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MR2395125 58E20 (53C43)

Loubeau, E.(F-BRET); Montaldo, S. (I-CAGL-MI) ; Oniciuc, C. (R-IASIM)The stress-energy tensor for biharmonic maps. (English summary)Math. Z.259(2008),no. 3,503–524.

A review for this item is in process.

References

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17. Montaldo, S., Oniciuc, C.: A short survey on biharmonic maps between Riemannian manifolds.Rev. Un. Mat. Argentina47(2), 1–22 (2006)MR2301373 (2008a:53063)

18. Oniciuc, C.: Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. Al.I. CuzaIasi Mat. (N.S.)48,237–248 (2002)MR2004799 (2004e:53097)

19. Petersen, P.: Riemannian Geometry. Springer, New York (1998)MR1480173 (98m:53001)20. Ruh, E., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc.149,569–573

(1970)MR0259768 (41 #4400)21. Sakai, T.: On Riemannian manifolds admitting a function whose gradient is of constant norm.

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attempt to correct errors.

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MR2341207 (2008f:53085)53C43 (58E20)

Loubeau, Eric (F-BRET); Montaldo, Stefano(I-CAGL-MI) ; Oniciuc, Cezar (R-IASIM)The biharmonic stress-energy tensor and the Gauss map. (English summary)Geometry, integrability and quantization, 234–245,Softex, Sofia, 2007.

Summary: “We consider the energy and bienergy functionals as variational problems on the setof Riemannian metrics and present a study of the biharmonic stress-energy tensor. This approachis then applied to characterise weak conformality of the Gauss map of a submanifold. Finally,working at the level of functionals, we recover a result of Weiner linking Willmore surfaces andpseudo-umbilicity.”For the entire collection seeMR2341394 (2008d:00011)

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MR2271198 (2007j:53066)53C43 (58E20)

Balmus, A. (I-CAGL) ; Montaldo, S. (I-CAGL) ; Oniciuc, C. (R-IASIM)Biharmonic maps between warped product manifolds. (English summary)J. Geom. Phys.57 (2007),no. 2,449–466.

One of the known effective methods for constructing new proper (meaning nonharmonic) bihar-monic maps is to furnish either the domain manifold or the target manifold of a harmonic mapwith a new conformal metric. Then, by carefully changing the conformal factor, although the orig-inal map would lose its harmonicity, this technique also provides a possibility for the originalmap to gain biharmonicity. In the paper under review, the authors generalize this idea to the caseof a warped productM ×f 2 N , which is a product manifoldM ×N endowed with a “partiallyconformal” metricGf 2: for X,Y ∈ T(x,y)(M ×N),

Gf 2(X,Y ) = g(dπ(X), dπ(Y ))+ (f π)2h(dη(X), dη(Y )),

whereM andN are Riemannian manifolds with metricsg andh, respectively;π:M ×N →Mandη:M ×N → N are projections andf is a suitable positive function onM , which is to bedefined from the context. In this setting, starting from the product of two harmonic mapsϕ =1M ×ψ:M ×N →M ×N and then warping the metric either on the domain product manifoldor the target product manifold, the authors obtain conditions for the biharmonicity of some relatedmaps, and hence construct a lot of new proper biharmonic maps which profoundly enlarge thefamily of proper biharmonic maps. In addition, from the viewpoint of the warped product, theauthors also deal with the biharmonicity of other maps such as inclusions, projections and axiallysymmetric maps.

Reviewed byGuo Ying Jiang

References

1. P. Baird, Harmonic Maps with Symmetry, Harmonic Morphisms and Deformations of Metrics,Pitman Books, 1983, pp. 27–39.MR0716320 (85i:58038)

2. P. Baird, D. Kamissoko, On constructing biharmonic maps and metrics, Ann. Global Anal.Geom. 23 (2003) 65–75.MR1952859 (2004c:58033)

3. P. Baird, J.C. Wood, Harmonic Morphisms between Riemannian Manifolds, Oxford SciencePublications, 2003.MR2044031 (2005b:53101)

4. A. Balmus, Biharmonic properties and conformal changes, An. Stiint. Univ. Al.I. Cuza IasiMat. (N.S.) 50 (2004) 361–372.MR2131943 (2006c:53065)

5. M. Bertola, D. Gouthier, Lie triple systems and warped products, Rend. Mat. Appl. (7) 21(2001) 275–293.MR1884948 (2003d:53049)

6. J.K. Beem, P.E. Ehrlich, Th.G. Powell, Warped product manifolds in relativity, in: SelectedStudies: Physics-Astrophysics, Mathematics, History of Science, North-Holland, Amsterdam,New York, 1982, pp.41–56.MR0662851 (83j:53062)

7. A.L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987.MR0867684 (88f:53087)8. R.L. Bishop, B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969)

1–49.MR0251664 (40 #4891)

9. R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds ofS3, Internat. J. Math. 12(2001) 867–876.MR1863283 (2002k:53123)

10. R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math. 130(2002) 109–123.MR1919374 (2003c:53090)

11. R. Caddeo, S. Montaldo, P. Piu, Biharmonic curves on a surface, Rend. Mat. Appl. 21 (2001)143–157.MR1884940 (2002k:58031)

12. R. Caddeo, C. Oniciuc, P. Piu, Explicit formulas for non-geodesic biharmonic curves of theHeisenberg group, Rend. Sem. Mat. Univ. Politec. Torino 62 (2004) 265–278.MR2129448(2006e:53113)

13. J. Eells, J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86(1964) 109–160.MR0164306 (29 #1603)

14. J. Eells, L. Lemaire, Selected topics in harmonic maps, Conf. Board. Math. Sci. 50 (1983)109–160.MR0703510 (85g:58030)

15. J. Eells, A. Ratto, Harmonic Maps and Minimal Immersions with Symmetries, PrincetonUniversity Press, 1993.MR1242555 (94k:58033)

16. B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Greno-ble) 28 (2) (1978) 107–144.MR0499588 (80h:58023)

17. N. Innami, Splitting theorems of Riemannian manifolds, Compos. Math. 47 (3) (1982) 237–247.MR0681608 (84e:53053)

18. J. Inoguchi, Submanifolds with harmonic mean curvature in contact 3-manifolds, Colloq. Math.100 (2004) 163–179.MR2107514 (2005h:53105)

19. G. Y. Jiang, 2-harmonic isometric immersions between Riemannian manifolds, Chinese Ann.Math. Ser. A 7 (1986) 130–144.MR0858581 (87k:53140)

20. G. Y. Jiang, 2-harmonic maps and their first and second variation formulas, Chinese Ann. Math.Ser. A 7 (1986) 389–402.MR0886529 (88i:58039)

21. E. Kamke, Differentialgleichungen I, 9th ed., Teubner, Leipzig, 1977.MR0466672 (57 #6549)22. D.-S. Kim, Y.H. Kim, Compact Einstein warped product spaces with nonpositive scalar curva-

ture, Proc. Amer. Math. Soc. 131 (2003) 2573–2576.MR1974657 (2004b:53063)23. E. Loubeau, C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (3)

(2005) 729–745.MR2135286 (2006b:53087)24. E. Loubeau, C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Trans.

