Evolving compact locally convex curves and convex hypersurfaces
Let \(M^n\) be a compact, smooth and locally convex hypersurface in the (\(n+1\))-dimensional Euclidean space \(\mathbb{E}^{n+1}\), and let \(X\) be a smooth immersion of \(M^n\) into \(\mathbb{E}^{n+1}\). In this paper, the authors consider the next flow of locally convex surfaces: \[ \left\{ \begin{array}{ll} \frac{\partial X}{\partial t}(x,t)=\bigl[R(x,t)+W(t)-(n+1)S(x,t)\bigr]N(x,t) & \text{ on }M^n\times(0,\omega),\\ X(x,0)=X_0(x) & \text{ on }M^n, \end{array} \right. \] where \(R\) represents the sum of the radii of curvature, \(S\) denotes the support function of the surface, \(W\) is its mean width and \(N\) is the outer normal vector. Then they prove that if \(M_0^n\) is an \(n\)-manifold immersed in \(\mathbb{E}^{n+1}\), the above flow with initial \(M_0^n\) exists globally and deforms the evolving curve into an \(m\)-fold circle in the plane if \(n=1\), and drives the evolving hypersurface into a standard sphere if \(n>1\), as \(t\to\infty\).
scientific article; zbMATH DE number 7457117
scientific article