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1. Lagrangian Mean Curvature Flow

    1. Uniqueness of Joyce-Lee-Tsui translator


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      Problem 1.05.

      Let $L_t$ be an ancient, almost calibrated, connected, exact LMCF. Assume that the tangent flow at $-\infty$ is equal to $P_1\cup P_2$ where $P_1, P_2$ are two Lagrangian planes in $\mathbb{C}^2$ meeting along a line. Must $L_t$ be the Joyce-Lee-Tsui translator or $P_1\cup P_2$? (What if we assume that $L_t$ is a translator?)

        • Weak LMCF

              Brakke flow is not a good notion of a weak solution in the Lagrangian setting because it does not preserve the Lagrangian condition through singularities.

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          Problem 1.1.

          Find a good notion of weak solutions to LMCF.

            • Examples of non-compact Special Lagrangians


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              Problem 1.15.

              Find new examples of non-compact Special Lagrangians. (For example in $T^{\star}S^n$)

                • Generic singularities of LMCF


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                  Problem 1.2.

                  Is the Lawlor neck a generic singularity for LMCF?

                    • Singularities of LMCF


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                      Problem 1.25.

                      Can we classify the singularities of LMCF in the equivariant setting?

                        • Lagrangians in Landau-Ginzburg models


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                          Problem 1.3.

                          Are there good conditions for the existence of calibrated Lagrangians in Landau-Ginzburg models?

                            • Characterize Translators for LMCF


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                              Problem 1.35.

                              Under what conditions will eternal solutions to LMCF be translators?

                                • Characterize flat Lagrangian shrinkers


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                                  Problem 1.4.

                                  Under what conditions must a non-compact complete Lagrangian shrinker be flat? (e.g. is one end and simply connected enough?)

                                    • Classification of Lagrangian tori


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                                      Problem 1.45.

                                      Can we classify Lagrangian shrinking tori in $\mathbb{C}^2$?

                                        • Blow-up of mean curvature at type II singularity


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                                          Problem 1.5.

                                          Does the mean curvature blow-up at a type II singularity? (Construct examples where it does not.)

                                            • Preserving HS conditions


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                                              Problem 1.55.

                                              Can we somehow couple the LMCF to the Kähler-Ricci flow to preserve the Hamiltonian stationary condition?

                                                • LMCF and holomorphic curves


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                                                  Problem 1.6.

                                                  Relate the behaviour of LMCF to the existence of J-holomorphic curves.

                                                    • Convergence of LMCF


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                                                      Problem 1.65.

                                                      Does the LMCF converge if the length of the path in the space of calibrated Lagrangian remains bounded?

                                                          Cite this as: AimPL: Stability in mirror symmetry, available at http://aimpl.org/stabmirrorv.