Harnack inequalities for curvature flows depending on mean curvature
Let \(M^n\) be a compact orientable surface, smoothly immersed in \(\mathbb{R}^{n +1}\), and \(H\) the mean curvature of \(M^n\). An immersion \(F:M^n \to\mathbb{R}^{n+1}\) is called admissible if \(H(x)\in \Omega\) for all \(x\in M^n\), where \(\Omega\) is an open subset in \(\mathbb{R}\). The author proves Harnack inequalities for parabolic flows of \(M^n\), where the normal velocity is given by a smooth function \(f\) depending only on \(H\). If \(F_0: M^n\to \mathbb{R}^{n+1}\) is an admissible smooth immersion and \(f:\Omega \to\mathbb{R}\) is smooth with \(f'\) strict positive, then the nonlinear parabolic equation \((\partial/ \partial t)F= -f\nu\), \(F(x,0)= F_0(x)\) for all \(x\in M^n\), has a smooth admissible solution on a maximal time interval \([0,T]\), \(T>0\); \(\nu\) denotes the outward pointing unit normal. By some computational effort, the author proves the following theorem: assume that \(F_0: M^n\to \mathbb{R}^{n+1}\) is an admissible smooth and convex immersion of an orientable compact manifold \(M^n\) and \(f:\Omega \to\mathbb{R}\) is a smooth function such that all \(x\in\Omega\) we have \(f'>0\), \((f''/f') x^2\geq ax\), \(((f''/f')x)' \leq 0\), \(ff'x+ff'- (f')^2x\geq 0\), where \(a\in \mathbb{R}\) is a constant. Then we can find a small positive constant \(d\) such that \((\partial/ \partial t)f+ 2\langle\nabla f, V\rangle+ h_{ij} V^iV^j+ cf'H\geq 0\), holds for all tangent vectors \(V\), as long as \(t<T\), \(d+(a+2) t>0\), and \(M_t\) stays convex, where we have set \(c(t) =1/[d+ (a+2)t]\). Finally, the author establishes that one can use the Harnack inequality to prove longtime existence of solutions for some highly nonlinear flows.
scientific article; zbMATH DE number 1141740