About the mass of certain second order elliptic operators

Let (M,g) be a closed Riemannian manifold of dimension n≥3 and let f∈C∞(M), such that the operator Pf:=Δg+f is positive. If g is flat near some point p and f vanishes around p, we can define the mass of Pf as the constant term in the expansion of the Green function of Pf at p. In this paper, we esta...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2016-05, Vol.294, p.596-633
Main Authors: Hermann, Andreas, Humbert, Emmanuel
Format: Article
Language:English
Subjects:
Positive mass theorem
Surgery
Yamabe operator
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Summary:Let (M,g) be a closed Riemannian manifold of dimension n≥3 and let f∈C∞(M), such that the operator Pf:=Δg+f is positive. If g is flat near some point p and f vanishes around p, we can define the mass of Pf as the constant term in the expansion of the Green function of Pf at p. In this paper, we establish many results on the mass of such operators. In particular, if f:=n−24(n−1)sg, i.e. if Pf is the Yamabe operator, we show the following result: assume that there exists a closed simply connected non-spin manifold M such that the mass is non-negative for every metric g as above on M, then the mass is non-negative for every such metric on every closed manifold of the same dimension as M.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2016.03.008