Asymptotic convergence for a class of anisotropic curvature flows

Li, Haizhong; Xu, Botong; Zhang, Ruijia

doi:10.1016/j.jfa.2022.109460

Cited by 2 publications

(2 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More general, Ding-Li proved that there exists a k-convex solution M t of (1.3) for all time and, after a proper rescaling converges exponentially to a sphere centered at the origin in the [6,7]. For more examples on anisotropic flows, see [20,21,28,29] and reference therein. There are some comparable results in [7] with Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

A Class of inverse curvature type flows in Rn+1

Sheng,

Xue

2024

Preprint

View full text Add to dashboard Cite

In this paper, we consider an expanding flow of smooth, closed, $(\eta,k)$-convex hypersurfaces in Euclidean $\mathbb{R}^{n+1}$ with speed $u^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$, where $u, \rho$ are the support function and radical function of the hypersurface, respectively, $\alpha,\delta\in\mathbb{R}^1$, $\beta>0$, $k$ is an integer and $1 \leq k \leq n$, $\eta=Hg-h$, the first Newton transformation of the second fundamental form $h$, $\lambda(\eta)$ denote the eigenvalues of $g^{-1}\eta$. For $\alpha+\delta+\beta\leq 1$, we prove that the flow has a unique smooth and $(\eta,k)$-convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $\alpha+\delta+\beta< 1$, we prove that the flow with the speed $fu^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$ exists for all time, and converges smoothly after normalisation to a soliton, which is a solution of $fu^{\alpha-1}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))=\gamma$, provided that $f$ is a smooth positive function on $\mathbb{S}^n$ and $\gamma$ is a positive constant. What's more, we can use a more general flow to prove the existence of solution to a class of Hessian quotient equations again.

show abstract

Section: Introductionmentioning

confidence: 99%

A Class of inverse curvature type flows in Rn+1

Sheng,

Xue

2024

Preprint

View full text Add to dashboard Cite

show abstract

“…For an anisotropic contracting curvature flow with speed given by f u α K β , Sheng and Yi [40] proved the existence and convergence after appropriate normalisation. For the flow with speed equals to r α E β k , Li, Xu and Zhang [32] studied the convergence of k-convex and star-shaped hypersurfaces. This kind of curvature flows can be used to deal with the Minkowski problems [15,17,30,35,37,40].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

Lv,

Wang

2021

Preprint

View full text Add to dashboard Cite

In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space R n+1 with the normal speed Φ given by r α F β or u α F β , where F is a monotone, symmetric, inverse-concave, homogeneous of degree one function of the principal curvatures, r is the distance from the hypersurface to the origin and u is the support function of hypersurface. If α ≥ β + 1 when Φ = r α F β or α > β + 1 when Φ = u α F β , we prove that the flow exists for all times and converges to the origin. After proper rescaling, we prove that the normalized flow converges exponentially in the C ∞ topology to a sphere centered at the origin. Furthermore, for special inverse concave curvature function), where K is Gauss curvature and F 1 is inverseconcave, we obtain the asymptotic convergence for the flow with Φ = u α F β when α = β + 1. If α < β + 1, a counterexample is given for the above convergence when speed equals to r α F β .

show abstract

Asymptotic convergence for a class of anisotropic curvature flows

Cited by 2 publications

References 26 publications

A Class of inverse curvature type flows in Rn+1

A Class of inverse curvature type flows in Rn+1

Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

Contact Info

Product

Resources

About