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Abstract Suppose that is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), , and a complete embedded minimal hypersurface of lying to one side of . If the density at infinity of is less than twice the density of , then we show that , where is the Hardt–Simon foliation of . This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of .more » « less
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Abstract Suppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in$$\mathbf {C} ^{n}$$ . We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in$$\mathbf {C} ^{2}$$ , all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.more » « less
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