Perturbation of harmonic structures and an index-zero theorem
Annales de l'Institut Fourier, Volume 20 (1970) no. 1, pp. 317-359.

In the framework of an axiomatic theory of sheaves of “harmonic” functions, a notion of perturbation of these sheaves is introduced which corresponds to the replacement of the operator Δ by an operator Δ+f, in the classical situation. The “harmonic” functions with which the paper is concerned are assumed to satisfy certain hypotheses (weaker than the axioms of Bauer); it is shown that the perturbed harmonic functions also satisfy these hypotheses. Moreover, the results obtained imply that the dimensions of the spaces H 0 (W,H) and H 1 (W,H) are (finite and) equal whenever the base space W of a sheaf H satisfying these hypotheses is compact. That fact generalizes the classical theorem that the index of any second order elliptic operator on a (trivial bundle over a) compact manifold is zero. Further, it implies that whenever H satisfies the Brelot axioms and its adjoint sheaf H * exists, the spaces H W and H W * (where W is again compact) have the same (finite) dimension.

Dans le cadre d’une théorie des faisceaux de fonctions “harmoniques”, on introduit une notion de perturbation de ces faisceaux, qui correspond au remplacement de l’opérateur Δ par Δ+f . dans la théorie classique. Les faisceaux pris au point de départ satisfont à certaines hypothèses, plus faibles que les axiomes de Bauer, et on trouve que les faisceaux perturbés satisfont encore à ces mêmes hypothèses. Les résultats entraînent la finitude et l’égalité des dimensions des espaces H 0 (W,H) et H 1 (W,H), dans le cas où la base W du faisceau H est compacte. Ceci est une généralisation du théorème classique qui dit que l’indice d’un opérateur elliptique du second ordre sur une variété compacte est nul. Comme conséquence, on trouve que les espaces linéaires H W et H W * (où W est encore compacte) ont la même dimension finie, si le faisceau H satisfait aux axiomes de Brelot et si son adjoint existe.

     author = {Walsh, Bertram},
     title = {Perturbation of harmonic structures and an index-zero theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {317--359},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {20},
     number = {1},
     year = {1970},
     doi = {10.5802/aif.344},
     zbl = {0187.04303},
     mrnumber = {43 #554},
     language = {en},
     url = {}
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Walsh, Bertram. Perturbation of harmonic structures and an index-zero theorem. Annales de l'Institut Fourier, Volume 20 (1970) no. 1, pp. 317-359. doi : 10.5802/aif.344.

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