论文标题
沿着稳定的超曲面的最小超曲面的通用疤痕
Generic scarring for minimal hypersurfaces along stable hypersurfaces
论文作者
论文摘要
令$ m^{n+1} $为Dimension $ 3 \ LEQ N+1 \ LEQ 7 $的封闭歧管。我们表明，对于任何连接，封闭，嵌入式，$ 2 $ 2 $ s的，稳定，稳定，最小的高度表面$ s \ subset（m，g）$，$ m $上的$ c^\ int $ m $ $ g $ a $ $ m $ $ g $对应于一系列封闭，嵌入式，嵌入式，最小的n s $ \ \ \} $ scarr and s $ c \} $ $σ_K$的摩尔斯指数均与无限分开，并且在正确重新归一化后，$σ_k$将$σ_K$收敛到$ s $作为varifolds。我们还表明，在大多数封闭的Riemannian $ 3 $ manifods中，会发生浸入稳定表面的浸泡最小表面的疤痕。
Let $M^{n+1}$ be a closed manifold of dimension $3\leq n+1\leq 7$. We show that for a $C^\infty$-generic metric $g$ on $M$, to any connected, closed, embedded, $2$-sided, stable, minimal hypersurface $S\subset (M,g)$ corresponds a sequence of closed, embedded, minimal hypersurfaces $\{Σ_k\}$ scarring along $S$, in the sense that the area and Morse index of $Σ_k$ both diverge to infinity and, when properly renormalized, $Σ_k$ converges to $S$ as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian $3$-manifods.
