The stability of hypersurfaces with constant shifted k ‐th mean curvature

2022 ◽  
Author(s):  
Yimin Chen ◽  
Haizhong Li
Keyword(s):  
Mean Curvature ◽  
The Stability

scholarly journals Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Heiko Kröner ◽  
Matteo Novaga
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Time Behavior ◽  
Long Time ◽  
Self Similar ◽  
The Stability ◽  
Lipschitz Graphs ◽  
Similar Solutions ◽  
Anisotropic Mean Curvature

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

scholarly journals Evolution of Brain Tumor and Stability of Geometric Invariants

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
K. Tawbe ◽  
F. Cotton ◽  
L. Vuillon
Keyword(s):  
Brain Tumor ◽  
Brain Tumors ◽  
Mean Curvature ◽  
Medical Diagnosis ◽  
Gauss Curvature ◽  
Mr Images ◽  
Geometric Invariants ◽  
The Stability ◽  
Discrete Mean Curvature ◽  
Discrete Gauss Curvature

This paper presents a method to reconstruct and to calculate geometric invariants on brain tumors. The geometric invariants considered in the paper are the volume, the area, the discrete Gauss curvature, and the discrete mean curvature. The volume of a tumor is an important aspect that helps doctors to make a medical diagnosis. And as doctors seek a stable calculation, we propose to prove the stability of some invariants. Finally, we study the evolution of brain tumor as a function of time in two or three years depending on patients with MR images every three or six months.

scholarly journals Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds

2017 ◽  
Vol 20 (01) ◽  
pp. 1650065
Author(s):  
Shiguang Ma
Keyword(s):  
Riemannian Manifolds ◽  
Stability Condition ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Asymptotically Flat ◽  
Cmc Surfaces ◽  
Ode Method ◽  
Nonlinear Ode ◽  
The Stability

In this paper, we introduce a nonlinear ODE method to construct constant mean curvature (CMC) surfaces in Riemannian manifolds with symmetry. As an application, we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like [Formula: see text]. The existence of unstable CMC spheres tells us that the stability condition in Qing–Tian’s work [On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds, J. Amer. Math. Soc. 20(4) (2007) 1091–1110] cannot be removed generally.

The stability index of hypersurfaces with constant scalar curvature in spheres

2014 ◽  
Vol 144 (3) ◽  
pp. 447-453
Author(s):  
Qing-Ming Cheng ◽  
Haizhong Li ◽  
Guoxin Wei
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
Stability Index ◽  
Dimensional Sphere ◽  
Weak Stability ◽  
Totally Umbilical ◽  
Compact Hypersurface ◽  
The Mean ◽  
The Stability

The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.

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