TL;DR
This paper develops divergence theorems in semi-Riemannian supergeometry, linking boundary integrals, superharmonic functions, and supersphere integration, providing new proofs for mean value theorems of harmonic superfunctions.
Contribution
It establishes divergence theorems in supergeometry using vector field flows and boundary integrals, and offers a new proof of mean value theorems for harmonic superfunctions.
Findings
Divergence theorems in semi-Riemannian supergeometry are formulated.
Boundary integrals are connected to conserved quantities in supergeometry.
New proofs for mean value theorems of harmonic superfunctions are provided.
Abstract
The transformation formula of the Berezin integral holds, in the non-compact case, only up to boundary integrals, which have recently been quantified by Alldridge-Hilgert-Palzer. We establish divergence theorems in semi-Riemannian supergeometry by means of the flow of vector fields and these boundary integrals, and show how superharmonic functions are related to conserved quantities. An integration over the supersphere was introduced by Coulembier-De Bie-Sommen as a generalisation of the Pizzetti integral. In this context, a mean value theorem for harmonic superfunctions was established. We formulate this integration along the lines of the general theory and give a superior proof of two mean value theorems based on our divergence theorem.
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