Ancient Lagrangian Mean-Curvature Flows with Planar Blow-Downs

Definitions and mathematical statement

Equip ${\mathbb{C}}^2$, with coordinates $z_j=x_j+iy_j$, with its Euclidean metric, symplectic form $\omega=\sum_{j=1}^2{\,\mathrm{d}} x_j\wedge{\,\mathrm{d}} y_j$, holomorphic volume form $\Omega={\,\mathrm{d}} z_1\wedge{\,\mathrm{d}} z_2$, and Liouville form $\lambda=\tfrac12\sum_{j=1}^2(x_j{\,\mathrm{d}} y_j-y_j{\,\mathrm{d}} x_j)$.

A smooth Lagrangian mean curvature flow is a family of immersions $F_t:M^2\to{\mathbb{C}}^2$, with images $L_t$, satisfying $F_t^*\omega=0$ and $(\partial_tF_t)^\perp=H_t$, where $H_t$ is the mean curvature vector. It is ancient if it is defined arbitrarily far backward in time, conventionally on an interval $( -\infty,T)$. This notation does not impose maximality or prescribe behavior at $T$. A slice is exact when $F_t^*\lambda={\,\mathrm{d}} f_t$ for a globally defined real function $f_t$ on $M$; an exact flow has this property at every time.

For an oriented Lagrangian slice, its phase is defined by $F_t^*\Omega=e^{i\theta_t}{\,\mathrm{d}}\mu_t$, where ${\,\mathrm{d}}\mu_t$ is its induced volume form and initially $e^{i\theta_t}$ is circle-valued. The usual quantitative meaning of almost calibrated is that the phase stays in a fixed open semicircle with a positive margin: for a reference angle $\alpha$ and $\varepsilon>0$, \[ \operatorname{Re}\left(e^{-i\alpha}e^{i\theta_t(p)}\right)\geq\varepsilon. \] For a uniform interpretation on the entire ancient flow, the quantifiers are $\exists\alpha\in{\mathbb{R}}\;\exists\varepsilon>0\;\forall t<T\;\forall p\in M$. The supplied record does not specify whether this uniform interpretation, or a slice-dependent bound, is intended.

A backward parabolic blow-down is obtained by taking scales $r_k\to\infty$ and considering, for a fixed spacetime center $(x_0,t_0)$, \[ L_s^{(k)}=r_k^{-1}\left(L_{t_0+r_k^2s}-x_0\right),\qquad s<0, \] whenever these slices are defined. The stated limiting configuration is a pair of real two-dimensional Lagrangian planes $P_1,P_2$ whose intersection is a real line, so that $\dim_{{\mathbb{R}}}(P_1\cap P_2)=1$. The record does not specify the convergence notion, multiplicities, choice or independence of center, whether convergence is required for the rescaled flow or only selected slices, or whether one such subsequential limit or every blow-down must have this form. Thus $P_1\cup P_2$ describes the supplied geometric support, without supplying a multiplicity convention.

Classification problem. Classify all exact, almost calibrated ancient solutions of Lagrangian mean curvature flow in ${\mathbb{C}}^2$ whose blow-down is a pair of planes intersecting in a real line. The class is to retain precisely the hypotheses stated above; completeness, embeddedness, connectedness, properness, area growth, curvature bounds, and a classification equivalence relation are not specified by the supplied record. These omissions and the unresolved blow-down and calibration conventions prevent a uniquely determined fully quantified classification statement without further source review.

The supplied dated curation labels the problem partially solved and reports that a theorem reduces the stated class to translators, while leaving classification of those translators incomplete. Here a translator means a flow whose images satisfy $L_t=L_{t_*}+(t-t_*)V$ for a fixed vector $V\in{\mathbb{C}}^2$. The reported reduction and its precise hypotheses have not been independently verified and are not asserted here as a theorem.

Short English statement

Classify exact, almost calibrated ancient Lagrangian mean curvature flows in ${\mathbb{C}}^2$ with a backward rescaling limit supported on two planes meeting in a real line. The source leaves several analytic conventions and classification criteria unspecified. Its supplied dated assessment reports a reduction to translators and labels the classification partially solved; this assessment requires independent verification.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004173$, source problem code catalogue entry; registry locations identified by the project are data/unsolvedmath/problems.json and catalog/problems.json. No individual proposer is supplied; the recorded proposed year is 2019.
• Supplied primary citation, also listed under literature sources as "OWR primary report": Differentialgeometrie im Großen (2020), Oberwolfach Reports, pp. 1791--1839, DOI 10.4171/owr/2019/30. https://doi.org/10.4171/owr/2019/30. The background attributes the collection to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source labeled "ancient solutions and translators": https://doi.org/10.1007/s10240-023-00143-5. The record supplies no authors, theorem number, or detailed theorem hypotheses.
• Literature assessment date: 2026-08-22, copied exactly from literature_checked_at. This is supplied dated curation, not an independently verified assessment. It reports that every almost calibrated exact ancient solution in the stated blow-down configuration is a translator, and that this reduction does not completely classify the translators. The supplied status is partially_solved.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial qualifications, not independent verification of mathematical completeness or current literature status.