Amer. Math. Soc. (in press).MR2327029 (2008g:53078)25. C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al.I. Cuza Iasi

Mat. (N.S.) 48 (2002) 237–248.MR2004799 (2004e:53097)26. C. Oniciuc, On the second variation formula for biharmonic maps to a sphere, Publ. Math.

Debrecen 61 (2002) 613–622.MR1943720 (2003i:58031)27. C. Oniciuc, New examples of biharmonic maps in spheres, Colloq. Math. 97 (2003) 131–139.

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J. Geom. Phys. 56 (2006) 358–374.MR2171890 (2006e:53117)29. T. Sakai, On Riemannian manifolds admitting a function whose gradient is of constant norm,

Kodai Math. J. 19 (1996) 39–51.MR1374461 (97a:53048)

30. T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors areeigenvectors, Publ. Math. Debrecen 67 (2005) 285–303.MR2162123 (2006c:53064)

31. T. Sasahara, Stability of biharmonic Legendre submanifolds in Sasakian space forms, preprint.32. M. Svensson, Holomorphic foliations, harmonic morphisms and the Walczak formula, J. Lon-

don Math. Soc. (2) 68 (3) (2003) 781–794.MR2010011 (2004g:53033)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR2301373 (2008a:53063)53C43 (31B30)

Montaldo, S. (I-CAGL) ; Oniciuc, C. (R-IASIM)A short survey on biharmonic maps between Riemannian manifolds.Rev. Un. Mat. Argentina47 (2006),no. 2,1–22 (2007).

A biharmonic map is a smooth mapϕ: (M, g) → (N,h) between Riemannian manifolds whichis a critical point of the bienergy functionalE2(ϕ) = 1

2

∫M |τ(ϕ)|2vg, whereτ(ϕ) is the tension

field of ϕ. The Euler-Lagrange equations are of fourth order; for maps to Euclidean spacesthey read∆2ϕ = 0. Harmonic maps satisfyτ(ϕ) = 0, and so are trivially biharmonic; otherbiharmonic maps are called proper. The paper under review is an attractive survey of biharmonicmaps, concentrating on the differential geometric aspects. The main topics covered are as follows:non-existence results for proper biharmonic maps; biharmonic curves, especially in surfaces andthe Heisenberg group; biharmonic submanifolds of spheres and Sasakian space forms; axiallysymmetric biharmonic maps between Euclidean spaces; conformally changing the metrics tomake harmonic maps proper biharmonic; biharmonic submersions; and biharmonic morphisms,i.e., maps which preserve the biharmonic equation.

Note that, although for the most part the functional under consideration is the bienergy, in thesecond paragraph of the paper ‘bienergy’ should read ‘energy’.

Reviewed byJohn C. Wood

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MR2274737 (2007g:58015)58E20Caddeo, Renzo(I-CAGL-MI) ; Montaldo, Stefano(I-CAGL-MI) ;Oniciuc, Cezar (R-IASIM) ; Piu, Paola[Piu, M. Paola] (I-CAGL-MI)The Euler-Lagrange method for biharmonic curves. (English summary)Mediterr. J. Math.3 (2006),no. 3-4,449–465.

This paper presents an alternative method to describe biharmonic curves. Letγ: I → (M, g) bean arclength parametrized curve in a Riemannian manifold defined byγ(t) = (x1(t), . . . , xn(t)).SettingT = γ′, the tension field isτ(γ) =∇TT and the biharmonic equation reduces to∇3

TT −R(T,∇TT )T = 0. Let

(∗) d2

dt2(∂L2

∂x′′k)− d

dt(∂L2

∂x′k) +

∂L2

∂xk= 0, k = 1, . . . , n,

associated to the bi-Lagrangian

L2(x(t), x′(t), x′′(t)) =12g(τ(γ), τ(γ)).

The system(∗) and the biharmonic equation are equivalent. Then a biharmonic curve inH2 shouldbe geodesic. The authors also characterize biharmonic curves in the so-called 3-dimensionalCartan-Vranceanu manifolds.

Reviewed byAngel Ferrandez

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MR2262416 (2007g:53007)53A10 (49Q10 53C42)

Mercuri, Francesco (BR-ECPM); Montaldo, Stefano(I-CAGL) ;Piu, Paola[Piu, M. Paola] (I-CAGL)A Weierstrass representation formula for minimal surfaces inH3 andH2×R. (Englishsummary)Acta Math. Sin. (Engl. Ser.)22 (2006),no. 6,1603–1612.

It is well known that the classical Weierstrass representation formula for minimal surfaces inthe 3-dimensional Euclidean space is a very useful tool for their study. In the present paper theauthors describe a method by which to derive a Weierstrass-type representation formula for simplyconnected immersed minimal surfaces when the ambient space is the 3-dimensional Heisenberggroup or the product of the 2-dimensional hyperbolic space by the real line. In fact, using thestandard harmonic map equation, they produce a Weierstrass-type representation formula forimmersed minimal surfaces in a Riemannian manifold. This general setting is then applied to thecase of 3-dimensional Lie groups endowed with a left-invariant metric, the main results beingobtained when the ambient space is the 3-dimensional Heisenberg group or the product of the

2-dimensional hyperbolic space by the real line. A good description of the classical Weierstrassrepresentation and its simple applications can be found in [J. L. M. Barbosa and A. G. Colares,Minimal surfaces inR3, Translated from the Portuguese, Lecture Notes in Math., 1195, Springer,Berlin, 1986;MR0853728 (87j:53010)]. We mention that, with similar methods, a Weierstrass-type formula for minimal surfaces in the hyperbolicn-space has been derived by M. Kokubu[Tohoku Math. J. (2)49 (1997), no. 3, 367–377;MR1464184 (98f:53008)].

