Lojasiewicz不等式及其应用
Lojasiewicz inequalities and applications
摘要：
在实代数几何中，Lojasiewicz定理断言，任何具有聚点的解析函数的梯度流的积分曲线都有唯一的极限。Lojasiewicz证明了这一结果在20世纪60年代初的后果，他的梯度不等式。变分学中的许多问题都是关于无穷维泛函的临界点或梯度流线的问题。也许令人惊讶的是，甚至许多非线性偏微分方程在奇点处的爆破，在某种意义上，可以被认为是解析泛函的无穷维梯度流的极限。因此，爆破的唯一性问题可以作为Lojasiewicz定理的无限维版本来处理。爆破的唯一性问题也许是关于奇点的最基本的问题。这种方法的独特性是开创了莱昂西蒙30年前的面积功能和许多相关的功能使用一个精心减少到一个有限的维度设置Lojasiewicz的论点适用。最近，作者证明了两个新的无限维Lojasiewicz不等式在非紧奇点，这是众所周知的，减少Lojasiewicz的参数是不可能的，而是需要全新的技术。因此，作者解决了一个主要的长期悬而未决的问题的唯一性爆破的中曲率流在所有一般的奇点和平均凸中曲率流在所有奇点。利用这一点，作者得到了具有一般奇点的MCF时空奇异集的一个比较完整的描述。特别地，$\bf{R}^{n+1}$中的MCF的奇异集只具有一般奇异性，并且包含在维数至多为$n-1$的多个紧致Lipschitz子流形中，以及维数至多为$n-2$的一个集合中。
英文摘要：
In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a consequence of his gradient inequality. Many problems in calculus of variations are questions about critical points or gradient flow lines of an infinite dimensional functional. Perhaps surprisingly, even blowups at singularities of many nonlinear PDE's can, in a certain sense, be thought of as limits of infinite dimensional gradient flows of analytic functionals. Thus, the question of uniqueness of blowups can be approached as an infinite dimensional version of Lojasiewicz's theorem. The question of uniqueness of blowups is perhaps the most fundamental question about singularities. This approach to uniqueness was pioneered by Leon Simon thirty years ago for the area functional and many related functionals using an elaborate reduction to a finite dimensional setting where Lojasiewicz's arguments applied. Recently, the authors proved two new infinite dimensional Lojasiewicz inequalities at noncompact singularities where it was well-known that a reduction to Lojasiewicz's arguments is not possible, but instead entirely new techniques are required. As a consequence, the authors settled a major long-standing open question about uniqueness of blowups for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities. Using this, the authors have obtained a rather complete description of the space-time singular set for MCF with generic singularities. In particular, the singular set of a MCF in $\bf{R}^{n+1}$ with only generic singularities is contained in finitely many compact Lipschitz submanifolds of dimension at most $n-1$ together with a set of dimension at most $n-2$.
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