AI 中文总结
研究伪欧几里得空间中带边界的类空平均曲率流，通过建立弱形式、引入相关向量，证明闭包和紧致性定理等，采用椭圆正则化程序证明存在性，为研究奇点外的弱类空平均曲率流提供变差设置。
AI 中文摘要
我们在伪欧几里得空间中建立了类空平均曲率流的弱形式。该框架基于类空整数可求长变差和一阶变分的伪欧几里得版本。特别关注固定类空边界的存在。我们引入了广义平均曲率向量和沿边界的弱类空余法向量。在自然的一致类空性和曲率假设下，我们证明了带边界类空变差的闭包定理。然后通过相应的伪欧几里得Brakke不等式定义类空Brakke流，并证明了具有固定边界的类空Brakke流序列的紧致性定理。建立了基本的单调性性质，反映了环境不定度量的符号结构。最后，我们采用Ilmanen的椭圆正则化程序来证明类空Brakke流的存在性，并将White的局部正则性定理应用于伪欧几里得情形。这些结果为研究奇点之外的弱类空平均曲率流提供了一个变差设置。
英文摘要
We develop a weak formulation of spacelike mean curvature flow in pseudo-Euclidean space. The framework is based on spacelike integer rectifiable varifolds and a pseudo-Euclidean version of first variation. Particular attention is paid to the presence of a fixed spacelike boundary. We introduce a generalized mean curvature vector and a weak spacelike conormal along the boundary. Under natural uniform spacelikeness and curvature assumptions, we prove a closure theorem for spacelike varifolds with boundary. We then define spacelike Brakke flows by means of the corresponding pseudo-Euclidean Brakke inequality, and prove a compactness theorem for sequences of spacelike Brakke flows with fixed boundary. Basic monotonicity properties are established, reflecting the sign structure of the ambient indefinite metric. Finally, we adapt Ilmanen's elliptic regularization procedure to prove existence of spacelike Brakke flows, and the local regularity theorem of White to the pseudo-Euclidean setting. The results provide a varifold setting for studying weak spacelike mean curvature flow beyond singularities.
Comments41 pages, 1 figure
