Interior estimates for Translating solitons of the $Q_k$-flows in $\mathbb{R}^{n+1}$

Abstract

The main result of this paper is that a 2-convex Q k -translator with principal curvatures in the cone Γ k+1 is convex. This is analogous to the theorems by Spruck-Xiao [SX17] and Spruck-Sun [SS19] on Mean Curvature Flow.In addition, we prove interior gradient and second order estimates for Q kflow and Q k -translators in R n+1 .

Rotationally symmetric translating solutions to extrinsic geometric flows

“…Now we show differentiability at r = 0. Estimating equation ( 13) from above and below using (17), (18) (1 + w 2 )g(γe ǫ , 1) ≤ v ′ (r) ≤ (1 + w 2 ǫ )g(γ, 1) . Letting ǫ → 0, (19) lim…”

“…Urbas [20] studied soliton solutions (including bowl-type solitons) to flows by powers of the Gauss curvature, exploiting techniques from the study of Monge-Ampère-type equations. Santaella [17,18] studied translating solutions to flows by ratios Q k = S k+1 /S k of consecutive elementary symmetric polynomials S k in the principal curvatures. In particular, he constructed a bowl-type soliton for the Q n−1 -flow (which is perhaps better known as the harmonic mean curvature flow ).…”

An example of rotationally symmetric $Q_{n-1}$-translators and a non-existence theorem in $\mathbb{R}^{n+1}$

“…[Die05], [CD16] and [TS20] for some related work on this topic. In this context, a Q k -translator is a solution F (x, t) to (1) of the form…”