[Paper Review] New Examples of Biharmonic Submanifolds in $CP^n$ and $S^{2n+1}$

Wei Zhang|ArXiv.org|May 27, 2007

Geometric Analysis and Curvature Flows2 references3 citations

TL;DR

This paper constructs new examples of proper biharmonic submanifolds in complex projective space $\mathbb{C}P^n$ and the sphere $S^{2n+1}$ using the Hopf fibration to lift generalized Clifford tori. By solving specific curvature and radius conditions derived from the biharmonic equation, it proves that certain Lagrangian and real hypersurfaces of Clifford torus type are biharmonic if and only if their radii satisfy non-trivial algebraic equations, yielding explicit new families of non-minimal biharmonic submanifolds.

ABSTRACT

We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .

Motivation & Objective

• To construct new examples of proper biharmonic submanifolds in $\mathbb{C}P^n$ and $S^{2n+1}$, which are rare despite extensive study.
• To address the generalized Chen’s conjecture by providing non-minimal biharmonic submanifolds in non-positively curved ambient spaces.
• To extend known results on biharmonic tori by characterizing biharmonic Clifford-type tori in $\mathbb{C}P^n$ via radius constraints.
• To analyze the stability of these new biharmonic submanifolds using second variation of the bienergy functional.

Proposed method

• Lift generalized Clifford tori $S^{2p+1}(r) \times S^{2q+1}(s)$ from $S^{2n+1}$ to $\mathbb{C}P^n$ via the Hopf fibration to construct real hypersurfaces and Lagrangian submanifolds.
• Derive biharmonic conditions by analyzing the bitension field and normal/tangential components of the tension field in $\mathbb{C}P^n$ and $S^{2n+1}$.
• Use the fact that the Hopf fibration preserves horizontal components of the second fundamental form and mean curvature to relate geometry in $S^{2n+1}$ to $\mathbb{C}P^n$.
• Solve algebraic equations involving radii $r$, $s$ and $a_i$ that arise from the biharmonic condition under parallel mean curvature assumption.
• Apply the second variation formula for bienergy to assess stability, computing the second derivative of $E_2$ along the mean curvature vector field.
• Use symbolic computation (Mathematica) to solve and verify the explicit radius solutions for specific $n$, $p$, $q$.

Experimental results

Research questions

• RQ1 Under what conditions on the radii $r$ and $s$ is the image of a generalized Clifford torus $S^{2p+1}(r) \times S^{2q+1}(s)$ in $S^{2n+1}$ biharmonic in $\mathbb{C}P^n$?
• RQ2 What are the necessary and sufficient conditions for a Lagrangian Clifford torus in $\mathbb{C}P^n$ to be biharmonic, given its radii $a_i$?
• RQ3 Can proper biharmonic submanifolds of Clifford torus type exist in $\mathbb{C}P^n$ without being minimal?
• RQ4 Is the biharmonic structure of these submanifolds stable under variations of the inclusion map?
• RQ5 How do the curvature and mean curvature conditions in $\mathbb{C}P^n$ differ for real hypersurfaces versus Lagrangian submanifolds?

Key findings

• The image $M^C_{p,q}(r,s)$ of the generalized Clifford torus $S^{2p+1}(r) \times S^{2q+1}(s)$ in $\mathbb{C}P^n$ is biharmonic if and only if $(s/r)^2(2p+1) + (r/s)^2(2q+1) = 2(n+2)$.
• The standard Clifford torus $T^{n+1} \subset S^{2n+1}$ with radii $a_i$ is biharmonic if and only if $a_i d - 1/a_i^3 = 2(n+1)((n+1)a_i - 1/a_i)$, where $d = \sum 1/a_i^2$.
• The quotient $T^n_C$ of $T^{n+1}$ by the $S^1$-action is Lagrangian in $\mathbb{C}P^n$ and biharmonic if and only if $a_i d - 1/a_i^3 = 2(n+3)((n+1)a_i - 1/a_i)$.
• Explicit solutions for the radii $r$ and $s$ are derived for $n=4$, yielding four distinct biharmonic tori for each $(p,q)$ pair.
• The second variation of the bienergy along the mean curvature vector is negative, proving that all constructed biharmonic tori are unstable.
• The Weingarten operator eigenvalues of the constructed submanifolds are non-negative, and such submanifolds cannot be stable if all eigenvalues are non-negative.