具有一般边界数据的调和映射的孤立奇点
Isolated singularities of harmonic maps with generic boundary data
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中文总结 AI 辅助
本文研究具有一般边界数据的调和映射孤立奇点，提出最小指标原理，证明在低维情况下奇点切映射为径向投影，并建立无穷远刚性定理。
中文摘要 AI 辅助
我们研究了具有一般边界数据的稳态调和映射的孤立奇点的行为。对于球面目标，我们证明了当 $4\leqslant n\leqslant7$ 时，从有界光滑 $n$ 维区域到圆 $(n-1)$-球面的每个稳定稳态调和映射，若具有一般光滑边界数据，则其所有切映射仅为径向投影与正交变换的复合；对于 $7$ 维区域和圆 $k$-球面目标（$k\geqslant7$），每个具有一般光滑边界数据的稳定稳态调和映射都是光滑的。这些结果源自一个针对闭实解析目标流形的通用最小指标原理。在适当的目标假设下（确保强紧性并排除低维奇点模型），我们证明在奇点可能出现的第一个域维度中，一般边界数据迫使每个奇异切映射的链环达到可能的最小 Morse 指标。该原理既适用于稳定稳态调和映射，也适用于在相应更强的目标假设下的所有稳态调和映射。如果相关类中没有非常数切映射的链环达到该最小值，则可得一般光滑性。作为应用，我们建立了一个无穷远刚性定理。对于 $3\leqslant n\leqslant7$，我们证明任何从 $n$ 维欧氏空间到圆 $(n-1)$-球面的能量极小映射，若其爆破极限为径向投影，则其自身必为径向投影的平移。
英文摘要
We study the behavior of isolated singularities of stationary harmonic maps with generic boundary data. For round sphere targets, we prove that, for $4\leqslant n\leqslant7$, every stable stationary harmonic map from a bounded smooth $n$-dimensional domain to round $(n-1)$-sphere with generic smooth boundary data has only radial projections composed with orthogonal transformations as tangent maps at its singularities; for 7-dimensional domains and round $k$-sphere targets with $k\geqslant7$, every stable stationary harmonic map with generic smooth boundary data is smooth. These results follow from a general minimum-index principle for closed real-analytic target manifolds. Under suitable target hypotheses ensuring strong compactness and excluding lower-dimensional singularity models, we show that, in the first domain dimension in which singularities can occur, generic boundary data force the link of every singular tangent map to attain the smallest possible Morse index. This principle applies both to stable stationary harmonic maps and, under the corresponding stronger target hypotheses, to all stationary harmonic maps. If no non-constant tangent map in the relevant class has a link attaining this minimum, generic smoothness follows. As an application, we establish a rigidity at infinity theorem. For $3\leqslant n\leqslant7$, we show that any energy-minimizing map from $n$-Euclidean space to round $(n-1)$-sphere admitting the radial projection as a blow-down limit must itself be a translate of the radial projection.
发表机构
- Cornell University（康奈尔大学）
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