Reviewed byJoao Lucas Marques Barbosa

References

1. Barbosa, J. L. M., Colares, A. G.: Minimal surfaces inR3, Lecture Notes in Mathematics, 1195,Springer-Verlag, Berlin, 1986MR0853728 (87j:53010)

2. Gray, A.: Modern differential geometry of curves and surfaces with Mathematica, Secondedition, CRC Press, Boca Raton, FL, 1998MR1688379 (2000i:53001)

3. Bekkar, M.: Exemples de surfaces minimales dans l’espace de Heisenberg.Rend. Sem. Fac.Sci. Univ. Cagliari, 61, 123–130 (1991)MR1193456 (94b:53017)

4. Bekkar, M., Sari, T.: Surfaces minimales reglees dans l’espace de HeisenbergH3. Rend. Sem.Mat. Univ. Politec. Torino, 50, 243–254 (1992)MR1249465 (94h:53009)

5. Caddeo, R., Piu, P., Ratto, A.:SO(2)-invariant minimal and constant mean curvature surfacesin 3-dimensional homogeneous spaces.Manuscripta Math., 87, 1–12 (1995)MR1329436(96b:53014)

6. Figueroa, C. B., Mercuri, F., Pedrosa, R. H. L.: Invariant surfaces of the Heisenberg groups.Ann. Mat. Pura Appl., 177(4), 173–194 (1999)MR1747630 (2000m:53089)

7. Piu, P., Sanini, A.: One-parameter subgroups and minimal surfaces in the Heisenberg group.Note Mat..18, 143–153 (1998)MR1759022 (2001d:53070)

8. Scott, D. P.: Minimal surfaces in the Heisenberg group, preprintcf. MR 2005g:350389. Nelli, B., Rosenberg, H.: Minimal surfaces inH2×R. Bull. Braz. Math. Soc. (N.S.), 33, 263–

292 (2002)MR1940353 (2004d:53014)10. Kokubu, M.: Weierstrass representation for minimal surfaces in hyperbolic space.Tohoku Math.

J., 49, 367–377 (1997)MR1464184 (98f:53008)11. Koszul, J. L., Malgrange, B.: Sur certaines fibrees complexes.Arch. Math., 9, 102–109 (1958)

MR0131882 (24 #A1729)12. Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds.Amer. J. Math., 86,

109–160 (1964)MR0164306 (29 #1603)13. Eells, J., Lemaire, L.: Selected topics in harmonic maps, AMS regional conference series in

mathematics, 50, 1983MR0703510 (85g:58030)14. Inoguchi, J., Kumamoto, T., Ohsugi, N., Suyama, Y.: Differential geometry of curves and

surfaces in 3-dimensional homogeneous spaces I and II.Fukuoka Univ. Sci. Rep., 29 and30,(1999 and 2000)MR1719108 (2000h:53018)

15. Figueroa, C. B.: Geometria das subvariedades do grupo de Heisenberg, Ph.d Thesis, Campinas16. Rosenberg, H.: Minimal surfaces inM2 × R. Illinois J. Math.. 46, 1177–1195 (2002)

MR1988257 (2004d:53015)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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Citations

From References: 2From Reviews: 2

MR2250208 (2007k:58020)58E20 (53C43)

Caddeo, R.(R-IASIM) ; Montaldo, S. (R-IASIM) ; Oniciuc, C. (R-IASIM) ;Piu, P. [Piu, M. Paola] (R-IASIM)The classification of biharmonic curves of Cartan-Vranceanu 3-dimensional spaces.(English summary)Modern trends in geometry and topology, 121–131,Cluj Univ.Press, Cluj-Napoca, 2006.

It is well known that the harmonic mapsϕ: (M, g)→ (N,h) between Riemannian manifolds arethe critical points of the energyE(ϕ) = 1

2

∫M |dϕ|2 vg, with the corresponding Euler-Lagrange

equation given by the vanishing of the tension fieldτ(ϕ) = tr∇dϕ. Continuing in a similarvein, one defines a biharmonic map as a critical point of the bi-energy functionalE2(ϕ) =12

∫M |τ(ϕ)| vg. The Euler-Lagrange equation forE2 is shown to be

τ2(ϕ):=−∆τ(ϕ)− trRN(dϕ, τ(ϕ))dϕ= 0.

Every harmonic map is clearly biharmonic, so of interest is an investigation of proper biharmonicmaps (those that are not harmonic).

If one restricts to the case of curvesγ: I → (Nn, h) parametrized by the arclength, the biharmonicequationτ(ϕ) = 0 reduces to∇3

TT −R(T,∇TT )T = 0,whereT = γ′. By introducing the FrenetframeFii=1,...,n of a curveγ the authors first obtain the following characterization of properbiharmonic curves in terms of the curvaturesk1, k2, . . . , kn−1 of γ:

k1 = const 6= 0,k2

1 + k22 =R(F1, F2, F1, F2),

k′2 =−R(F1, F2, F1, F3),k2k3 =−R(F1, F2, F1, F4),R(F1, F2, F1, Fj) = 0, j = 5, . . . , n.

They then apply this to classify biharmonic curves in Cartan-Vranceanu spaces(M,ds2l,m) de-

scribed as follows:Let M = R3 if m ≥ 0 andM = (x, y, z);x2 + y2 < −1/m otherwise. Define onM a 2-

parameter family of Riemannian metrics by

ds2l,m =

dx2 + dy2

[1+m(x2 + y2)]2+(dz+

l

2ydx−xdy

[1+m(x2 + y2)]

)2

, l,m ∈ R.

This family includes all 3-dimensional homogeneous metrics whose group of isometries is ofdimension 4 or 6, except those of constant negative curvature.

The main result of the paper is that a proper biharmonic curveγ: I → (M,ds2l,m) has both

constant geodesic curvature and geodesic torsion, i.e. it is a helix. Whenm 6= 0 andl2− 4m 6= 0,the authors give explicit parametric equations (divided into three types) of all proper biharmoniccurves of(M,ds2

l,m). It turns out that any proper biharmonic curve of the Cartan-Vranceanu spacesis a geodesic on a surface which is invariant under the action of a 1-parameter group of isometries.For the entire collection seeMR2238874 (2007c:53001)

Reviewed byIvko Dimitric

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MR2196776 (2007d:53102)53C42 (53A10 58E20)

Loubeau, Eric (F-BRET); Montaldo, Stefano(I-CAGL)Examples of biminimal surfaces of Thurston’s three-dimensional geometries.Mat. Contemp.29 (2005), 1–12.

