Compactness of immersions with local Lipschitz representation (Q445001)

The results in the paper under review are in the spirit of Cheeger's finiteness theorem. They are a natural generalization which can be traced back to \textit{J. Langer} [Math. Ann. 270, 223--234 (1985; Zbl 0564.58010)] where the compactness of immersed surfaces in \({\mathbb R}^3\) admitting uniform bounds on the second fundamental form and the area of the surfaces is studied. The finiteness of topological types was generalized to arbitrary dimension and codimension by \textit{K. Corlette} [Geom. Dedicata 33, No. 2, 153--161 (1990; Zbl 0717.53035)] and the compactness theorem was generalized to hypersurfaces of arbitrary dimension by \textit{S. Delladio} [J. Geom. Anal. 11, No. 1, 17--42 (2001; Zbl 1034.49043)].NEWLINENEWLINELet \(\mathcal{F}^1_v(r,\lambda)\) be the set of \(C^1\)-immersions \(f:M^m\to {\mathbb R}^{m+1}\) which may locally be written over an \(m\)-space as the graph of a \(\lambda\)-Lipschitz function \(u:B_r\to{\mathbb R}\), and with vol\((M)\leq v\). Analogously, \(\mathcal{F}^0(r,\lambda)\) denotes the set of Lipschitz immersions.NEWLINENEWLINEThe first main result is the compactness of \((r,\lambda)\)-immersions in codimension one. \(\mathcal{F}^1_v(r,\lambda)\) is relatively compact in \(\mathcal{F}^0(r,\lambda)\) in the following sense: For a any sequence \(f^i:M^i\to {\mathbb R}^{m+1}\) there exist a subsequence \(f^{i_k}\), an \(f:M\to {\mathbb R}^{m+1}\) in \(\mathcal{F}^0(r,\lambda)\), and diffeomorphisms \(\phi^{i_k}:M\to M^{i_k}\), such that \(f^{i_k}\circ\phi_{i_k}\) is uniformly Lipschitz bounded and converges uniformly to \(f\). Moreover, up to diffeomorphism, there are only finitely many manifolds in \(\mathcal{F}^1_v(r,\lambda)\).NEWLINENEWLINENext, the compactness of \((r,\lambda)\)-immersions in arbitrary codimension is shown, but only for \(\lambda\leq\frac14\). In the end of the last section the author discuss some possibilities of how to prove the same result for a bigger Lipschitz constant.