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Comparing heat operators through local isometries or fibrations. (English) Zbl 0968.58021
Let \((M,g)\) be any connected Riemannian manifold of finite dimension \(n\). Let us denote by \(\Delta_M= \Delta_{(M,g)}\) the Laplace-Beltrami operator acting on functions and let us consider the heat equation: \[ \left( \Delta_M+ {\partial\over \partial t}\right) u=0, \] with Dirichlet or Neumann condition on the boundary if \(M\) has a nonempty boundary \(\partial M\). The corresponding heat kernel will be denoted by \(p_M(t,x,y)\) when the boundary is empty, \(p^D_M (t,x,y)\) or \(p^N_M (t,x,y)\) resp. when the boundary condition is Dirichlet’s or Neumann’s one. For a noncompact manifold, the author considers \(p_M\) to be the unique minimal heat kernel, i.e. the limit of the Dirichlet heat kernels of regular compact domains exhausting \(M\); if \(M\) is complete and if its Ricci curvature is bounded from below, then \(p_M\) is the unique heat kernel on \(M\). If \(M\) is compact, the spectrum of \(\Delta_M\) is a discrete sequence \(\{ \lambda_i(M)\}_{i=0,1,2, \dots}\) (each eigenvalue is repeated according to its finite multiplicity); in this case the author also considers the trace \(Z_M(t)\) of the heat operator \(e^{-\Delta_Mt}\) (with positive \(t)\): \[ Z_M(t)= \sum^{+ \infty}_{i=0} e^{-\lambda_i (M)t} \] and similar expressions for \(Z^D_M(t)\) and \(Z^N_M(t)\).
Given a mapping \(f:(M',g') \to(M,g)\), the aim of the author is to compare the heat kernels of the two manifolds, under suitable assumptions for \(f\).
If \(f\) is a fibration of compact manifolds with typical fiber \(F\), the so-called Kato’s inequality compares the trace of the heat operator on \((M',g')\) with the one of the trivial fibration with the same typical fiber \(F\). If \(f\) is a Riemannian submersion of compact boundaryless manifolds, whose fibers are totally geodesic submanifolds of \(M'\), then \[ Z_{M'} (t)\leq Z_{M \times F}(t)= Z_m(t) \cdot Z_F(t); \] in particular, if \(f\) is a regular \(\ell\)-sheeted Riemannian covering, one obtains: \[ Z_{M'}(t)\leq \ell\cdot Z_M(t). \] The author generalizes and improves Kato’s inequality. A comparison with the heat kernel of a suitable space-form gives, as a consequence, an analogue of Kato’s inequality for non compact manifolds, which improves the classical inequality when the manifolds are compact. The author gets another generalization for local isometries, which are no more supposed to be covering maps. Last he considers Riemannian submersions with minimal fibers.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
53C20 Global Riemannian geometry, including pinching
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