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Geometric properties of the space of Lagrangian self-shrinking tori in ℝ⁴ Ma, Man Shun


We prove that any sequence {Fn : ∑ → ℝ⁴} of conformally branched compact Lagrangian self-shrinkers to the mean curvature flow with uniform area upper bound has a convergent subsequence, if the conformal structures do not degenerate. When ∑ has genus one, we can drop the assumption on non-degeneracy the conformal structures. When ∑ has genus zero, we show that there is no branched immersion of ∑ as a Lagrangian self-shrinker, generalizing the rigidity result of [52] in dimension two by allowing branch points. When the area bound is small, we show that any such Lagrangian self-shrinking torus in $\mathbb R^4$ is embedded with uniform curvature estimates. For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori, along which the Lagrangian condition is preserved, area is decreasing, and the compact type I singularities with a fixed area upper bound can be perturbed away in finitely many steps. This is a Lagrangian version of the construction for embedded surfaces in ℝ³ in [17]. In the noncompact situation, we derive a parabolic Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on ℓ-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental form. This generalizes a result of Wang [53] for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in [8].

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