Integral formulas in Riemannian geometry

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

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

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 dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric.

The stress-energy tensor for biharmonic maps

Biharmonic maps between warped product manifolds

In this paper, using the framework of equivariant differential geometry, we study proper $$\hbox {SO}(p+1) \times \hbox {SO}(q+1)$$
 -invariant biconservative hypersurfaces into the Euclidean space $${\mathbb {R}}^n$$
 (
 $$n=p+q+2$$
 ) and proper $$\hbox {SO}(p+1)$$
 -invariant biconservative hypersurfaces into the Euclidean space $${\mathbb {R}}^n$$
 (
 $$n=p+2$$
 ). Moreover, we show that, in these two classes of invariant families, there exists no proper biharmonic immersion.

/pdf/proper-biconservative-immersions-into-the-euclidean-space-z5a8g144rp.pdf

Proper biconservative immersions into the Euclidean space

In this article we consider the Euler-Lagrange method associated to a suitable bilagrangian to study biharmonic curves of a Riemannian manifold. We apply this method to characterize biharmonic curves of the three-dimensional Lie group Sol. We also classify, using a geometric method, the biharmonic curves of the three-dimensional Cartan-Vranceanu manifolds.

The Euler-Lagrange Method for Biharmonic Curves

In this paper, using the framework of equivariant differential geometry, we study proper $SO(p+1) \times SO(q+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+q+2$) and proper $SO(p+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+2$). Moreover, we show that, in these two classes of invariant families, there exists no proper biharmonic immersion.

Stefano Montaldo

Papers

The stress-energy tensor for biharmonic maps

Biharmonic maps between warped product manifolds

Proper biconservative immersions into the Euclidean space

The Euler-Lagrange Method for Biharmonic Curves

Proper Biconservative immersions into the Euclidean space