Perturbation of harmonic structures and an index-zero theorem

@article{Walsh1970,
abstract = {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 $\Delta $ by an operator $\Delta +f$, 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 $H^0(W,\{\bf H\})$ and $H^1(W,\{\bf H\})$ are (finite and) equal whenever the base space $W$ of a sheaf $\{\bf H\}$ 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 $\{\bf H\}$ satisfies the Brelot axioms and its adjoint sheaf $\{\bf H\}^*$ exists, the spaces $\{\bf H\}_W$ and $\{\bf H\}^*_W$ (where $W$ is again compact) have the same (finite) dimension.},
author = {Walsh, Bertram},
journal = {Annales de l'institut Fourier},
keywords = {partial differential equations},
language = {eng},
number = {1},
pages = {317-359},
publisher = {Association des Annales de l'Institut Fourier},
title = {Perturbation of harmonic structures and an index-zero theorem},
url = {http://eudml.org/doc/74005},
volume = {20},
year = {1970},
}

TY - JOUR
AU - Walsh, Bertram
TI - Perturbation of harmonic structures and an index-zero theorem
JO - Annales de l'institut Fourier
PY - 1970
PB - Association des Annales de l'Institut Fourier
VL - 20
IS - 1
SP - 317
EP - 359
AB - 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 $\Delta $ by an operator $\Delta +f$, 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 $H^0(W,{\bf H})$ and $H^1(W,{\bf H})$ are (finite and) equal whenever the base space $W$ of a sheaf ${\bf H}$ 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 ${\bf H}$ satisfies the Brelot axioms and its adjoint sheaf ${\bf H}^*$ exists, the spaces ${\bf H}_W$ and ${\bf H}^*_W$ (where $W$ is again compact) have the same (finite) dimension.
LA - eng
KW - partial differential equations
UR - http://eudml.org/doc/74005
ER -