We prove an extension of T. Takahashi's result on minimal submanifolds in Euclidean spaces and in spheres, and as a corollary obtain support for B. Y. Chen's conjecture which claims that the round spheres are the only compact surfaces of finite type in Euclidean 3-space.
In [10] we introduced a new first order language for valued fields. This language has three sorts of variables, namely variables for elements of the valued field, variables for elements of the residue field and variables for elements of the value group. contains symbols for the standard field, residue field, and value group operations and a function symbol for the valuation. Essential in our language is a function symbol for an angular component map modulo P, which is a map from the field to the residue field (see Definition 1.2).For this language we proved a quantifier elimination theorem for Henselian valued fields of equicharacteristic zero which possess such an angular component map modulo P [10, Theorem 4.1]. In the first section of this paper we give some partial results on the existence of an angular component map modulo P on an arbitrary valued field.By applying the above quantifier elimination theorem to ultraproducts ΠQp/D, we obtained a quantifier elimination, in the language , for the p-adic field Qp; and this elimination is uniform for almost all primes p [10, Corollary 4.3]. In §2 we prove that our language is essentially stronger than the natural language for p-adic fields in the sense that the angular component map modulo P cannot be defined, uniformly for almost all p, in terms of the natural language for p-adic fields.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.