# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Hypersurfaces of ${𝔼}^{4}$ with harmonic mean curvature vector. (English) Zbl 0944.53005

Let $x:{M}^{n}\to {E}^{m}$ be an isometric immersion into Euclidean space. According to the well known Beltrami’s formula, we have ${\Delta }x=H$, where ${\Delta }$ is the Laplacian of $\left(M,x\right)$ and $H$ its mean curvature vector, so that Euclidean minimal submanifolds are characterized as those with harmonic position vector. B.-Y. Chen [Soochow J. Math. 17, 169-188 (1991; Zbl 0749.53037)]posed the problem of classifying Euclidean submanifolds with harmonic mean curvature vector, that is, those submanifolds satisfying ${{\Delta }}^{2}x=0$, which are therefore called biharmonic submanifolds. He also conjectured that the only biharmonic Euclidean submanifolds are the minimal ones. The conjecture is known to be true in some special cases: surfaces in ${E}^{3}$ (B.-Y. Chen), curves and submanifolds with constant mean curvature (I. Dimitric), and hypersurfaces in ${E}^{4}$ (T. Hasanis and T. Vlachos), for instance.

In this paper, the author gives a new proof of the Hasanis-Vlachos result (who gave the result in a more ample context) by using a method which is coordinate independent and does not appeal to the use of the computer, in the hope that it can be easier generalized to higher dimensional hypersurfaces.

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean $n$-space 53A10 Minimal surfaces, surfaces with prescribed mean curvature