Biharmonic mapsϕ: (M, g) → (N,h) between Riemannian manifolds are critical points of thebienergy functionalE2(ϕ) = 1

2

∫M |τ(ϕ)|2vg, whereτ(ϕ) = tr∇dϕ is the tension field. There is

a paucity of examples of such maps, so that, for example, there exists no biharmonic surface ineither Euclidean or hyperbolic 3-space, whereas the only such example inS3 is S2(1/

√2). To

enlarge the class of surfaces which satisfy a similar condition, the authors consider biminimalmapsϕ: (Mm, g) → (Nn, h) (m ≤ n), defined to be the critical points of the bienergyE2 forvariations normal to the imageϕ(M) ⊂ N . In the case of an isometric immersionϕ:M → N ,biminimal submanifolds are characterized by the Euler-Lagrange equation, which takes the form

[∆ϕH − traceRN(dϕ,H)dϕ]⊥ = 0,

whereH is the mean curvature vector field,⊥ denotes the normal component andRN is thecurvature tensor ofN. More generally, an immersion is calledλ-biminimal if it is a critical pointwith respect to normal variations of fixed energy of the functional

E2,λ(ϕ) =12

∫M

|τ(ϕ)|2vg +λ

∫M

|dϕ|2vg.

The main portion of the paper is devoted to finding some examples of biminimal surfaces inThurston’s 3-dimensional geometries (all but one), that is to say three spaces of constant curvatureR3, S3, H3, two Riemannian productsS2 ×R,H2 ×R, and two line bundles, the HeisenberggroupH3 and ˜SL2(R). Namely, the right cylinder whose directrix is a biminimal curve inR2 isa biminimal surface inR3, as well as a cone on a biminimal curve inS2. A Hopf cylinder ofS3

is biminimal if and only if the base curveγ in S2(1/2) is a 2-biminimal curve. A right cylinder

in hyperbolic 3-space (half-space model) whose directrix is a biminimal curve inR2 (plane atinfinity) is a (−1)-biminimal surface inH3.

In the spaceN 2×R, whereN 2 stands for eitherS2 or H2, the cylinderS = π−1(γ) with respectto the canonical projectionπ:N 2×R→N 2, is a biminimal surface if and only ifγ is a biminimalcurve onN 2. In the Heisenberg groupH3, the flat cylinderS = π−1(γ) ⊂ H3 is a biminimalsurface if and only ifγ is a 1/2-biminimal curve ofR2,whereπ:H3 → R2 stands for the projection

π:[ 1 x z

0 1 y0 0 1

]→ (x, y).

Similarly, the flat cylinderS = π−1(γ) ⊂ ˜SL2(R) is a biminimal surface if and only ifγ is a1/2-biminimal curve ofH2, where ˜SL2(R) is identified withR3

+ = (x, y, z) ∈ R3; z > 0 andπ: (x, y, z)→ (y, z) is the projection.

Reviewed byIvko Dimitric

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Article

Citations

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MR2162419 (2006g:53093)53C42 (53C40)

Montaldo, Stefano(I-CAGL) ; Onnis, Irene I. (BR-ECPM)Invariant surfaces of a three-dimensional manifold with constant Gauss curvature. (Englishsummary)J. Geom. Phys.55 (2005),no. 4,440–449.

In this paper, the authors investigate surfaces with constant Gauss curvature in a three-dimensional(complete) Riemannian manifold which are invariant under a one-parameter transformation groupconsisting of isometries of the ambient space. LetX be a Killing vector field on a three-dimensional(complete) Riemannian manifoldN ,GX be the one-parameter transformation group generated byX andNr be the set of all points belonging to principal orbits of the transformation group. For aGX-invariant surfaceM contained inNr, the projectionγ ofM to the orbit spaceNr/Gx is calledthe profile curve ofM . Let γ be a lift ofγ toNr and define a functionω by ω(t) = ‖Xγ(t)‖. First

the authors show thatM is of constant Gauss curvatureK if and only if d2ωdt2 +Kω = 0 holds under

the assumption thatd2ωdt2 has no zeroes. Also, in the case whenN =H2×R (H2 the2-dimensional

hyperbolic space) orH3 (the three-dimensional Heisenberg group), they parametrize the profilecurveγ byω explicitly for some special Killing vector fieldsX. After this, for some special Killingvector fieldsX onH2×R andH3, the profile curves ofGX-invariant surfaces of constant Gausscurvature are parametrized by the arclength explicitly. Therefore, examples of surfaces of constantGauss curvature inH2 ×R andH3, which are invariant under a one-parameter transformationgroup consisting of isometries of the ambient space, are obtained.

Reviewed byNaoyuki Koike

References

1. R. Caddeo, A. Gray, Curve e Superfici, vol. I, CUEC, 2000.2. R. Caddeo, P. Piu, A. Ratto, Rotational surfaces inH3 with constant Gauss curvature, Boll. Un.

Mat. Ital. B (7) (1996) 341–357.MR1397352 (97e:53008)3. C.B. Figueroa, F. Mercuri, R.H.L. Pedrosa, Invariant surfaces of the Heisenberg groups, Ann.

Mater. Pure Appl. 177 (4) (1999) 173–194.MR1747630 (2000m:53089)4. P.J. Olver, Application of Lie Groups to Differential Equations, GTM 107, Springer-Verlag,

New York, 1986.MR0836734 (88f:58161)5. I.I. Onnis, Ph.D. Thesis, University of Campinas, 2005.6. R.S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. Math. (73)

(1961) 295–323.MR0126506 (23 #A3802)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR2113588 (2005m:53010)53A10Montaldo, Stefano(I-CAGL)On minimal surfaces in the Heisenberg space. (English summary)Recent advances in geometry and topology, 249–258,Cluj Univ.Press, Cluj-Napoca, 2004.

The Heisenberg spaceH3 is the 3-dimensional Lie group represented inGL(3,R) by(1 x z+ 1

2xy0 1 y0 0 1

), x, y, z ∈ R.

The paper presents a review of the literature on minimal surfaces inH3 that includes the equationof minimal graphs, a conjecture for Bernstein’s theorem (it cannot be true in general, as inR3,since the paraboloidf(x, y) = xy/2 is a minimal graph inH3), and a characterization of thetranslational and helicoidal minimal surfaces inH3. It also discusses the work of Montaldo, F.Mercuri and P. Piu, in which a sort of Weierstrass representation for minimal surfaces inH3 isobtained, and computes some special cases of this Weierstrass representation.For the entire collection seeMR2112604 (2005i:53004)

Reviewed byM. do Carmo

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From References: 5From Reviews: 0

MR2062613 (2005c:53012)53A10Montaldo, Stefano(I-CAGL) ; Onnis, Irene I. (BR-ECPM)Invariant CMC surfaces in H2×R. (English summary)Glasg. Math. J.46 (2004),no. 2,311–321.

