Critical metrics of the trace of the heat kernel on a compact manifold.
Let \(M\) be a compact connected manifold without boundary. For any Riemannian metric \(g\) on \(M\), one obtains the spectrum \(Sp(g)=\{\lambda_0(g)\leq \lambda_1(g)\leq \dots \lambda_i(g)\leq \dots\}\). If \(\{f_i\}_{i\in \mathbb{N}}\) is an orthonormal basis consisting of eigenfunctions of \(\Delta\), then the heat kernel \(K_g\) of \((M,g)\) is defined on \((0,\infty)\times M\times M\) by \(K_g(t,x,y)=\sum_{i\in \mathbb{N}} e^{-\lambda_i t}f_i(x)f_i(y)\). Then its trace \(Z_g\) is defined for any \(t>0\) as \(Z_g(t)=\int_M K_g(t,x,x)v_g\). The authors investigate the critical metrics of \(Z(t)\) considered as a functional on the space of metrics, or a conformal class of metrics, of fixed volume. The metrics on strongly harmonic manifolds and on homogeneous Riemannian spaces with irreducible isotropy representation are examples of critical metrics for \(Z(t)\). Next the authors study the conditions under which the product metric of two critical metrics is critical too. Finally, the authors obtain a classification of the critical metrics on surfaces of genus \(0\) and \(1\).
scientific article; zbMATH DE number 1975964
scientific article