Nonlinear estimates for hypersurfaces in terms of their fundamental forms
The paper deals with a sufficiently regular hypersurface immersed in the \((n+1)\)-dimensional Euclidean space. Such a hypersurface is determined up to a proper isometry of \(\mathbb R^{n+1}\) by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. In this article other versions of these uniqueness results are established, by means of inequalities showing that an appropriate distance between two immersions from a domain \(\omega\) of \(\mathbb R^n\) into \(\mathbb R^{n+1}\) is bounded by the \(L^p\)-norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion. Some of the results of this article generalize or are based on previous results of \textit{P. G. Ciarlet} et al. [C. R., Math., Acad. Sci. Paris 355, No. 2, 226--231 (2017; Zbl 1361.53004)] and \textit{P. G. Ciarlet} and the second author [J. Math. Pures Appl. (9) 83, No. 1, 1--15 (2004; Zbl 1050.74030)]. The proofs of these theorems are only sketched and they are supposed to appear in a submitted paper of the authors.
scientific article; zbMATH DE number 6812090