Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

Alessandro Savo; Debora Impera; Michele Rimoldi

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Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

Debora Impera, Michele Rimoldi, Alessandro Savo

Differential Geometry · Submitted 22 Mar 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number.

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arXiv ID1803.08268 (version 1)

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CitationAlessandro Savo, Debora Impera, Michele Rimoldi. "Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers." arXiv:1803.08268 (2018). https://pith.science/paper/YNSPRESO

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.88 · compares with prior results 0.88 · mentions code or data availability 0.01.

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	contribution 0.91	0.87	0.06	asserted 0.78
s1	When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number.	result 1.00	0.95	0.62	asserted 1.00
s2	When the first Betti number is large, this improves index estimates known in literature.	interpretation 0.95	0.66	0.13	asserted 0.99
s3	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.	result 0.99	0.91	0.18	asserted 1.00

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Original abstract

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.

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5 arXiv papers cite this in 12 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Uses its method or tool 1 paper

As in [ref] , in order to get our results we will produce good test-functions for the quadratic form $Q_f$ using $f$-harmonic 1-forms of the $f$-minimal hypersurface.

Index and first Betti number of f-minimal hypersurfaces: general ambients · Main $f$-index estimate · 1.00

Cites it as background 4 papers

For a more detailed discussion about stability properties and estimates on the $f$-index of $f$-minimal hypersurfaces we refer the reader to the introduction in [ref] and [ref] and references therein.

Index and first Betti number of f-minimal hypersurfaces: general ambients · Introduction · 1.00
We recall that the following Weitzenböck formula holds in the weighted setting: [equation] where $∇^*_f=∇^*+i_∇f$ and $Ric_f=Ric+Hess(f)$; see e.g. [ref] .

Quantitative index bounds for translators via topology · Basic equations · 0.83
There are also results that give lower bounds for the index of self-shrinkers in terms of their genus [ref] .

Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori · Introduction · 0.80

What it cites · 2 sentences without a confident label

Cites as background 1 paper

For other basic concepts and results about self-shrinkers and their stability properties we refer to [ref] and the very recent [ref] .

Rigidity and gap results for low index properly immersed self--shrinkers in $\mathbb{R}^{m+1}$ · Self-shrinkers · 1.00

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