arXiv 1911.091662019-11math.DG

有界面积极小超曲面的Morse指标、贝蒂数和奇异集

Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces

Antoine Song

备注：: v2: Section 4 improved, minor corrections, results unchanged. v3: Corrections suggested by referees, correction suggested by Giada Franz and Santiago Cordero Misteli. To appear in Duke Math. J

摘要：

我们引入了一个组合的论点来研究有界面积和高Morse指数的闭极小超曲面。设$(M^{n+1},g)$是闭黎曼流形，$Σ\subset M$是面积至多为$A>0$且Hausdorff维数至多为$n-7$的闭嵌入极小超曲面.我们给出以下边界：存在仅依赖于$n$、$g$和$A$的$C_A>0$，使得$$\sum_{i=0}^n b^i(Σ) \leq C_A \big(1+index(Σ)\big) \quad \text{ if $3\leq n+1\leq 7$},$$其中$b^i$表示任何域上的贝蒂数，$\mathcal{H}^{n-7}$是$(n-7)$维Hausdorff测度，而$Sing(Σ)$是$Σ$的奇异集。事实上，在维度$n+1=3$中，$C_A$线性地依赖于$A$。在文章的最后我们列出了一些有待解决的问题。

英文摘要：

We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $Σ\subset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$, $g$, and $A$ so that $$\sum_{i=0}^n b^i(Σ) \leq C_A \big(1+index(Σ)\big) \quad \text{ if $3\leq n+1\leq 7$},$$ $$\mathcal{H}^{n-7}\big(Sing(Σ)\big) \leq C_A \big(1+index(Σ)\big)^{7/n} \quad \text{ if $n+1\geq 8$},$$ where $b^i$ denote the Betti numbers over any field, $\mathcal{H}^{n-7}$ is the $(n-7)$-dimensional Hausdorff measure and $Sing(Σ)$ is the singular set of $Σ$. In fact in dimension $n+1=3$, $C_A$ depends linearly on $A$. We list some open problems at the end of the paper.

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摘要：

英文摘要：

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