The Laplace spectrum and Hermitian spaces (Q2703565)

The purpose of the article under review is to study the influence of \(p\)-isospectrability (i.e. the equality of the spectrums of the Laplacian on \(p\)-forms), for \(p=0,1\) or \(2\), on the different classes of almost Hermitian manifolds. The principal implements used are, on the one hand, the expression of the norm of the Bochner tensor in terms of the Riemann curvature, the Ricci tensor and the scalar curvature, and, on the other hand, the formulas, in each of the three cases \(p=0,1,2\), for the first three coefficients of the asymptotic formula for the trace of the heat operator of Minakshisundaram-Pleijel-Gaffney. On Bochner-flat manifolds, \(p\)-isospectrability between a complex space form and an \(AH_{3}\) manifold (a class of almost Hermitian manifolds defined by the \(J\)-invariance of the Riemann curvature) forces, under dimension conditions, the second to be also Kähler. A slightly different approach is that disjoint classes of almost Hermitian manifolds can be made to coincide under \(p\)-isospectrability, e.g. if a Hermitian semi-Kähler Einstein manifold and an \(AH_{3}\) Bochner-flat manifold are \(p\)-isospectral then they are, in some dimensions, holomorphically isometric space forms. Replacing Bochner-flat by semi-Kähler Einstein, the author proves that if a Hermitian manifold of constant curvature is \(p\)-isospectral to a Hermitian semi-Kähler Einstein manifold then they are both flat \(H_{1}\) and locally Kähler.