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 prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.

New results toward the classification of biharmonic submanifolds in S^n

Rotationally symmetric biharmonic maps between models

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

https://www.emis.de/proceedings/Varna/vol11/Balmus.pdf

Properties of Biharmonic Submanifolds in Spheres

The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups $$\mathbf{SU}(n)$$
 , $$\mathbf{SO}(n)$$
 and $$\mathbf{Sp}(n)$$
 . We work in a geometric setting which connects our study with the theory of submersive harmonic morphisms. We develop a general duality principle and use this to interpret our new examples on the Euclidean sphere $${\mathbb S}^3$$
 and on the hyperbolic space $${\mathbb H}^3$$
 .

Biharmonic Functions on the Classical Compact Simple Lie Groups

In this paper we survey the known results on the classification of biharmonic submanifolds in space forms and construct a family of new examples of proper biharmonic submanifolds in the Euclidean n-dimensional sphere.

/pdf/classification-results-and-new-examples-of-proper-biharmonic-3fl2s8s1rb.pdf

Stefano Montaldo

Papers

New results toward the classification of biharmonic submanifolds in S^n

Rotationally symmetric biharmonic maps between models

Properties of Biharmonic Submanifolds in Spheres

Biharmonic Functions on the Classical Compact Simple Lie Groups

Classification results and new examples of proper biharmonic submanifolds in spheres