Indices and Relative Indices in Contact and CR-Geometry

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AbstractProposal: DMS-9970487Principal Investigator: Charles EpsteinLet (X,H) denote a compact, odd dimensional manifold with acontact structure, H. The basic examples are (1) The co-spherebundle of a compact manifold, (2) The strictly pseudoconvexboundary of a complex manifold. Any contact manifold has adistinguished algebra of pseudo-differential operators called theHeisenberg algebra. Epstein and Melrose, building on work ofBoutet de Monvel and Guillemin, identified a family of order zeroprojection operators in the Heisenberg algebra, generalizing theSzego projector defined on the boundary of a complex manifold.They established that if S, S' are two such projectors then theprojection defined by S from the image of S' to the image of S isa Fredholm map. Let i(S,S') denote the index of thisrestriction. We call this the relative index. By consideringHeisenberg operators which act on vector bundles, many classicalindex problems can be re-phrased as relative index problems forpairs of generalized Szego projectors. The main thrust of theproposed research is to obtain formulae for the relative index interms of more classical geometric invariants, such as indices ofDirac operators and holomorphic Euler characteristics. A generalrelative index formula of this type will allow Epstein andMelrose to completely settle the general case of theAtiyah-Weinstein problem, asking for a geometric formula for theindex of an FIO defined by a contact diffeomorphism between twoco-sphere bundles. Beyond this the investigator plans to searchfor analytic invariants of contact structures. Using an approachsuggested by Lempert and an extension of the Heisenberg algebrahe hopes to obtain a second formula for i(S,S'), which should beuseful in studying the small embeddable deformations of theCR-structure on a strictly pseudoconvex, compact CR-manifold.Partial differential equations are the principal tool formathematical descriptions of problems in physics, chemistry,economics, etc., and it is a fundamental problem to decide howmany solutions, if any, such an equation has. In general thisproblem is too delicate to solve exactly, but there is a numbercalled the Fredholm index which can estimate the number ofsolutions. The main result in this subject is the "Atiyah-SingerIndex Theorem." The research proposed by Epstein is concernedwith extending these ideas to more general classes of equationswhich arise in the context of "contact manifolds," a naturalarena for problems in mechanics and control theory. Analyticaltools are being developed which should have many applicationsbeyond the formula for the Fredholm index.