Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions (Q2352240)

This paper gives heat-trace asymptotics for Hodge Laplacian with algebraic boundary conditions on particular stratified spaces, called edge manifolds: the expansion contains logarithmic terms, as in the exotic expansions obtained by \textit{K. Kirsten} et al. [J. Math. Phys. 47, No. 4, 043506, 27 p. (2006; Zbl 1111.58025)] for abstract regular-singular operator. The particular case of the space with isolated cone-like singularities has been studied by \textit{J. Cheeger} [Proc. Symp. Pure Math., Vol. 36, 91--146 (1980; Zbl 0461.58002); Proc. Natl. Acad. Sci. USA 76, 2103--2106 (1979; Zbl 0411.58003)] for Laplacian on functions and by \textit{E. A. Mooers} [J. Anal. Math. 78, 1--36 (1999; Zbl 0981.58022)] for Hodge Laplacian on forms of any degree. The case considered here concerns a Riemannian stratified space \((\overline M,g_M)\) with two strata, a top-dimensional open stratum \(M\) and a lower dimensional stratum \(B\): there exists an open neighborhood \(U\) of \(B\) such that \(U\) is a fibration with base \(B\) and with fibre a truncated cone \([(0,1)_x\times F]/{\sim}\) where the equivalence relation \(\sim\) identifies all \((0,f), f\in F\) together and \(F\) is a smooth manifold. The space \(\overline M\) admits a blow-up \(\varphi:\widetilde M\to \overline M\) where \(\widetilde M\) is a manifold with boundary : \(\partial \widetilde M=\varphi^{-1}(\partial\overline M)\) (\textit{resp.} \(\widetilde U=\varphi^{-1}(U)\)) is the total space of a fibration with base \(B\) and with fibre \(F\) (\textit{resp.} a truncated cylinder \([0,1)_x\times F\)). The metric \(g_M\) restricted to \(U\setminus B\) is supposed to be asymptotic to the edge fibre metric \(dx^2+x^2g_F+\varphi^*g_B\), with an admissible condition requiring a Riemannian submersion property: the metric \(g_M\) is not complete and the induced Hodge Laplacian with core domain the space \(\Omega_0^*(M)\) of smooth compactly supported forms on \(M\) is not essentially self-adjoint; the choice of boundary condition is necessary. In this general setting, the description of the self-adjoint extension domains defined between the minimal and maximal extensions domain of the Hodge Laplacian is not so easily achieved as in the case of singular spaces with conical singularities (the case when \(B\) is 0-dimensional) where the Hodge Laplacian self-adjoint extensions are parametrized by a matrix whose coefficients are indexed by the low spectrum \([0,1)\cap\sigma_F\) of the Hodge Laplacian \(\Delta_{g_F}\). However, such a definition of a boundary condition through a matrix \(S\) is valid for the Gauß-Bonnet operator \(D=d+\delta\) self-adjoint extensions on the edge manifold \(\overline M\), \(B\) being 0-dimensional or not. If \(D_S\) is such an self-adjoint algebraic extension, the operator of Hodge Laplace type \(\Delta_S=D_S^*D_S\) will be said obeying an algebraic boundary condition, as in the title of the paper. Similarly to the work by \textit{R. Mazzeo} and the author [Adv. Math. 231, No. 2, 1000--1040 (2012; Zbl 1255.58012)], methods of microlocal analysis permit to build the heat kernel on a convenient resolution of the heat space \(\widetilde M^2\times\mathbb{R}\) obtained by a sequence of real blowups : this heat kernel lives in a class of polyhomogeneous distributions with nice functional calculus. The different steps of the construction of the heat kernel for the Hodge Laplacian \(\Delta_S\) include study of the heat kernel for the Friedrichs extension, solution of a signaling problem (as in [Mooers, loc.cit.]) and corrections added to the Friedrichs extension heat kernel. The heat-trace asymptotics is obtained then through push-forward properties of polyhomogeneous distributions : they contain logarithmic terms in the case of non-Friedrichs extension.