• Corpus ID: 221172823

Biharmonic hypersurfaces in hemispheres

@article{Vieira2020BiharmonicHI,
  title={Biharmonic hypersurfaces in hemispheres},
  author={Matheus Vieira},
  journal={arXiv: Differential Geometry},
  year={2020}
}
  • Matheus Vieira
  • Published 19 August 2020
  • Mathematics
  • arXiv: Differential Geometry
In this paper, we consider the Balmus-Montaldo-Oniciuc's conjecture in the case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface in a hemisphere of $S^{n+1}$ must be the small hypersphere $S^{n}\left(\frac{1}{\sqrt{2}}\right)$, provided that $n^{2}-H^{2}$ does not change sign. 
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References

SHOWING 1-10 OF 18 REFERENCES
Biharmonic hypersurfaces with bounded mean curvature
We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $\phi:(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$
Classification results for biharmonic submanifolds in spheres
We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii)
Biharmonic hypersurfaces with constant scalar curvature in space forms
Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\leq0$,
Biharmonic submanifolds in spheres
We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereSn. In the case of curves inSn we solve explicitly the biharmonic equation.
Biharmonic hypersurfaces in Riemannian manifolds
We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms
Biharmonic submanifolds with parallel mean curvature vector field in spheres
We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean
Biharmonic submanifolds in space forms
In this paper we outline some recent results concerning the classification of biharmonic submanifolds in a space form.
2-harmonic maps and their first and second variational formulas
In [1], J. Eells and L. Lemaire introduced the notion of a k-harmonic map. In this paper we study the case k = 2, derive the first and second variational formulas of the 2-harmonic maps, give
Biharmonic Maps Between Riemannian Manifolds
points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the
The stress-energy tensor for biharmonic maps
Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of
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