Elliptic operators on manifolds with nonisolated singularities
by
Boris Sternin
Independent University of Moscow

We consider elliptic operators on manifolds with edges. A manifold with edges is defined as the quotient space of a smooth manifold M, whose boundary is the total space of a smooth fibration \pi:\partialM --> X (with smooth closed base and fibers) by identifying the points in the fibers. The quotient is a singular space consisting of two strata: the top dimensional stratum of smooth points and a lower dimensional stratum X called the edge. Near a point on the edge the space is isomorphic to a neighborhood of the point (0, 0) in the product Rⁿ×K_\Omega, where K_\Omega is the cone with base \Omega - the fiber of \pi. Typical differential operators of order m on manifolds with edges have the form (in a neighborhood of the singular set):

D=  1 rm å |\alpha|+l <= m  a\alphal(x, r) æ è ir  \partial \partialr ö ø l   æ è -ir  \partial \partialx ö ø \alpha   ,

(1)

where x are coordinates on the edge, \omega - coordinates on \Omega, r - radial coordinate on the cone, while a_\alphal(x, r) are smooth families of differential operators on \Omega.

The ellipticity condition (e.g. see []) for operators of this type consists of the invertibility of the principal symbol \sigma(D) defined on the compressed cotangent bundle T^*₀M on the smooth stratum and the invertibility of the edge symbol \sigma_\Lambda(D), which is an operator family on T^*₀X acting in special weighted Sobolev spaces on the infinite cone K_\Omega (for the operator D as in (1) the edge symbol is obtained by freezing the coefficients a_\alphal(x, r) at r=0 and formally replacing -i\partial/\partialx --> \xi). An elliptic operator is Fredholm in the weighted wedge Sobolev spaces.

For elliptic operators on manifolds with edges, we consider the problem of determining the contributions of the strata to the index formula. For several classes of elliptic operators we obtain the contributions of the strata as homotopy invariant functionals of the corresponding symbols. Let us state one of the index formulas.

Consider some splitting

\partialT*M =~ \pi*T*X\oplusT*\Omega\oplusR

Let \alpha:T^*X --> T^*X be a linear involution and [(\alpha)\tilde] be an extension of this involution to \partialT^*M by adding the identity in the complementary bundle. One can glue two copies of the Atiyah-Bott-Patodi spaces D^*M=S(T^*M\oplusR) along their boundaries twisting by the involution [(\alpha)\tilde]. The resulting closed manifold is denoted by N_\alpha.

Theorem 1 If \alpha is orientation reversing (det  \alpha = -1 ) and the principal symbol of an elliptic operator D is equivariant under [(\alpha)\tilde] then the index has a decomposition:

ind D =  1 2 (ind(H\otimes\gamma-11/2(T*2M)\otimes[\sigma(D)])+ind [(\alpha*\sigma\Lambda(D))-1\sigma\Lambda(D)]),

as a sum of the index of an elliptic operator on a closed manifold N_\alpha and an index of an operator on the edge X with operator-valued symbol (\alpha^*\sigma_\Lambda(D))^-1\sigma_\Lambda(D). Here H is the Hirzebruch (signature) operator on the oriented manifold N_\alpha, \gamma_t are the Grothendieck operations in K-theory, while [\sigma(D)] in Vect(N_\alpha) is the vector bundle defined by \sigma(D). The index of the first operator can be computed by the Atiyah-Singer index formula:

ind(H\otimes\gamma-11/2(T*2M)\otimes[\sigma(D)]) = < ch[\sigma(D)]Td(T*(2M)\otimesC), [N\alpha] > ,

while the index of the second term can be expressed by the Luke's theorem [].

Remark 1 1. For orientation preserving involutions a similar result is valid for anti-equivariant symbols: [(\alpha)\tilde]^*\sigma(D)|_\partialM=\sigma(D)^-1|_\partialM;

2. The above stated result remains valid for general edge-elliptic problems with boundary and coboundary conditions along the edge.

There is a topological obstruction to the existence of elliptic edge problems for a given elliptic operator D. We give an explicit formula for this obstruction.

Theorem 2 Let D be a differential operator with elliptic principal symbol. Then this operator has an elliptic edge problem if and only if its principal symbol satisfies the equality:

\pi*[\sigma( D)|\partialT*M]=0 in K1(T*X),

where [\sigma(D)|_\partialT^*M] in K¹( T^*\partialM) is the difference element of the restriction of the principal symbol to the boundary of the compressed cotangent bundle, while \pi_* is the direct image map induced by the projection \pi:\partialM --> X.

The results are a joint work with V.E. Nazaikinskii, A. Savin and B.-W. Schulze [].

References

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B. -W.Schulze. Pseudodifferential operators on manifolds with singularities. North Holland, Amsterdam, 1991.

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G. Luke. Pseudodifferential operators on Hilbert bundles. J. Differential Equations, 12:566-589, 1972.

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V. Nazaikinskii, A. Savin, B.-W. Schulze, and B. Sternin, Elliptic Theory on Manifolds with Nonisolated Singularities IV. Index Formulas for Elliptic Operators on Manifolds with Edges. Preprint 2003/02, ISSN 1437-739X, Institut fur Mathematik, Uni Potsdam, 2003.

Date received: April 2, 2003