We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, 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.

中文翻译：

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$ f $-最小超曲面和自收缩器的索引和第一个Betti数

我们研究了在具有凸权重的加权欧几里得空间中，对于平均曲率流，以及更普遍而言，$ f $最小超曲面的自收缩的莫尔斯指数。当超曲面是紧致的时，我们表明该索引从下方受到其第一个Betti数的仿射函数的限制。当第一个Betti数很大时，这会改善文献中已知的指数估计。在完全非紧实的情况下，下界是根据加权平方可加的$ f $-调和1形式的空间的尺寸而言的；特别地，在维度2中，该过程根据表面的属类给出了索引估计。