Anisotropic Curvature Flow of Immersed Networks

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $$L^2$$ L 2 -norm of the anisotropic curvature blows up.

A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem

<p style='text-indent:20px;'>In this paper, a generalitzation of the <inline-formula><tex-math id="M2">\begin{document}$ L_{p} $\end{document}</tex-math></inline-formula>-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for <inline-formula><tex-math id="M3">\begin{document}$ c = 1 $\end{document}</tex-math></inline-formula>.</p>

On a curvature flow in a band domain with unbounded boundary slopes

<p style='text-indent:20px;'>This paper is devoted to an anisotropic curvature flow of the form <inline-formula><tex-math id="M1">\begin{document}$ V = A(\mathbf{n})H + B(\mathbf{n}) $\end{document}</tex-math></inline-formula> in a band domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega : = [-1,1]\times {\mathbb{R}} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \mathbf{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> denote respectively the unit normal vector, normal velocity and curvature of a graphic curve <inline-formula><tex-math id="M6">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula>. We require that the curve <inline-formula><tex-math id="M7">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula> contacts <inline-formula><tex-math id="M8">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> with slopes equaling to the heights of the contact points (which corresponds to a kind of Robin boundary conditions). In spite of the unboundedness of the boundary slopes, we are able to obtain the <i>uniform interior gradient estimates</i> for the solutions by using the zero number argument. Furthermore, when <inline-formula><tex-math id="M9">\begin{document}$ t\to \infty $\end{document}</tex-math></inline-formula>, we show that <inline-formula><tex-math id="M10">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula> converges to a traveling wave with cup-shaped profile and <i>infinite</i> boundary slopes in the <inline-formula><tex-math id="M11">\begin{document}$ C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}}) $\end{document}</tex-math></inline-formula>-topology.</p>

Deforming a Convex Hypersurface by Anisotropic Curvature Flows

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem.

Anisotropic tensor calculus

We introduce the anisotropic tensor calculus, which is a way of handling tensors that depends on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection and we introduce the anisotropic curvature tensor. We also relate the concept of anisotropic connection with the classical concept of linear connections in the vertical bundle. Furthermore, we also introduce the concept of anisotropic Lie derivative.