Evolution of hypersurfaces by powers of the scalar curvature

Abstract

We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.

Entire downward solitons to the scalar curvature flow in Minkowski space

“…Entire translating solitons thus readily furnish natural examples of solutions for the entire parabolic problem (2). It is moreover plausible that an entire solution of (2) necessarily converges to a translating soliton if the initial hypersurface has bounded curvature and is such that sup R n |Du| < 1 : this was proved for the mean curvature flow in [1].…”

“…It is moreover plausible that an entire solution of (2) necessarily converges to a translating soliton if the initial hypersurface has bounded curvature and is such that sup R n |Du| < 1 : this was proved for the mean curvature flow in [1]. So the study of the soliton equation ( 4) is certainly important for the study of the entire flow (2). We are especially interested here in the existence and uniqueness of entire solutions of (4), and in their asymptotic properties.…”

“…Translating solitons are solutions of (2) moving by vertical translations, i.e. such that u = a for some constant a ∈ R : they are solutions of the so-called soliton equation 2 .…”

“…for some constants α, β > 0. Such a pinching on the curvatures of the initial data was used in [2,7] for the study of the euclidean scalar curvature flow. We do not know if (12) holds for the more general solitons obtained in Theorem 3.…”

Volume-preserving flow by powers of the mth mean curvature

“…In fact, the result of [22] has been generalized to a large class of σ homogeneous of degree one (see in particular [2]). When the degree is greater than one, the analysis is less complete and often restricted to dimension two, see [6,5,34]; the results in general dimension concern specific choices of the speed [13,33,1] and all of them require a pinching condition on the initial hypersurface. The case where the degree is less than one is even more difficult.…”