This paper studies translational and helicoidal constant mean curvature surfaces inH2×R, thatis, CMC surfaces that are invariant under the action of a 1-parameter subgroupG of the isometrygroupIsom(H2×R) generated by translations alongR, respectively by compositions of rotationsand translations alongR. The authors give a complete classification of these surfaces. The resultsare explicitly stated and clearly presented. Most computations are elementary and straightforward,making this report accessible to students with a basic foundation of differential geometry.

Reviewed byMagdalena Daniela Toda

References

1. A. Back, M. P. do Carmo and W. Y. Hsiang, On the fundamental equations of equivariantgeometry (unpublished manuscript).

2. W. T. Hsiang and W. Y. Hsiang, On the uniqueness of isoperimetric solutions and embeddedsoap bubbles in non-compact symmetric spaces,Invent. Math.89 (1989), 39–58.MR1010154(90h:53078)

3. W. T. Hsiang and W. Y. Hsiang, On the existence of codimension one minimal spheres in com-pact symmetric spaces of rank 2,J. Diff. Geom.17 (1982), 583–594.MR0683166 (84a:53057)

4. W. Y. Hsiang and H. B. Lawson Jr., Minimal submanifold of low cohomogeneity,J. Diff. Geom.5 (1971), 1–38.MR0298593 (45 #7645)

5. R. H. L. Pedrosa and M. Ritore, Isoperimetric domains in the Riemannian product of a circlewith a simply connected space form and applications to free boundary problems,Indiana Univ.Math. J.48 (1999), 1357–1394.MR1757077 (2001k:53120)

6. P. Piu, Sur les flots riemanniens des espaces de D’Atri de dimension 3,Rend. Sem. Mat. Univ.Politec. Torino46 (1988), 171–187.MR1084565 (92a:53054)

7. P. Tompter, Constant mean curvature surfaces in the Heisenberg group,Proc. Symp. Pure Math.54 (1993), 485–495.MR1216601 (94a:53098)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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MR1922040 (2003f:53115)53C43Caddeo, R.(I-CAGL) ; Montaldo, S. (I-CAGL) ; Oniciuc, C. (R-IASIM)Biharmonic immersions into spheres.Differential geometry, Valencia, 2001, 97–105,World Sci.Publ., River Edge, NJ, 2002.

The bi-energy of a mapϕ: (M, g) → (N,h) between two Riemannian manifolds is defined byE2(ϕ) = 1

2

∫M |τ(ϕ)2|vg, whereτ(ϕ) = trace∇dϕ is the tension field. The critical points of the

bi-energy functional are called biharmonic maps. In this paper the authors focus on isometricimmersions into spheres. The first interesting result states the following: Letγ: I → S3 be a non-geodesic biharmonic curve parametrized by arclength. Then eitherkg = 1 andγ is a circle ofradius1/

√2 or 0 < kg < 1 andγ is a geodesic of the Clifford torusS1(1/

√2)× S1(1/

√2) ⊂

S3. In looking for nonharmonic biharmonic submanifolds, the authors consider a harmonic sub-manifoldM in Sn(a)×b, such thata2 + b2 = 1 and0 < a < 1. Then they show thatM is anonharmonic biharmonic submanifold inSn+1 if and only if a= 1/

√2 andb=±1/

√2. Finally,

as for surfaces, it is shown that there are closed orientable embedded nonminimal biharmonicsurfaces of arbitrary genus inS4. This is a nice result when comparingS3 andS4.For the entire collection seeMR1919551 (2003b:53002)

Reviewed byAngel Ferrandez

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Article

Citations

From References: 14From Reviews: 4

MR1919374 (2003c:53090)53C43 (58E20)

Caddeo, R.(I-CAGL) ; Montaldo, S. (I-CAGL) ; Oniciuc, C. (R-IASIM)Biharmonic submanifolds in spheres. (English summary)Israel J. Math.130(2002), 109–123.

The authors prove some new non-existence theorems for nonharmonic biharmonic maps into man-ifolds with constant negative sectional curvature−1. It is also shown in the paper that any minimalsubmanifold of a certain parallel hypersphere of the unit sphereSn is a nonharmonic biharmonicsubmanifold ofSn. Thus many new nonharmonic biharmonic submanifolds are constructed forSn (n > 3). As regards nonharmonic biharmonic curves inSn, by deriving the correspondingordinary differential equation, the authors obtain its explicit solutions.

Reviewed byGuo Ying Jiang

References

1. R. Caddeo, S. Montaldo and C. Oniciuc,Biharmonic submanifolds ofS3, International Journalof Mathematics, to appear.MR1863283 (2002k:53123)

2. B.Y. Chen,Some open problems and conjectures on submanifolds of finite type, SoochowJournal of Mathematics17 (1991), 169–188.MR1143504 (92m:53091)

3. B. Y. Chen and S. Ishikawa,Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu Journal of Mathematics52 (1998), 167–185.MR1609044(99b:53078)

4. B. Y. Chen and K. Yano,Minimal submanifolds of a higher dimensional sphere, Tensor (N.S.)22 (1971), 369–373.MR0287495 (44 #4699)

5. I. Dimitric, Submanifolds ofEm with harmonic mean curvature vector, Bulletin of the Instituteof Mathematics. Academia Sinica20 (1992), 53–65.MR1166218 (93g:53087)

6. J. Eells and J.H. Sampson,Harmonic mappings of Riemannian manifolds, American Journalof Mathematics86 (1964), 109–160.MR0164306 (29 #1603)

7. H. Gluck, Geodesics in the unit tangent bundle of a round sphere, L’EnseignementMathematique34 (1988), 233–246.MR0979640 (90b:53051)

8. T. Hasanis and T. Vlachos,Hypersurfaces inE4 with harmonic mean curvature vector field,Mathematische Nachrichten172(1995), 145–169.MR1330627 (96c:53085)

9. G. Y. Jiang,2-harmonic isometric immersions between Riemannian manifolds, Chinese Annalsof Mathematics. Series A7 (1986), 130–144.MR0858581 (87k:53140)

10. G. Y. Jiang,2-harmonic maps and their first and second variational formulas, Chinese Annalsof Mathematics. Series A7 (1986), 389–402.MR0886529 (88i:58039)

11. H. B. Lawson,Complete minimal surfaces inS3, Annals of Mathematics (2)92 (1970), 335–374.MR0270280 (42 #5170)

12. C. Oniciuc,Biharmonic maps between Riemannian manifolds, Analele Stiintifice ale UniversityAl. I. Cuza Iasi. Mat. (N.S.), to appear.cf. MR 2004e:53097

13. J. Simons,Minimal varieties in Riemannian manifolds, Annals of Mathematics88 (1968),62–105.MR0233295 (38 #1617)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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Citations

From References: 5From Reviews: 0

MR1884940 (2002k:58031)58E20 (53A04)

Caddeo, R.(I-CAGL) ; Montaldo, S. (I-CAGL) ; Piu, P. [Piu, M. Paola] (I-CAGL)Biharmonic curves on a surface. (English, Italian summaries)Rend. Mat. Appl. (7)21 (2001),no. 1-4,143–157.

