Ancient gradient flows of elliptic functionals and Morse index
Ancient gradient flows of elliptic functionals and Morse index
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DOI：
10.1353/ajm.2022.0010
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发表时间：
2019-02
影响因子：
1.7
通讯作者：
K. Choi;Christos Mantoulidis
中科院分区：
文献类型：
--
作者：
K. Choi;Christos Mantoulidis
摘要：我们研究了黎曼流形中椭圆泛函梯度流的封闭古代解，包括平均曲率流和调和映射热流。我们的工作会产生各种后果。在所有维度和余维度中，我们将古代平均曲率流分类为低面积的 ${\bf S}^n$ ：它们是稳定或收缩的赤道球体。在 ${\bf S}^3$ 中的平均曲率流情况下，我们对具有更宽松区域边界的古代流进行分类：它们是稳定或收缩的赤道或 Clifford tori。在 ${\bf S}^2$ 中嵌入曲线缩短的情况下，我们对古代有界长度流进行了完全分类：它们是稳定的或收缩的圆。
abstract:We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in ${\bf S}^n$ with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in ${\bf S}^3$, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in ${\bf S}^2$, we completely classify ancient flows of bounded length: they are steady or shrinking circles.