Title: Quantization and the Hessian of Mabuchi energy
Abstract: Let L→X be an ample bundle over a compact complex manifold. Fix a Hermitian metric in L whose curvature defines a Kähler 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 Pk∗Pk of balancing energy, a function appearing in the study of balanced embeddings. Pk∗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 Pk∗Pk is D∗D. We next show that if Aut(X,L)/C∗ is discrete, then the eigenvalues and eigenspaces of Pk∗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 Kähler 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.