TY - JOUR

T1 - Asymptotic Convergence for a Class of Fully Nonlinear Curvature Flows

AU - Li, Qi Rui

AU - Sheng, Weimin

AU - Wang, Xu Jia

N1 - Publisher Copyright:
© 2019, Mathematica Josephina, Inc.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space Rn + 1 with speed rασk, where σk is the k-th elementary symmetric polynomial of the principal curvatures, α∈ R1, and r is the distance from the hypersurface to the origin. If α≥ k+ 1 , we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If α< k+ 1 , a counterexample is given for the above convergence. In the case k= 1 and α≥ 2 , we also prove that the flow converges to a round point if the initial hypersurface is weakly mean-convex and star-shaped.

AB - In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space Rn + 1 with speed rασk, where σk is the k-th elementary symmetric polynomial of the principal curvatures, α∈ R1, and r is the distance from the hypersurface to the origin. If α≥ k+ 1 , we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If α< k+ 1 , a counterexample is given for the above convergence. In the case k= 1 and α≥ 2 , we also prove that the flow converges to a round point if the initial hypersurface is weakly mean-convex and star-shaped.

KW - Asymptotic behaviour

KW - Nonlinear parabolic equation

KW - σ-Flow

UR - http://www.scopus.com/inward/record.url?scp=85062721267&partnerID=8YFLogxK

U2 - 10.1007/s12220-019-00169-4

DO - 10.1007/s12220-019-00169-4

M3 - Article

SN - 1050-6926

VL - 30

SP - 834

EP - 860

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 1

ER -