Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$

Question

A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a compact manifold. This is a special class of non-compact manifolds where elliptic theory of differential operators is well-understood, Sobolev embeddings, Hodge theory, Fredholm theory are already know, standard references are:

i) R.B. Lockhart and R.C. McOwen. Elliptic differential operators on noncompact manifolds. Ann.Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 409–447.

ii)R.B. Lockhart. Fredholm, Hodge and Liouville theorems on noncompact manifolds.Trans. Amer. Math. Soc., 301:1–35, 1987.

iii) V.G. Maz’ya and B.A. Plamenevski ̆ Estimates in Lp and Holder classes and Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr., 81:25–82, 1978.

I am looking for references of elliptic operators on non compact manifolds such that outside a set $K$ (possibly non compact) the space $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)\times \mathbb{R}$. This case is more general than the cylindrical case but close to it and it seems natural to consider such spaces, however I have not been able to find any references.

Any reference or comment you could provide about elliptic operators on this class of manifolds would be very useful.

Thanks!!

Answer 1

After looking carefully for this type of manifolds I found that these manifolds are called manifolds with edges. Consider the manifold $M=\Omega\times[0,\infty)\times\mathbb{R}^{N}$ the boundary of this manifold is a fibration over $\mathbb{R}^{N}$ i.e. $\partial M=\Omega \times \left\{ 0 \right\}\times\mathbb{R}^{N}$. This is the total space of a fibration over $\mathbb{R}^{N}$ with fiber $\Omega$. Also each $y\in\mathbb{R}^{N}$ has a conical fiber $\Omega\times[0,\infty)$ so at infinity the manifold $M$ looks like a fibration of conical submanifold over $\mathbb{R}^{N}$. The stretched manifold associated to $M$ is the blow up of the tip of the cone i.e $\tilde M=\Omega\times(0,\infty)\times\mathbb{R}^{N}$ and at infinity it is a fibration of cylindrical submanifolds. It seems there are several approaches to this kind of manifolds and differential operators on them. On one hand we have the approach of Richard Melrose:

i) Differential analysis on manifolds with corners. Richard Melrose. (unfinished book). This book explain the setting of manifold with corners and edges. Unfortunately it is not finished and it does have the analysis of edge PDO yet. However some theorems are spread in many paper of R. Mazzeo.

On the other hand we have the approach of B-W Schulze.

ii) B.-W. Schulze. Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.

This looks to me a more complete and developed theory of PDO (actually $\psi$DO) on manifolds with edge singularities. Moreover B.-W. Schulze has many other books dealing with BVP and lots of applications of his theory:

iii)Ju.V. Egorov and B.-W. Schulze. Pseudo-Differential Operators, Singularities, Applications. Birkhäuser Verlag, Basel, 1997.

iv) B.-W. Schulze. Boundary Value Problems and Singular Pseudo-Differential Operators. J. Wiley, Chichester, 1998.

v)G. Harutyunyan and B.-W. Schulze. Elliptic Mixed, Transmission, and Singular Crack Problems. EMS Tracts in Mathematics Vol.4, European Math. Soc

Just to mention some of them