奇偶维Dirac算子谱函数的极值度量
Extremal metrics for spectral functions of Dirac operators in even and odd dimensions
摘要：
设（M^n，g）是闭光滑黎曼自旋流形，用D表示其Atiyah-Singer-Dirac算子.本文研究了D^2的zeta函数和泛函行列式的黎曼度量的变化，证明了Morse指标在平稳度量上的有限性，以及在度量的一般变化（即不仅是共形变化）下在平稳度量上的局部极值性.在偶数维，这也是一个新的情况下，共形拉普拉斯算子，相关的稳定性算子的日志-多齐次伪微分型，我们证明了新的结果的独立利益，对这种运营商的频谱。我们用它来证明黎曼度量变分下的局部极值，在重要的例子中，当（M^n，g）是圆n-球面时，给出了布兰森关于极值模式的猜想的部分验证。因此，当维数分别为（4k，4k + 1，4k + 2，4k + 3）时，det（D^2）具有局部（max，max，min，min）。
英文摘要：
Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of D^2, and prove finiteness of the Morse index at stationary metrics, and local extremality at such metrics under general, i.e. not only conformal, change of metrics. In even dimensions, which is also a new case for the conformal Laplacian, the relevant stability operator is of log-polyhomogeneous pseudodifferential type, and we prove new results of independent interest, on the spectrum for such operators. We use this to prove local extremality under variation of the Riemannian metric, which in the important example when (M^n, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det(D^2) has a local (max, max, min, min) when the dimension is (4k, 4k + 1, 4k + 2, 4k + 3), respectively.
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