子流形几何的整体性质研究

Studying the geometric properties of self-shrinker by using V-harmonic maps, the existing literature is rarely involved. But this mthod is very important for us to slove the global properties of sunmanifolds; this project plans to study rigidity theorems by using V-harmonic maps. On the other hand, since the result of non proper self-shrinker is little, such that its geometric understanding is not sufficient, we plan to study its Weierstrass representation by using DPW method, and then construct the non proper self-shrinker. In addition, it is conjectured that the mean curvature blow up at the first singularity time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. The conjecture has to be yet resolved, the project plans to prove the conjecture is true under the condition that the mean curvature flow is F-stable. Meanwhile, the maximum number of components of minimal graphs in M^nxR(the generalization of Meek's conjecture to M^nxR), in the absence of the mean value theorem on minimal submenifolds in M^nxR, the existing methods are no longer effective, thus the problem becomes more difficult, and it is a problem needs to be solved, we shall treat this study. The approach of this subject will reveal the global properties of submanifolds geometry more comprehensive, the results will have important theoretical value in this area.

关于用V-调和映照方法研究self-shrinker的几何性质，现有文献很少涉及，而这种方法对解决子流形整体性问题非常重要，本课题计划用V-调和映照研究其刚性定理。另一方面，由于非逆紧self-shrinker的研究结果不多，使得我们对它的几何了解并不充分，我们计划通过DPW方法研究其Weierstrass表示，进而给出非逆紧的例子。此外，平均曲率流中有著名的猜想：R^n中平均曲率流在第一奇点处平均曲率一定blow up，该猜想至今尚未得到解决，本课题计划在F-稳定的条件下证明猜想成立。另外，M^nxR中极小图的连通分支的最大个数问题（Meeks猜想在M^nxR中的推广），由于其极小子流形上没有平均值定理，使得已有方法不再有效，从而问题变得比较困难，是令人关注、有待解决的问题，我们将对此进行研究。本课题采用的研究方法能更全面的揭示子流形几何的整体性质，其研究结果对该领域有重要的理论价值。

本项目主要利用V-调和映照研究子流形的几何性质及其相关问题。具体如下：.1) 利用新建立的Omori-Yau极值原理，我们证明了伪欧氏空间中任何完备类空自收缩子(self-shrinker)一定是仿射平面，而且不存在完备类空平移孤立子(translating soliton)。在此研究路径上，已达最优结果。.2) 证明从完备非紧黎曼流形出发的V-调和映照热流的整体存在性，将Chen- Jost-Wang [JGA, 2015]的结果从紧致带边情形推广到完备非紧情形。进一步，我们的方法可以改进我们先前得到的刘维尔型定理，从而覆盖了Brighton [JGA, 2013]的结果。.3) 研究了非线性抛物方程u_t=\Dealta_V u + au\log u正解的梯度估计和Harnack不等式，推广了Huang-Huang-Li [AGAG, 2013]和Li-Xu [AIM, 2011]的结果。.4) 当外围流形是具有有界几何的完备黎曼流形时，初始流形是闭的超曲面，考虑它在幂平均曲率流下的形变。我们在一定的曲率条件下建立了幂平均曲率流的延拓定理，推广了李逸的结果。.5) 在一定的条件下，证明了组合曲率流的长时间存在性和收敛性，推广了Chow-Luo的结果。

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