7 problems

Sharp upper bound conjecture. The metric $g_0$ is the unique extremal metric on a Klein bottle and is the $\lambda_1$-maximal metric. Moreover,

Uniqueness conjecture. An extremal metric for the first eigenvalue on a Klein bottle is unique, and consequently determines a sharp upper bound for $\lambda_1$ among Klein-bottle m…

Evans's conjecture. There is no Lagrangian Klein bottle in the homology class $\beta$ when $\lambda\geq 2$.

Let $\tilde\tau_{3,1}$ be Lawson's bipolar surface, and let $\varphi:\mathbb{S}^4\to\mathbb{R}^4$ be a stereographic projection. For $n\geq 4$, regard…

Let $\tilde\tau_{3,1}$ be the bipolar surface of Lawson's $\tau_{3,1}$-surface, and let $\varphi:\mathbb{S}^4\to\mathbb{R}^4$ be a stereographic projection. For $n\geq 4$, regard…

Let $K$ be a Klein bottle immersed into $\mathbb{R}^3$, and let Lawson's minimal Klein bottle in $\mathbb{S}^3$ be mapped to $\mathbb{R}^3$ by inverse stereographic projection. Kus…

Let $\mathbb{K}$ be a Finsler Klein bottle. Its Holmes–Thompson volume is denoted by ${\rm vol}_{HT}(\mathbb{K})$, and its systole by ${\rm sys}(\mathbb{K})$. Finsler Klein-bottle…