Quantitative index bounds for translators via topology

TL;DR

This paper provides quantitative bounds on the index of translators in mean curvature flow using topology and weighted harmonic forms, with implications for stability and geometric properties.

Contribution

It introduces new quantitative estimates for the index of translators based on topology and harmonic form spaces, extending Li-Tam theory to weighted settings.

Findings

Estimates involving the dimension of weighted square integrable f-harmonic 1-forms.

Bounds on the nullity of the stability operator when principal curvatures are distinct.

Quantitative stability index estimates for translators with bounded second fundamental form.

Abstract

We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed in previous works, this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction. When there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator. This permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two-dimensional or (in higher dimension) have finite topological type and are contained in a upper halfspace.