Summary: “In this paper we consider the biharmonic curves on a surface. These curves are criticalpoints of the bienergy functional, and generalize the harmonic curves (geodesics). We first findconditions on the Gaussian curvature of the surface along a nongeodesic biharmonic curve. Thenwe study biharmonic curves on a surface of revolution, giving the explicit solutions in the case ofsurfaces of revolution with constant Gaussian curvature.”

Reviewed byGuo Ying Jiang

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MR1871019 (2002m:53099)53C43 (58E20)

Caddeo, Renzo(I-CAGL) ; Montaldo, Stefano(I-CAGL) ;Piu, Paola[Piu, M. Paola] (I-CAGL)On biharmonic maps. (English summary)Global differential geometry: the mathematical legacy of Alfred Gray(Bilbao, 2000), 286–290,Contemp.Math., 288,Amer.Math.Soc., Providence, RI, 2001.

A smooth mapϕ: (M, g) → (N,h) between Riemannian manifolds is said to be harmonic if itis a critical point of the energyE(ϕ) = 1

2

∫M |dϕ|2 dv. The Euler-Lagrange equation of energy is

given by vanishing of the tension fieldτ = trace∇dϕ. Likewise, one defines the bienergy ofϕ byE2(ϕ) = 1

2

∫M |τ |2 dv, calling the mapϕ biharmonic if it is a critical point of the bienergy. The

Euler-Lagrange equation forE2 is τ2 = J(τ) = 0,whereJ is the Jacobi operator ofϕ [G. Y. Jiang,Chinese Ann. Math. Ser. A7 (1986), no. 4, 389–402;MR0886529 (88i:58039)]. In the case whenN = Rn is a Euclidean space andϕ an isometric immersion, this definition agrees with that of B.Y. Chen, who defined biharmonic submanifolds of Euclidean space as those satisfying∆2ϕ = 0.Since∆ϕ = −nH (H being the mean curvature vector), every minimal submanifold is clearlybiharmonic. The question is whether, apart from minimal ones, one can find other (nonminimal)biharmonic immersions. All efforts so far to find such an example invariably showed that anybiharmonic submanifold belonging to certain classes of submanifolds has to be minimal. Forexample, the reviewer showed that a biharmonic submanifold inRn must be minimal in eachof the following cases: (a) it is a curve; (b) it has at most two principal curvatures; (c) it hasconstant mean curvature [I. Dimitric, Bull. Inst. Math. Acad. Sinica20 (1992), no. 1, 53–65;MR1166218 (93g:53087)]. The only biharmonic hypersurfaces inR4 are the minimal ones [T.Hasanis and T. Vlachos, Math. Nachr.172(1995), 145–169;MR1330627 (96c:53085)]. This led

Chen to conjecture that there are no nonminimal biharmonic submanifolds inRn. The situationappears to be similar when the ambient space has non-positive curvature, thus the generalizedChen conjecture states that the only biharmonic submanifolds of a Riemannian manifoldN withRiemN ≤ 0 are the minimal ones. On the other hand, in the spaces of positive curvature thesituation is different, since examples of proper biharmonic submanifolds do exist.

This paper is devoted to a study of biharmonic curves on 2-dimensional surfaces. For a curveγ: I → (M 2, g) the Euler-Lagrange equation of bienergy splits into two conditions (of interesthere is the case whereγ is not a geodesic):kg = const 6= 0 andk2

g =G, wherekg is the geodesiccurvature of the curve andG the Gauss curvature of the surface along the curve. Thus, ifγ is a non-geodesic biharmonic curve on a surfaceM 2, then alongγ the Gauss curvature is constant, positiveand equal to the square of the geodesic curvature ofγ. As a consequence, ifM 2 has nonpositiveGauss curvature, any biharmonic curve is necessarily a geodesic. A separate section of the paperis devoted to biharmonic curves on surfaces of revolution. The biharmonic parallels (circles oflatitude) on such surfaces are characterized. Examples of proper biharmonic curves on surfaces ofconstant curvature inR3 are produced. For example, on the 2-sphereS2(r), the circle of latitudeof radiusr/

√2 (i.e. the geodesic circle of radiusπr/4) is an example of a biharmonic curve.

For the entire collection seeMR1870993 (2002g:53002)Reviewed byIvko Dimitric

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Citations

From References: 18From Reviews: 3

MR1863283 (2002k:53123)53C43 (58E20)

Caddeo, R.(I-CAGL) ; Montaldo, S. (I-CAGL) ; Oniciuc, C. (R-IASIM)Biharmonic submanifolds ofS3. (English summary)Internat. J. Math.12 (2001),no. 8,867–876.

An important survey of J. Eells, Jr. and J. H. Sampson [Amer. J. Math.86 (1964), 109–160;MR0164306 (29 #1603)] presented many useful results on harmonic mapsϕ: (M, g) → (N,h)between Riemannian manifolds known at that time. These maps appear as critical points of theenergyE(ϕ) = 1

2

∫M |dϕ|2vg, and they are determined by the Euler-Lagrange equation, which,

in this case, is given by the vanishing of the tension fieldτ(ϕ) = trace∇dϕ. Eells and Sampsonalso proposed a study ofk-harmonic maps. Whenk = 2, the bienergy ofϕ is defined byE2(ϕ) =12

∫M |τ |2dvg, and consideration of its critical points, which one calls biharmonic maps, leads to

the Euler-Lagrange equationτ2(ϕ) = J(τ(ϕ)) = 0, whereJ is the Jacobi operator ofϕ, whichtakes on the expression

τ2(ϕ) =−∆τ(ϕ)− traceRN(dϕ, τ(ϕ))dϕ.

Note that ifM is not compact, the requirement is thatϕ be a critical point of the corresponding

functional on any compact domainΩ ⊂M. WhenN is Euclidean space, this definition agreeswith that of B. Y. Chen according to which a mapϕ is biharmonic if∆2ϕ= ∆(∆ϕ) = 0, and theLaplacian (onM ) is taken componentwise.

