A characterization of the Ejiri torus in $S^{5}$

TL;DR

This paper characterizes the Ejiri torus in S^5, proving it minimizes the Willmore functional among tensor product Willmore tori and analyzing their stability properties.

Contribution

It establishes the minimal Willmore functional value for the Ejiri torus and classifies all tensor product constrained Willmore surfaces in S^3.

Findings

Ejiri torus attains the minimum Willmore functional of 2π^2√3 in S^5.

All tensor product Willmore tori are unstable in high co-dimension.

Ejiri torus is unstable in S^5.

Abstract

Ejiri's torus in $S^5$ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in $S^{n}$ by reducing them into elastic curves in $S^3$, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in $S^5$ attains the minimum $2\pi^2\sqrt{3}$. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable in $S^5$. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in $S^3$. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough. We conjecture that a Willmore torus having Willmore functional between $2\pi^2$ and $2\pi^2\sqrt{3}$ is either the Clifford torus, or the Ejiri torus.