Remarks on mean curvature flow solitons in warped products

Abstract

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

Recent rigidity results for graphs with prescribed mean curvature

“…Our results also apply to Schwarzschild, ADS-Schwarzschild and Reissner-Nordström-Tangherlini spaces, considered in [37], that we now briefly recall. Having fixed a mass parameter > 0 and a compact Einstein manifold ( , ) with Ric = ( − 1) , the Schwarzschild space is the product…”

“…Theorem 2.11 (Thm. E in [37]). There exists no entire graph in the Schwarzschild and ADS-Schwarzschild space (with spherical, flat or hyperbolic topology), over a complete , that is a soliton with respect to the field √ ( ) .…”

Poincaré type inequality for hypersurfaces and rigidity results

“…So, we consider hypersurfaces satisfying the equation S α r+1 = δ G(ρ)∇ρ, η . These surfaces have been object of research in recent years as self-similar solutions of curvature flows in ambient spaces other than R m+1 , see, for example, [6] and [26]. If…”

Erratum to: Remarks on mean curvature flow solitons in warped products

“…The purpose of this note is to provide full details of the rather involved proof of the following result, [2,Theorem 3.4] (also, Theorem C in the Introduction). Indeed, the one appearing in [2] was incomplete, especially in the concluding topological part. At the same time, we simplify various arguments and reorganize the proof to make it more readable.…”

“…In what follows, (P m , , P ) is a complete m-dimensional Riemannian manifold, M = I × h P is the product of the interval I ⊂ R and of P, endowed with the warped product metric ḡ = dt 2 + h(t) 2 , P , and π I : I × h P → I is the standard projection. With our chosen normalization, a soliton hypersurface with respect to the vector field X = h(t)∂ t , with soliton constant c ∈ R, is an isometric immersion ψ : M m → M which solves…”