Quantitative index bounds for translators via topology

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.2010 Mathematics Subject Classification. 53C42, 53C21.

Rigidity and vanishing theorems for complete translating solitons

“…In fact, the above inequality was proved in [6]. However, the author pointed out in [7] that there is a gap in their proof of this inequality. Here, we assume that this inequality holds true.…”

“…As noted in [19], the curvature condition in Theorem 1.1 is weaker than that in Theorem 7.1 in [21]. If translators is located in a halfspace, in [6,7], Impera and Rimoldi proved a weighted Sobolev inequality by using the bijective corresponding found out by Smoczyk [17] between translators and minimal hypersurfaces in a suitble warped product. Apply their Sobolev inequality, we are able to obtain the following theorem.…”

“…Some of Xin's results then were extended by Wang, Xu, and Zhao by using integral curvature pinching conditions of the trace-free second fundamental form (see [19]). Recently, in [6,7] Impera and Rimoldi studied rigidity results and topology at infinitity of translating solitons of the mean curvature flows. Their approachc relies on the theory of f -minimal hypersufaces.…”

“…Using Sobolev inequalities discovered by Impera and Rimoldi, Kunikawa and Sato [9] pointed out that any complete f -stable translating soliton admits no codimension one cycle which does not disconnect M. As a consequence, any two dimensional complete f -stable translator has genus zero. For further discussion on translators, we refer to [2,3,5,6,7,8,9,13,14,17,19,20,21] and the references therein.…”

Simply connected translating solitons contained in slabs

“…Remark 5. Simply connected translating solitons are particularly interesting because it is known (see [28], [29], [30]) that complete -dimensional stable translaters in R must be simply connected. By stable translaters we mean translaters which are linearly stable as minimal surfaces w.r.t.…”