On the second variation of the biharmonic Clifford torus in S^4

TL;DR

This paper computes the second variation properties of a specific biharmonic immersion of a Clifford torus in S^4, analyzing its index, nullity, and kernel structure, and extends the results to more general minimal immersions.

Contribution

It provides explicit calculations of the biharmonic index and nullity for the Clifford torus in S^4 and establishes conditions for nonnegativity of second variations in more general settings.

Findings

Computed biharmonic index and nullity of the Clifford torus immersion.

Identified a direction in the kernel with vanishing derivatives up to third order.

Established a sufficient condition for nonnegative second variation in general minimal immersions.

Abstract

The flat torus ${\mathbb T}={\mathbb S}^1\left (\frac{1}{2} \right ) \times {\mathbb S}^1\left (\frac{1}{2} \right )$ admits a proper biharmonic isometric immersion into the unit $4$-dimensional sphere ${\mathbb S}^4$ given by $\Phi=i \circ \varphi$, where $\varphi:{\mathbb T} \to {\mathbb S}^3(\frac{1}{\sqrt 2})$ is the minimal Clifford torus and $i:{\mathbb S}^3(\frac{1}{\sqrt 2}) \to {\mathbb S}^4$ is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion $\Phi$. After, we shall study in the detail the kernel of the generalised Jacobi operator $I_2^\Phi$. We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper we shall analyse the specific contribution of $\varphi$ to the biharmonic index and nullity of $\Phi$. In this context, we shall study a more general composition $\tilde{\Phi}=\tilde{\varphi} \circ i$, where $\tilde{\varphi}: M^m \to {\mathbb S}^{n-1}(\frac{1}{\sqrt 2})$, $ m \geq 1$, $n \geq {3}$, is a minimal immersion and $i:{\mathbb S}^{n-1}(\frac{1}{\sqrt 2}) \to {\mathbb S}^n$ is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of $\tilde{\Phi}$ is nonnegatively defined on $\mathcal{C}\big (\tilde{\varphi}^{-1}T{\mathbb S}^{n-1}\big )$. Then we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the $p$-harmonic index and nullity of $\varphi$. In the final section we compare our general results with those which can be deduced from the study of the equivariant second variation.