2009 # Biharmonic submanifolds of $${\mathbb{C}P^n}$$

**Abstract:** We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifoldM in CP n and the bitension field of the inclusion of the corresponding Hopf-tube in S 2n+1 . Using this relation we produce new families of proper-biharmonic submanifold…

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“…Results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space were also proved (see [28]). There was obtained the relation between the bitension field of the inclusion ȷ : M → CP n of a submanifold in CP n and the bitension field of the inclusion ȷ : M → S 2n+1 of the corresponding Hopf cylinder in S 2n+1 ,…”

confidence: 99%

“…Results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space were also proved (see [28]). There was obtained the relation between the bitension field of the inclusion ȷ : M → CP n of a submanifold in CP n and the bitension field of the inclusion ȷ : M → S 2n+1 of the corresponding Hopf cylinder in S 2n+1 ,…”

confidence: 99%

“…There exist many non-minimal biharmonic submanifolds in a sphere or a complex projective space (see, for example, [5] and [25]). On the other hand, the following conjecture proposed by Chen [12] is still open.…”

confidence: 99%

“…[2]∼ [12], [15], [17], [18], [19], [22], [24], [32], etc...). Interestingly, their examples and classification results suggest that "any biharmonic submanifold in spheres has constant mean curvature".…”

confidence: 99%