Gauss maps of the Ricci-mean curvature flow
Gauss maps of the Ricci-mean curvature flow
复制标题
里奇平均曲率流的高斯图
DOI：
10.1007/s10711-017-0271-8
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发表时间：
2018
影响因子：
0.5
通讯作者：
Hikaru Yamamoto
中科院分区：
文献类型：
--
作者：
Naoyuki Koike;Hikaru Yamamoto
在本文中，我们研究了里奇平均曲率流的高斯图。里奇平均曲率流是平均曲率流和环境流形上的里奇流的耦合方程。 Ruh 和 Vilms (Trans Am Math Soc 149: 569–573, 1970) 证明了欧几里得空间中最小子流形的高斯映射是调和映射，Wang (Math Res Lett 10(2–3):287–299, 2003) 通过证明其高斯映射满足调和映射，将此结果扩展到欧几里得空间中的平均曲率流热流方程。在本文中，我们推导了Ricci平均曲率流高斯图的演化方程，并作为直接推论证明了当子流形的余维数为1时，Ricci平均曲率流高斯图满足垂直调和图热流方程。
In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms (Trans Am Math Soc 149: 569–573, 1970) proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang (Math Res Lett 10(2–3):287–299, 2003) extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.