Integral formulas in Riemannian geometry

We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereS
 n
 . In the case of curves inS
 n
 we solve explicitly the biharmonic equation.

Biharmonic submanifolds in spheres

and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

A short survey on 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 mathematical scene since the early ’60, when some of its analytical aspects have been discussed, and regularity of its critical points is nowadays a well-developed fleld, a systematic study of the geometry of biharmonic maps has started only recently. In this lecture we focus on the geometric properties of biharmonic maps and describe some recent achievements on the subject: (a) We give the explicit classiflcations of biharmonic curves and surfaces of some Thurston’s geometries [2, 3, 4]. (b) We describe the biharmonicity of maps between warped products and using this setting we study three classes of axially symmetric biharmonic maps [1]. (c) Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy, show it derives from a variational problem on metrics, exhibit the peculiarity of dimension four, and use the stress-energy tensor to construct new examples of biharmonic maps [5].

http://icm2006.org/v_f/AbsDef/Posters/abs_0057.pdf

Biharmonic maps between 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 studied by Jiang, Chen, Caddeo, Montaldo, and Oniciuc. We then apply the equation to show that the generalized Chen conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a 2-parameter family of conformally flat metrics and a 4-parameter family of multiply warped product metrics, each of which turns the foliation of an upper-half space of R m by parallel hyperplanes into a foliation with each leaf a proper biharmonic hypersurface. We also study the biharmonicity of Hopf cylinders of a Riemannian submersion.

/pdf/biharmonic-hypersurfaces-in-riemannian-manifolds-3nvy8kw9ff.pdf

Biharmonic hypersurfaces in Riemannian manifolds

We consider the biharmonicity condition for maps between Riemann- ian manifolds (see (BK)), and study the non-geodesic biharmonic curves in the Heisenberg group H3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.

Renzo Ilario Caddeo

Papers

Biharmonic maps between Riemannian manifolds

Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group