arXiv 2207.136352022-07-27math.DGmath.SP

高维调和映射和eigenvalue optimization的存在性

Existence of harmonic maps and eigenvalue optimization in higher dimensions

Mikhail Karpukhin, Daniel Stern

备注：: 60 pages

摘要：

证明了从任意维数为$n>2$的闭流形$(M^n,g)$到任意不含稳定极小二球面的闭非球面流形$N$的最优正则的非常数调和映射的存在性.特别是，这给出了从高维流形到一大类正弯曲目标的调和映射的第一个一般存在性结果。在圆球$N=\mathbb{S}^k$，$k\geq 3$的特殊情况下，我们得到了一类指数至多为$k+1$的非常数调和映射$M\to \mathbb{S}^k$，其中对于充分大的$k$，余维数的奇异集至少为$7$.此外，如果$3\leq n\leq 5$，我们表明，这些光滑的调和映射稳定的$k$变大，并对应于$M$上的特征值优化问题的解决方案，推广的第一拉普拉斯特征值的共形最大化的表面。

英文摘要：

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^n,g)$ of dimension $n>2$ to any closed, non-aspherical manifold $N$ containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^k$, $k\geq 3$, we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^k$ of index at most $k+1$, with singular set of codimension at least $7$ for $k$ sufficiently large. Furthermore, if $3\leq n\leq 5$, we show that these smooth harmonic maps stabilize as $k$ becomes large, and correspond to the solutions of an eigenvalue optimization problem on $M$, generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

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