Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian
metric in L whose curvature defines a K\"ahler metric on X. The Hessian of
Mabuchi energy is a fourth-order elliptic operator D on functions which arises
in the study of scalar curvature. We quantise D by the Hessian E(k) of
balancing energy, a function appearing in the study of balanced embeddings.
E(k) is defined on the space of Hermitian endomorphisms of H^0(X, L^k), endowed
with the L^2-innerproduct. We first prove that the leading order term in the
asymptotic expansion of E(k) is D. We next show that if Aut(X,L) is discrete
modulo scalars, then the eigenvalues and eigenspaces of E(k) converge to those
of D. We also prove convergence of the Hessians in the case of a sequence of
balanced embeddings tending to a constant scalar curvature K\"ahler metric. As
consequences of our results we prove that a certain estimate of Phong-Sturm is
sharp and give a negative answer to a question of Donaldson. We also discuss
some possible applications to the study of Calabi flow.Comment: 42 pages. Latest version is substantial revision. Main results now
hold with no assumptions on spectral gaps. Applications and potential
applications now included. Introduction rewritten to provide more context. To
appear in Duke Mathematical Journa
