Quantitative Stability for Fractional Yamabe minimizers

Abstract

We study quantitative stability for the fractional Yamabe functional introduced by Gonz\'alez and Qing. Its minimizers correspond to conformal metrics with constant fractional curvature. We prove that the deficit of the fractional Yamabe functional controls, with a suitable power, the distance to the set of minimizers. Our approach is based on a Lyapunov--Schmidt reduction for the nonlocal functional, together with the extension characterization of the fractional conformal Laplacian, which provides the regularity theory needed for the local analysis. As a consequence, we also obtain an analogous quantitative stability estimate for the associated extension functional.