Index and first Betti number of \(f\)-minimal hypersurfaces and self-shrinkers (Q783782)

Let \(M^{m+1}\) be a Riemannian manifold and \(x:\Sigma^m\to M\) an immersed hypersurface. Given a smooth function \(f\) on \(M\), one can consider the \(f\)-volume functional \[ \mathrm{vol}_f(\Sigma )=\int_\Sigma e^{-f}d\mu , \] where \(d\mu \) is the Riemannian measure associated to the induced metric on \(\Sigma \). A hypersurface \(\Sigma \) is said to be \(f\)-minimal if it is a critical point of \(f\)-volume. It is easy to see that a codimension one self-shrinker \(x:\Sigma^m\to \mathbb{R}^{m+1}\) is an \(f\)-minimal hypersurface of \((\mathbb{R}^{m+1},g_{\mathrm{can}},e^{-f}d\mu )\) with the weight function \(f=-|x|^2/2\). The paper under review studies the Morse index of self-shrinkers and, more generally, of \(f\)-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, the authors show that the \(f\)-index is bounded from below by an affine function of its first Betti number, which improves index estimates known in literature when the first Betti number is large. In the complete noncompact case, the authors show that the lower bound is in terms of the dimension of the space of weighted square summable \(f\)-harmonic \(1\)-forms; in particular, in dimension \(2\), the procedure gives an index estimate in terms of the genus of the surface.