Enclosure theorems for generalized mean curvature flows
A regular hypersurface is an immersion \(X:M\rightarrow \mathbb{R}^{n+1}\) of a differentiable, \(n\)-dimensional, oriented manifold without boundary, \(M\), into Euclidean space \(\mathbb{R}^{n+1}\). The author defines a new kind of mean curvature for regular hypersurfaces in \(\mathbb{R}^{n+1}\). To this end, he firstly considers a positive weight \(\omega\), i.e., a differentiable endomorphism field. Then, he defines a new notion of mean curvature \(H_{\omega }= \text{trace}(S_\omega)\), called the mean \(\omega \)-curvature. Here \(S_\omega\) is the weighted Weingarten map. Actually, \(\omega\) and \(X\) determine a differential operator \(\Delta_\omega\), called weighted Laplacian to \(X\), which generalizes the usual Laplace-Beltrami operator and which has the property \(\Delta_\omega(X)=H_\omega N\) (here \(N\) denotes the Gauss mapping of \(X\) ). The flow \(X:M\times [0,T)\rightarrow R^{n+1}\) that satisfies the equation \(\frac{\partial X}{\partial t}(p,t)=H_\omega(p,t)N(p,t)\), is called a mean \(\omega \)-curvature flow. In the present paper, the author extends the results of \textit{M. A. Grayson} [Duke Math. J. 58, No. 3, 555--558 (1989; Zbl 0677.53059)] and \textit{K. Ecker} [Proc. Cent. Math. Anal. Aust. Natl. Univ. 26, 107--119 (1991; Zbl 0741.53003)] to mean \(\omega \)-curvature flows. Using the methods of \textit{U. Dierkes} [Manuscr. Math. 69, No. 2, 203--218 (1990; Zbl 0709.53043)] and \textit{S. Hildebrandt} [Math. Z. 128, 253--269 (1972; Zbl 0253.53005)], he proves for mean \(\omega \)-curvature flows several enclosure theorems which contain the results of \textit{M. A. Grayson} [see loc. cit.], \textit{K. Ecker} [see loc. cit.], \textit{K. Räwer} [Bonn: Univ. Bonn, 24 S. (1993; Zbl 0850.49005)] and \textit{U. Clarenz} [Theorems on extremals to parametric functionals (German) Bonner Math. Schriften 322 (1999; Zbl 1161.58304)] as special cases. In particular, he obtains ``neck-pinching'' results that generalize previous observations by \textit{K. Ecker} [loc. cit.] concerning the classical mean curvature flow. Finally, the Dirichlet problem that corresponds to such a flow is discussed.
scientific article; zbMATH DE number 1969048