Extensions to Cartan–Hadamard manifolds and higher codimension

Determine whether the optimal extinction-time estimates, optimal arrival-time estimates, and optimal parabolic isoperimetric estimates established for weak mean curvature flow extend to general-dimensional Cartan–Hadamard manifolds and to mean curvature flows of arbitrary codimension.

Background

The principal results are formulated in Euclidean hypersurface settings. The paper separately discusses expected analogues for two-dimensional surfaces in Cartan–Hadamard manifolds, but the explicitly posed problem asks more broadly whether the three main Euclidean estimates extend both to general-dimensional Cartan–Hadamard manifolds and to general codimension.

References

Do Theorem \ref{Optimal extinction time estimates for weak mean curvature flow}, Theorem \ref{lpestimates arrival time}, and Theorem \ref{sharp parabolic estimates} hold for the situation of general dimensional Cartan-Hadamard manifolds and of general codimensional case?

— Sharp dynamical isoperimetric principle for mean curvature flow  (2610.10461 - Choi et al., 7 Oct 2026) in Remark labeled problems, item 2

Is the following extinction time estimates \begin{equation*} T_{\mathrm{ext}(\Omega)\leq \frac{1}{2k}\left(\frac{\omegak_1(\partial\Omega)}{|\mathbb{S}k|}\right){\frac{2}{k} \end{equation*} valid and optimal for certain class of mean curvature flows (for example, convex closed mean curvature flows)?

— Sharp dynamical isoperimetric principle for mean curvature flow  (2610.10461 - Choi et al., 7 Oct 2026) in Remark labeled problems, item 4