.
• To establish conditions under which biminimal submanifolds must be minimal, particularly focusing on curvature growth and proper immersion.
• To provide affirmative partial answers to the generalized Chen’s conjecture using maximum principle techniques.

Proposed method

• Derives an elliptic inequality for the squared mean curvature |H|² using the biminimal condition and curvature assumptions.
• Applies a generalized maximum principle (Cheng-Yau type) to the inequality Δ|H|² ≥ c|H|⁴ under curvature bounds.
• Imposes a polynomial growth bound on the sectional curvature of the ambient manifold: K^N ≥ -L(1 + dist²)^{α/2} with 0 ≤ α < 2.
• Uses the proper immersion condition to ensure the maximum principle applies globally on the submanifold.
• Applies the maximum principle to show |H|² ≡ 0, hence H ≡ 0, implying minimality.
• Extends results to hypersurfaces and cases with bounded Ricci curvature, using similar analytic techniques.

Experimental results

Research questions

• RQ1 Under what curvature conditions on the ambient manifold are non-negative biminimal submanifolds necessarily minimal?
• RQ2 Can the global generalized Chen’s conjecture be affirmed for biminimal submanifolds in non-positively curved spaces?
• RQ3 Does the proper immersion condition allow the application of maximum principles to deduce minimality?
• RQ4 How does polynomial curvature decay (order < 2) affect the behavior of biminimal submanifolds?
• RQ5 Can the minimality result be extended to submanifolds with bounded Ricci curvature?

Key findings

• Any non-negative biminimal properly immersed submanifold in a complete Riemannian manifold with non-positive sectional curvature and polynomial curvature bound of order α < 2 is minimal.
• The mean curvature vector H vanishes identically on such submanifolds, implying minimality.
• The result confirms that biharmonic properly immersed submanifolds in hyperbolic space are minimal.
• For hypersurfaces, the same conclusion holds under the same curvature and immersion conditions.
• If the Ricci curvature of the submanifold is bounded from below, then non-negative biminimal submanifolds are minimal.
• The proof relies on a generalized maximum principle applied to the elliptic inequality Δ|H|² ≥ 2m|H|⁴ in the hypersurface case and a similar inequality in the general case.