Generalized Liu–Terng conjecture for mean curvature flows in compact symmetric spaces

Let $M$ be a compact hypersurface in a simply-connected irreducible symmetric space $N$ of compact type. Let ${\boldsymbol \xi}$ be a unit normal vector field on $M$. Assume that the Einstein constant of $N$ is equal to $\kappa$. Let $\{M_t\}_{t\in(-T_{\min},T_{\max})}$ denote the maximal solution of the mean curvature flow starting from $M$, and let $A(t)$ and $H(t)$ denote the shape operator and mean curvature of $M_t$, respectively. Assume that $T_{\min}=\infty$.

Generalized Liu–Terng conjecture. (i) If $A(t)$ and $H(t)$ satisfy the inequality referred to as for some $\epsilon\in(0,1/(2\kappa))$ instead of $\epsilon_F$, then $M$ is an equifocal hypersurface satisfying $\epsilon_F=\epsilon$, where $\epsilon_F$ is as in the tangential focal data of the equifocal hypersurface.

(ii) If $A(t)$ and $H(t)$ satisfy one of the inequalities referred to as -- for some $\epsilon\in(0,1/(2\kappa))$ instead of $\epsilon_F$ and $h\in(0,1)$, then $M$ is an equifocal hypersurface satisfying $\epsilon_F=\epsilon$.

This extends the Liu–Terng conjectures for mean curvature flows in the sphere to mean curvature flows in simply-connected irreducible symmetric spaces of compact type. The conjecture concerns characterizing equifocal hypersurfaces from curvature inequalities along ancient mean curvature flows.