三球子域上的Biot-Savart算子与电动力学
The Biot-Savart operator and electrodynamics on subdomains of the three-sphere
摘要：
我们研究了三维球子域上正曲率几何条件下的稳态磁场。通过将毕奥-萨伐尔定律推广到作用于所有向量场的积分算子BS，我们证明了在这样的环境中电动力学的行为与欧几里得电动力学相当相似。例如，对于电流J和磁场BS（J），我们表明麦克斯韦方程自然成立。在所有情况下，我们给出的公式都是几何意义的：它们被三球面的保向等距所保持。本文描述了BS的几个性质：我们证明了它是自伴的，有界的，并推广到希尔伯特空间上的紧算子。对于行为类似电流的向量场，我们证明了旋度算子是BS的左逆算子，因此Biot-Savart算子在旋度本征值的研究中是重要的，并应用于几何和物理中的能量最小化问题.最后，我们用两个例子，这表明我们的界限通常是在一个数量级的尖锐。
英文摘要：
We study steady-state magnetic fields in the geometric setting of positive curvature on subdomains of the three-dimensional sphere. By generalizing the Biot-Savart law to an integral operator BS acting on all vector fields, we show that electrodynamics in such a setting behaves rather similarly to Euclidean electrodynamics. For instance, for current J and magnetic field BS(J), we show that Maxwell's equations naturally hold. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. This article describes several properties of BS: we show it is self-adjoint, bounded, and extends to a compact operator on a Hilbert space. For vector fields that act like currents, we prove the curl operator is a left inverse to BS; thus the Biot-Savart operator is important in the study of curl eigenvalues, with applications to energy-minimization problems in geometry and physics. We conclude with two examples, which indicate our bounds are typically within an order of magnitude of being sharp.
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