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

We explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere ${\mathbb S}^3$.

BIHARMONIC SUBMANIFOLDS OF ${\mathbb S}^3$

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 submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.

https://people.unica.it/montaldo/files/2008/09/3561_mon-pdf.pdf

Stefano Montaldo

Papers

Biharmonic submanifolds in spheres

BIHARMONIC SUBMANIFOLDS OF ${\mathbb S}^3$

A short survey on biharmonic maps between Riemannian manifolds

Biharmonic maps between Riemannian manifolds

Classification results for biharmonic submanifolds in spheres