Translating Solitons to Flows by Powers of The Gaussian Curvature in Riemannian Products

TL;DR

This paper studies special translating solutions called solitons for geometric flows driven by powers of Gaussian curvature in product spaces, demonstrating their existence in various classical Riemannian manifolds.

Contribution

It establishes the existence of complete rotational translating solitons for $K^eta$-flows in product spaces like Euclidean, spherical, and hyperbolic spaces.

Findings

Existence of solitons in Euclidean space.

Existence of solitons in spherical space.

Existence of solitons in hyperbolic space.

Abstract

We consider translating solitons to flows by positive powers $\alpha$ of the Gaussian curvature -- called $K^\alpha$-flows -- in Riemannian products $M\times\mathbb R.$ We prove that, when $M$ is the Euclidean space $\mathbb R^n,$ the sphere $\mathbb S^n,$ or one of the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m,$ there exist complete rotational translating solitons to $K^\alpha$-flow in $M\times\mathbb R$ for certain values of $\alpha.$