论文标题
在较高维度中存在谐波图和特征值优化
Existence of harmonic maps and eigenvalue optimization in higher dimensions
论文作者
论文摘要
我们证明，从任意闭合的歧管$（m^n，g）$ dimension $ n> 2 $的最佳规律性的非稳定谐波图中，对于任何封闭的，非拼态的coplolt $ n $，包含不包含稳定的最小两杆。特别是，这给出了谐波图的第一个一般存在结果，从高维流形到一大批积极弯曲的靶标。在圆形的特殊情况下，$ n = \ mathbb {s}^k $，$ k \ geq 3 $，我们获得了一个杰出的非稳定的谐波映射$ m \ to \ mathbb {s}^k $ of Mathbb {s}^k $，最多$ k+1 $，至少是$ k $ k $ k $ k $ $ k $ k $ k $ k $ $ $ $。此外，如果$ 3 \ leq n \ leq 5 $，我们表明这些平滑的谐波映射稳定在$ k $变大，并且对应于$ m $上的特征值优化问题的解决方案，从而概括了第一个Laplace Eigenvalue在表面上的结构最大化。
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.
