Index estimates for free boundary \(f\)-minimal hypersurfaces (Q6053999)

The authors prove that for any compact free boundary \(f\)-minimal hypersurface in certain positively curved weighted manifolds the index is bounded from below by a linear function of its first Betti number. This generalizes results on minimal hypersurfaces due to \textit{L. Ambrozio} et al. [J. Differ. Geom. 108, No. 3, 379--410 (2018; Zbl 1385.53051); Math. Ann. 370, No. 3--4, 1063--1078 (2018; Zbl 1391.53007)]. To estimate the index, the authors use harmonic 1-forms to construct the test functions. This idea can be traced back to \textit{A. Savo} [Indiana Univ. Math. J. 59, No. 3, 823--838 (2010; Zbl 1209.53052)].