论文标题
平均曲率流中的质量下降和多样性
Mass Drop and Multiplicity in Mean Curvature Flow
论文作者
论文摘要
Brakke流的定义是用变异不平等的，这意味着它可能会随着时间的流逝而不连续的质量，即质量下降。长期以来，人们一直猜想与非凹陷水平集流相关的brakke流量没有质量下降，并且在布拉克不等式中达到平等。在自然假设下，我们表明当且仅当它满足多重性时，流量没有质量下降。一种应用是，没有平均凸的奇异域的水平集流没有质量下降，并且通用流程没有质量下降，直到有较高的多重性平面切线流量为止。同样，如果不润滑的流量没有较高的多样性平面作为极限流，则每个极限流都没有质量下降。对于某些重要情况，我们将这些结果升级为Brakke不平等的平等。我们表明，具有三凸爆炸类型的非凹入流是具有平等性的布拉克流动。这包括在第三维中具有通用奇异性的流，以及在第四维度中具有平均凸的近凸社区的流动。
Brakke flow is defined with a variational inequality, which means it may have discontinuous mass over time, i.e. have mass drop. It has long been conjectured that the Brakke flow associated to a nonfattening level set flow has no mass drop and achieves equality in the Brakke inequality. Under natural assumptions, we show that a flow has no mass drop if and only if it satisfies the multiplicity one conjecture $\mathcal{H}^n$-a.e. One application is that there is no mass drop for level set flows with mean convex neighborhoods of singularities, and a generic flow has no mass drop until there is a higher multiplicity planar tangent flow. Also, if a nonfattening flow has no higher multiplicity planes as limit flows, then each limit flow has no mass drop. We upgrade these results to equality in the Brakke inequality for certain important cases. We show that nonfattening flows with three-convex blow-up type are Brakke flows with equality. This includes flows with generic singularities in dimension three and flows with mean convex neighborhoods of singularities in dimension four.
