Tangent Flows and Type-Two Blow-Ups

Definitions and mathematical statement

The supplied question concerns Lagrangian mean curvature flow, locally written as a family of Lagrangian immersions $F_t:L\to M$ satisfying $(\partial_tF_t)^\perp=H_t$, where $H_t$ is the mean curvature vector. The ambient manifold, dimension, initial conditions, and singularity hypotheses are not specified in the record.

For terminology, in the standard flat model $M={\mathbb{C}}^n$, let $\omega$ be the standard symplectic form and $\Omega={\,\mathrm{d}} z_1\wedge\cdots\wedge{\,\mathrm{d}} z_n$. An oriented immersion is Lagrangian if $F_t^*\omega=0$. Its circle-valued Lagrangian angle satisfies $F_t^*\Omega=e^{i\theta_t}\,\mathrm{vol}_{L_t}$. Zero Maslov class means that this angle admits a globally defined real-valued lift. This flat model explains the terminology; the source does not explicitly select it as the intended ambient setting.

In the usual finite-time setting, a singular time $T<\infty$ is of Type II when \[ \limsup_{t\uparrow T}(T-t)\sup_{L_t}|A_t|^2=\infty, \] where $A_t$ is the second fundamental form. In the flat model, a tangent flow at a singular spacetime point $(x_0,T)$ is a subsequential limit, in an appropriate flow topology, of \[ \lambda_j\left(L_{T+s/\lambda_j^2}-x_0\right),\qquad \lambda_j\to\infty,\quad s<0. \] A curvature-normalized Type II blow-up is obtained, when the required limit exists, from rescalings \[ \mathcal M^{(j)}_s=\lambda_j\left(L_{t_j+s/\lambda_j^2}-x_j\right), \qquad \lambda_j=|A_{t_j}|(x_j)\to\infty, \qquad \lambda_j^2(T-t_j)\to\infty, \] with suitable choices of points and times. For a resulting limiting flow $\mathcal M_s$, a parabolic blow-down is a subsequential limit of \[ \rho_k\mathcal M_{s/\rho_k^2},\qquad \rho_k\downarrow0, \] for times for which these expressions are defined. These are conventional finite-time, flat-space interpretations; their existence, the convergence topology, and their applicability to the intended problem are not supplied hypotheses.

Requested research problem. Determine under what hypotheses tangent flows of a zero-Maslov Lagrangian mean curvature flow can be related to blow-downs of its Type II blow-up limits. The source does not specify whether the desired relation is equality, identification up to geometric symmetries, correspondence between selected subsequential limits, or a uniqueness assertion. Consequently, it does not determine a single quantified conjecture. In particular, no equality, uniqueness, or existence of the displayed limits is asserted here.

Short English statement

The question asks when a singularity model obtained by taking a tangent flow can be connected to a large-scale limit of a Type II blow-up in zero-Maslov Lagrangian mean curvature flow. The ambient setting, convergence notion, and intended meaning of this connection remain unspecified. The supplied partially solved label is a dated curation assessment, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004172$, exact source problem code catalogue entry, title "Tangent Flows and Type-Two Blow-Ups," in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2019 and supplies no individual proposer.
• Supplied primary citation: Differentialgeometrie im Großen (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/30, pp. 1791--1839. OWR primary report: https://doi.org/10.4171/owr/2019/30. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "ancient solutions and translators": https://doi.org/10.1007/s10240-023-00143-5.
• Supplied literature source, labeled "Type II singularities with zero Maslov class": https://doi.org/10.1016/j.jfa.2025.111032.
• Literature assessment dated 2026-08-22: supplied dated curation, not independently verified. It labels the problem partially solved and reports strong results relating Type II blow-ups, tangent flows, and blow-downs, including exactness and rigidity theorems for zero-Maslov or almost calibrated flows, while reporting that no general criterion for the requested relation or uniqueness is established. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. Those are supplied provenance claims and do not establish mathematical completeness or constitute verification in this draft.