arXiv subjects

Ruijie Ni

Publications and source records attributed to Ruijie Ni.

2 recordsLinked to original sources

arXiv2026

On the Willmore energy of flat $n$-tori in $\mathbb{R}^N$ and Chen's conjecture for $n$-tori

This paper establishes the sharp lower bound $(4nπ^2)^{n/2}$ for the Willmore energy $\mathcal{W}$ of flat $n$-tori in the Euclidean space. Up to Möbius transformations, the Clifford $n$-torus $\mathbb{S}^1\bigl(\sqrt{1/n}\,\bigr) \times \cdots \times \mathbb{S}^1\bigl(\sqrt{1/n}\,\bigr) \subset \mathbb{S}^{2n-1} \subset \mathbb{R}^{2n}$ is shown to be the unique minimizer attaining this bound. This also confirms Chen's conjecture for flat $n$-tori. However, when $n \geq3 $, we show that Chen's conjecture fails on the total mean curvature of general immersed $n$-tori: certain Möbius transformations of the Clifford $n$-torus strictly decrease the total mean curvature.

math.DG↗

arXiv2026

Willmore Energy Estimates for Klein Bottles

In this note, we provide an estimate for the Willmore energy of Klein bottles in $\mathbb R^N$, building on the ideas of Li-Yau and Montiel-Ros for $2$-tori. In particular, we prove that the Willmore energy $\mathcal{W}(ϕ)> 6π$ for any conformal branched immersion $ϕ:\mathbb{K}^2_b\rightarrow \mathbb R^N$, where $\mathbb{K}^2_b$ is a flat Klein bottle with $0.350\lesssim b \lesssim 0.755$. This confirms partially a conjecture of Kusner. Moreover, for a flat immersion $ψ:\mathbb{K}^2_b\rightarrow \mathbb R^N$, we derive a lower bound of $\mathcal{W}(ψ)$, which confirms a conjecture of Hirsch-Mäder-Baumdicker for flat Klein bottles in $\mathbb R^N$. We also construct a smooth family of flat Klein bottles in $\mathbb R^5$, with Willmore energy between $6.912π$ and $8π$.

math.DG↗