Four-dimensional pseudo-umbilical biharmonic submanifolds in the six-dimensional Euclidean space

Abstract: In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangential part of the biharmonic equation vanishes identically in dimension four. Therefore, we consider $λ$-biminimal submanifolds, whose defining equation generalizes the normal part of the biharmonic equation. In this paper, we show that any connected $λ$-biminimal pseudo-umbilical submanifold of codimension two and dimension at least three in a Euclidean space has constant mean curvature. Furthermore, when $λ$ is nonnegative, we prove that it is minimal. As a corollary, we show that any four-dimensional pseudo-umbilical biharmonic submanifold in the six-dimensional Euclidean space is minimal. This result also provides a partial affirmative answer to Chen's conjecture.