In this paper the authors classify nonminimal biharmonic submanifolds of the unit 3-sphereS3.Namely, ifm = 1, a nongeodesic curveM is biharmonic if and only ifM is either a circle ofradius1/

√2 or a geodesic of the Clifford torusS1(1/

√2)×S1(1/

√2)⊂ S3. Whenm= 2, then

the surfaceM is (a piece of) a small hypersphereS2(1/√

2)⊂ S3.Reviewed byIvko Dimitric

References

1. B. Y. Chen,Some open problems and conjectures on submanifolds of finite type, Soochow J.Math.17 (1991), 169–188.MR1143504 (92m:53091)

2. B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math.52 (1998), 167–185.MR1609044 (99b:53078)

3. M. Dajczer,Submanifolds and Isometric Immersions, Mathematics Lecture Series 13, Publishor Perish, 1990.MR1075013 (92i:53049)

4. J. Eells and J. H. Sampson,Harmonic mappings of Riemannian manifolds, Amer. J. Math.86(1964), 109–160.MR0164306 (29 #1603)

5. Z. H. Hou,Hypersurfaces in a sphere with constant mean curvature, Proc. Amer. Math. Soc.125(1997), 1193–1196.MR1363169 (97f:53096)

6. G. Y. Jiang, 2-harmonic isometric immersions between Riemannian manifolds, Chinese Ann.Math. Ser.A7(2) (1986), 130–144.MR0858581 (87k:53140)

7. G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann.Math. Ser.A7(4) (1986), 389–402.MR0886529 (88i:58039)

8. C. Oniciuc,Biharmonic maps between Riemannian manifolds, preprint.cf. MR 2004e:53097Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR1891166 (2002m:53104)53C43 (58E20)

Montaldo, S. (I-CAGL) ; Wood, J. C. (4-LEED-PM)Harmonic morphisms and the Jacobi operator. (English, Italian summaries)Rend. Sem. Fac. Sci. Univ. Cagliari70 (2000),no. 2,21–28.

Between Riemannian manifolds, harmonic maps are defined as critical points of the energy func-tional. The first variation yields the tension field, a generalisation of the Laplace-Beltrami operatorand of the geodesic equation.

To classify harmonic maps, Mazet and Smith, some thirty years ago, computed the secondvariation of the energy functional, and obtained the Jacobi operator,Jϕ = ∆ϕ − trR(dϕ, ·)dϕ(note the sign convention of the curvature), a linear elliptic self-adjoint operator. The crucialinformation onJϕ is the index (number of negative eigenvalues) and the nullity (the dimension ofits kernel).

Results on the stability of harmonic maps are few and far between, so the authors propose here tospecialise the study to harmonic morphisms, a subclass of harmonic maps with a strong geometricflavour.

Harmonic morphisms are, by definition, maps which preserve, by composition, local harmonicfunctions, and are characterised as horizontally weakly conformal (a geometrical condition gen-eralising Riemannian submersions) harmonic maps. Harmonic morphisms are also known topreserve harmonic maps, so it is logical to study their influence on stability.

The main result of this paper is that composing a harmonic map with a harmonic morphismsimply multiplies the Jacobi operator of the former by a positive function depending solely on thelatter. As a first consequence, composing by a harmonic morphism will only increase the nullity orindex of a harmonic map. As the identity map is always harmonic, one can ask when a harmonicmorphism has exactly the nullity and index of the identity. If we assume that the manifolds arecompact and the identity is stable (a nontrivial condition), this will happen when the harmonicmorphism considered is a Riemannian submersion with totally geodesic fibres and integrablehorizontal distribution.

Finally, it is shown that a non-constant harmonic morphismϕ from a compact manifold to asphere is infinitesimally and locally rigid, in the sense of Toth; that is, first, any Jacobi vector fieldV (i.e. inKer Jϕ) with 〈dϕ,∇V 〉= 0 can be writtenV =X ϕ withX ∈ so(n+1) and, second,for any harmonic variationV (i.e. exp (tV ) is harmonic for allt ∈ R), there exists agt ∈ O(n+1) such thatexp (tV ) = gt ϕ.

Reviewed byEric Loubeau

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MR1735683 (2001e:53069)53C43Montaldo, Stefano(I-CAGL)On the stability of harmonic morphisms.Harmonic morphisms, harmonic maps, and related topics(Brest, 1997), 31–38,Chapman &Hall/CRC Res.Notes Math., 413,Chapman & Hall/CRC, Boca Raton, FL, 2000.

The paper is a survey of results due mainly to the author [Algebras Groups Geom.16(1999), no. 3,277–286;MR1727731 (2000j:53088); Internat. J. Math.9 (1998), no. 7, 865–875;MR1651053(2000b:53096)] on the stability of harmonic morphisms for the energy functional.

Let (M, g), (N,h) and(P, k) be compact Riemannian manifolds without boundary. The authorgives the form of the Jacobi operator for a harmonic morphism, showing that whenRicciM ≤ 0,any harmonic morphismϕ:M →N is energy-stable.

He also shows that the index and nullity of any harmonic morphismψ:N → P increase whenψis composed with a harmonic morphism with minimal fibresϕ:M →N . In particular, the identityonN gives a lower bound for the index and nullity of any harmonic morphismϕ:M → N withminimal fibres.

He then studies the links between volume- and energy-stability of harmonic morphisms into asurface. Ifϕ:Mm→N 2 (m≥ 3) is a submersive harmonic morphism with minimal and volume-stable fibres into a surface, thenϕ is energy-stable. In particular, whenRiemM ≤ 0, any submersiveharmonic morphismϕ:Mm→N 2 with totally geodesic fibres is energy-stable.For the entire collection seeMR1735679 (2000j:53001)

Reviewed byAnne-Joelle Vanderwinden

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MR1739695 (2001c:53089)53C43 (58E20)

Loubeau, Eric (F-BRETT); Montaldo, Stefano(4-LEED-PM)A note on exponentially harmonic morphisms. (English summary)Glasg. Math. J.42 (2000),no. 1,25–29.

A smooth mapϕ: (M, g)→ (N,h) between Riemannian manifolds is exponentially harmonic if itis a critical point of the exponential energyE(ϕ) =

∫Ω exp(e(ϕ)) vg on every compact domainΩ

in M . It is an exponentially harmonic morphism if it pulls back germs of exponentially harmonicfunctions to exponentially harmonic functions.

The authors show that exponentially harmonic morphisms are exactly the Riemannian submer-sions with minimal fibres.

