Comparison Theorems for Fractional GJMS Operators

Abstract

In this paper, we mainly focus on the fractional GJMS operators $P_{2{\hat{I}}^3}$ which are defined on the conformal infinity of a Poincaré-Einstein manifold. We derive two comparison inequalities of the fractional Yamabe constants associated to the fractional GJMS operators. One is between $P_1$ and $P_{2{\hat{I}}^3}$ for ${\hat{I}}^3\in (1/2,1)$, and the other is between $P_2$ and $P_{2{\hat{I}}^3}$ for ${\hat{I}}^3\in (1,2)$. They both imply the rigidity theorems by characterizing the equalities. Together with the result in \cite{WZ1}, we partially provide some evidence for the monotonicity of the fractional Yamabe constants.

Fractional GJMS operators

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