Rotationally symmetric translating solutions to extrinsic geometric flows

Abstract

Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a very large class of extrinsic curvature flows, namely those whose speeds are α-homogeneous (α > 0), elliptic and symmetric with respect to the principal curvatures. We show that these solutions are necessarily convex, and give precise criteria for the speed functions which determine whether these translators are defined on all of R n or contained in a cylinder. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we are also able to describe the asymptotics of the translator at infinity. Contents 1. Introduction 1 2. Preliminaries 5 3. Existence and uniqueness 11 4. Smoothness at the origin 15 5. Low homogeneities 17 6. Degenerate speeds 17 7. Nondegenerate speeds 19 8. Convexity of solutions 20 References 21

Convexity estimate for translating solitons of concave fully nonlinear extrinsic geometric flows in $\mathbb{R}^{n+1}$

“…translator Σ 1 is entire, and thus would not poses a sub solution with this characteristic. In fact, this phenomena agree with the results given in [SR21].…”

“…Actually, the author in [TS20] constructs an example of Q n−1 -translator in R n+1 , which is strictly convex, complete and asymptotic to a cylinder. More recently, in [SR21], the author has shown the existence of bowl type solutions of generic α-homogeneous (α > 0) curvature functions without any convex or concave assumption. These solutions are entire or asymptotic to a round cylinder.…”

Maximum principles and consequences for γ$\gamma$‐translators in Rn+1${\mathbb {R}}^{n+1}$

“…Consequently, by the uniqueness result exposed in [13], we obtain the main result in following corollary.…”

“…To see this, we note that these functions satisfy properties (a)-(c) on Γ + . Moreover, by the characterization given in [13], we see that both functions vanish at (0, 1, … , 1) and…”

“…As for the applications of the tangential principle, we consider an important class of 𝛾translators in ℝ 𝑛+1 called "bowl"-type solutions, rigorously constructed and characterized in [13,Theorems 1.3 and 1.4]. These hypersurfaces are complete rotationally symmetric strictly convex graphs that are defined on a ball of radius 𝑟 0 = 𝛼 √ 𝛾(1, … , 1), or defined on all ℝ 𝑛 and being asymptotic † to…”