arXiv 1811.005152018-11math.APmath.DG

关于极小调和映射奇异集的大小

On the size of the singular set of minimizing harmonic maps

Katarzyna Mazowiecka, Michał Miśkiewicz, Armin Schikorra

备注：: This is a merged version of our preprints arXiv:1902.03161 and arXiv:1811.00515

摘要：

本文考虑从$Ω\subset \mathbb{R}^n$到闭黎曼流形$\mathcal{N}$的极小调和映射$u$，证明了：（1）Almgren和Lieb线性律对$n \geq 4$的推广.也就是说，如果目标流形$\mathcal{N}$的基本群是有限的，我们有\[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d \mathcal{H}^{n-1}; \]（2）Hardt和Lin稳定性定理的推广。即假设目标流形是$\mathcal{N}=\mathbb{S}^2$，我们得到了$u$的奇异集在边界数据的小$W^{1,n-1}$扰动下是稳定的.在维度$n=3$中，这两个结果都被证明与较弱的假设保持一致，即，仅假设我们的映射的迹位于分数空间$W^{s,p}$中，其中$s \in (\frac{1}{2},1]$和$p \in [2,\infty)$满足$sp \geq 2$。我们还讨论了锐度。

英文摘要：

We consider minimizing harmonic maps $u$ from $Ω\subset \mathbb{R}^n$ into a closed Riemannian manifold $\mathcal{N}$ and prove: (1) an extension to $n \geq 4$ of Almgren and Lieb's linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$ is finite, we have \[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d \mathcal{H}^{n-1}; \] (2) an extension of Hardt and Lin's stability theorem. Namely, assuming that the target manifold is $\mathcal{N}=\mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s \in (\frac{1}{2},1]$ and $p \in [2,\infty)$ satisfying $sp \geq 2$. We also discuss sharpness.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文