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Ancient solutions of codimension two surfaces with curvature pinching in \(\mathbb{R}^4\). (English) Zbl 1457.53073

Summary: We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces, which is different from the conditions of Risa and Sinestrari in [26] and we also remove the condition that the second fundamental form is uniformly bounded when \(t\in(-\infty, -1)\).

MSC:

53E10 Flows related to mean curvature
35K55 Nonlinear parabolic equations
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[27] B.
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