Quantization and the Hessian of Mabuchi energy
Quantization and the Hessian of Mabuchi energy
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Mabuchi 能量的量子化和 Hessian 矩阵
DOI：
10.1215/00127094-1813524
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发表时间：
2010
影响因子：
2.5
通讯作者：
J. Fine
中科院分区：
文献类型：
--
作者：
J. Fine
设L→X是紧致复流形上的一个样本丛.在L中固定一个埃尔米特度量，其曲率定义了X上的卡勒度量。Mabuchi能量Hessian算子是在研究标量曲率时产生的函数上的一个四阶椭圆算子D*D。我们用平衡能量的Hessian P*k Pk来表示D*D，这是一个在平衡嵌入研究中出现的函数。P* kPk定义在H_0（X，L_k）的Hermite自同态空间上，并赋予L_2-内积.我们首先证明了P*k Pk的渐近展开式中的首阶项是D*D。接下来我们证明，如果Aut（X，L）/λ * 是离散的，则P*k Pk的特征值和特征空间收敛于D*D的特征值和特征空间。我们还证明了收敛的Hessians的情况下，一系列的平衡嵌入往往是一个常数的标量曲率Kahler度量。作为我们的结果的后果，我们证明了估计的Phong和Sturm是尖锐的，并给出了否定的回答所提出的问题唐纳森。我们还讨论了一些可能的应用研究卡拉比流。© 2012年。
Let L→X be an ample bundle over a compact complex manifold. Fix a Hermitian metric in L whose curvature defines a Kahler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D*D on functions which arises in the study of scalar curvature. We quantize D*D by the Hessian P*k Pk of balancing energy, a function appearing in the study of balanced embeddings. P*k Pk is defined on the space of Hermitian endomorphisms of H0(X,Lk) endowed with the L2-inner product. We first prove that the leading order term in the asymptotic expansion of P*k Pk is D*D. We next show that if Aut (X,L)/ℂ* is discrete, then the eigenvalues and eigenspaces of P*k Pk converge to those of D*D. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature Kahler metric. As consequences of our results we prove that an estimate of Phong and Sturm is sharp and give a negative answer to a question posed by Donaldson. We also discuss some possible applications to the study of Calabi flow. © 2012.