Reviewed byAnne-Joelle Vanderwinden

References

1. S. Alinhac and P. Gerard,Operateurs pseudo-differentiels et theoreme de Nash-Moser, SavoirsActuels (InterEditionsEditions du C.N.R.S., 1991).MR1172111 (93g:35001)

2. J. Eells and L. Lemaire,Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math.50 (Amer. Math. Soc., Providence, R.I., 1983).MR0703510 (85g:58030)

3. J. Eells and L. Lemaire, Some properties of exponentially harmonic maps,Banach CenterPubl.27 (1992), 129–136.MR1205818 (94d:58045)

4. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds,Amer. J. Math.86

(1964), 109–160.MR0164306 (29 #1603)5. D. Gilbarg and N. S. Trudinger,Elliptic partial differential equations of second order, A

Series of Comprehensive Studies in Mathematics224 (Springer-Verlag, 1977).MR0473443(57 #13109)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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Citations

From References: 1From Reviews: 1

MR1727731 (2000j:53088)53C43Montaldo, Stefano(4-LEED-PMS)Stability of harmonic morphisms. (English summary)Algebras Groups Geom.16 (1999),no. 3,277–286.

Harmonic morphisms between Riemannian manifolds are, by definition, maps which preserve(local) harmonic functions. They were shown, independently, by B. Fuglede [Ann. Inst. Fourier(Grenoble)28 (1978), no. 2, vi, 107–144;MR0499588 (80h:58023)] and T. Ishihara [J. Math.Kyoto Univ. 19 (1979), no. 2, 215–229;MR0545705 (80k:58045)], to be a subclass of harmonicmaps satisfying a geometrical condition called horizontally weakly conformal. It therefore makessense to study the stability of harmonic morphisms.

Exploiting the geometrical properties of harmonic morphisms, the author gives an expression forthe Jacobi operator of a harmonic morphism and shows that since on any manifold of dimensionat least three there exists a complete metric for which the Ricci curvature is negative, a result of J.Lohkamp [Ann. of Math. (2)140(1994), no. 3, 655–683;MR1307899 (95i:53042)], any harmonicmorphism mapping into a manifold of dimension at least three is stable, i.e. the Jacobi operator ispositive definite.

Contrary to the case for harmonic maps, the composition of two harmonic morphisms is againa harmonic morphism and the author proves that the Jacobi operator of a harmonic morphismcomposed with a submersive harmonic morphism with minimal fibres is proportional to the Jacobioperator of the first map.

This result is then exploited to draw conclusions on the indices (the number of negative eigen-values of the Jacobi operator) and the nullities (the dimension of the kernel) of the composition ofharmonic morphisms and the identity map of the target.

The particularly simple case of the projection from a Riemannian product to one factor concludesthe paper. Note that here all manifolds are assumed to be compact. The result on the Jacobioperator of the composition, and its corollaries, have recently been shown to stand even withoutthe condition of minimal fibres [S. Montaldo and J.C. Wood, “Harmonic morphisms and the Jacobi

operator”, Preprint, 1999; per revr.].Reviewed byEric Loubeau

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Citations

From References: 0From Reviews: 0

MR1706588 (2000h:58032)58E20 (53C43)

Montaldo, Stefano(4-LEED-PM)p-minimising tangent maps and harmonick-forms. (Italian summary)Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)2 (1999),no. 2,331–339.

The author studies thep-tangent maps ofRm to Sn of harmonick-homogeneous maps. Forp-harmonick-formsSm−1 → Sn, a lower bound fork is derived, assuming that the tangent map isp-minimizing. A complete classification is given for the tangent mapsR8 → S4.

Reviewed byG. Toth

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MR1794446 (2002b:58022)58E20Montaldo, S. (I-CAGL)A minimising property of the radial projection. (English summary)Rend. Sem. Mat. Univ. Politec. Torino56 (1998),no. 2,55–58 (2000).

LetB be a geodesic ball, centered atp0, in a1-connected space formE3 (which, up to rescaling,is isometric to the standardS3, R3 or H3), and setB0 = Br p0. LetG be the set ofC∞-mapsψ:B0 → S2 of degree zero atp0.

It was proved in [H. R. Brezis, J.-M. Coron and E. H. Lieb, Comm. Math. Phys.107 (1986),no. 4, 649–705;MR0868739 (88e:58023)] that the radial projectionπ:B0 ⊂R3 → S2:x 7→ x/|x|is energy-minimizing inG. In the paper under review, the author extends this result to the otherspace forms: in all three cases, the minimum of the energy inG is attained by the radial projection.

The idea of the proof is to realize the pointed space formsE3 r p0 as warped products of the2-sphereS2 with a suitable 1-dimensional mani-

fold.Reviewed byAnne-Joelle Vanderwinden

c© Copyright American Mathematical Society 2002, 2008

Article

Citations

From References: 1From Reviews: 1

MR1651053 (2000b:53096)53C43 (58E20)

Montaldo, Stefano(4-LEED-PMS)Stability of harmonic morphisms to a surface. (English summary)Internat. J. Math.9 (1998),no. 7,865–875.

A mapϕ: (M, g) → (N,h) is called a harmonic morphism if, for any locally defined harmonicfunctionf , f ϕ is also a harmonic function. The famous characterization of harmonic morphismsby T. Ishihara and B. Fuglede reads as follows:ϕ is a harmonic morphism if and only ifϕ is a har-monic map and is horizontally (weakly) conformal. The latter concept of horizontal conformalitymeans thatdϕp|ker(dϕp)⊥ is conformal and surjective ifp ∈M rCϕ, whereCϕ is the set of pointsp ∈M such thatdϕp is not of maximal rank. In casedimM ≥ 2 anddimN = 2, if ϕ:M →N isa horizontally (weakly) conformal map, thenϕ is a harmonic morphism if and only ifϕ|MrCϕ hasminimal fibres. (This is a result due to Baird and Eells and was generalized to the case of allowingcritical point setCϕ by J. C. Wood.) The author studies the relation between the volume stabilityof minimal fibers and the energy stability of harmonic morphisms to a surface. This study is par-tially motivated by a result of J. Chen [J. Differential Geom.43 (1996), no. 1, 42–65;MR1424419(98c:58032)] which states that any stable harmonic map from a compact Riemannian manifoldinto a standard 2-sphere is a harmonic morphism. The author proves: Letϕ: (M, g) → (N 2, h)be a submersive harmonic morphism to a surface with volume-stable minimal fibers. Thenϕ isenergy-stable. This, together with a result of Y. Ohnita and S. Udagawa [Math. Z.205 (1990),no. 4, 629–644;MR1082880 (91k:58108)], gives a corollary: If the sectional curvature ofM isnon-positive, then any submersive harmonic morphismϕ: (M, g)→ (N 2, h) with totally geodesicfibers is energy-stable. In the last section, the author gives a result about the stability for harmonicmorphisms from a compact Riemannian manifold with piecewise smooth boundary using a re-sult by H. Takeuchi [Comment. Math. