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

Polyharmonic, or $r$-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper $r$-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion $i: S^{n-1}(R)\to S^n$ is a proper $r$-harmonic submanifold of $S^n$ if and only if the radius $R$ is equal to $1/ \sqrt{r}$. We shall also prove the existence of proper $r$-harmonic generalized Clifford's tori into the sphere.

New examples of r-harmonic immersions into the sphere

In this paper we shall assume that the ambient manifold is a pseudo-Riemannian space form $N^{m+1}_t(c)$ of dimension $m+1$ and index $t$ ($m\geq2$ and $1 \leq t\leq m$). We shall study hypersurfaces $M^{m}_{t'}$ which are polyharmonic of order $r$ (briefly, $r$-harmonic), where $r\geq 3$ and either $t'=t$ or $t'=t-1$. Let $A$ denote the shape operator of $M^{m}_{t'}$. Under the assumptions that $M^{m}_{t'}$ is CMC and $Tr A^2$ is a constant, we shall obtain the general condition which determines that $M^{m}_{t'}$ is $r$-harmonic. As a first application, we shall deduce the existence of several new families of proper $r$-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper $r$-harmonic hypersurfaces ($r \geq 3$). Finally, we shall obtain the complete classification of proper $r$-harmonic isoparametric pseudo-Riemannian surfaces into a $3$-dimensional Lorentz space form.

Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic hypersurfaces), where r ≥ 3 is an integer, into a space form Nm+1 (c) of curvature c. For this class of hypersurfaces we shall prove that, if c ≤ 0, then any r-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is $${\mathbb{S}^{m + 1}}$$ , we shall obtain the geometric condition which characterizes the r-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the bounds for these two constants. In particular, we shall prove the existence of several new examples of proper r-harmonic isoparametric hypersurfaces in $${\mathbb{S}^{m + 1}}$$ for suitable values of m and r. Finally, we shall show that all these r-harmonic hypersurfaces are also ES-r-harmonic, i.e., critical points of the Eells-Sampson r-energy functional. 

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Polyharmonic hypersurfaces into space forms

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional ...

Index and nullity of proper biharmonic maps in spheres

We study subelliptic biharmonic maps, i.e., smooth maps ϕ:M→N from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of the energy functional $E_{2,b} (\phi ) = \frac{1}{2} \int_{M} \| \tau_{b} (\phi ) \|^{2} \theta \wedge (d \theta)^{n}$. We show that ϕ:M→N is a subelliptic biharmonic map if and only if its vertical lift ϕ∘π:C(M)→N to the (total space of the) canonical circle bundle $S^{1} \to C(M) \stackrel{\pi}{\longrightarrow} M$ is a biharmonic map with respect to the Fefferman metric Fθ on C(M).

Stefano Montaldo

Papers

New examples of r-harmonic immersions into the sphere

Polyharmonic hypersurfaces into pseudo-Riemannian space forms

Polyharmonic hypersurfaces into space forms

Index and nullity of proper biharmonic maps in spheres

Subelliptic Biharmonic Maps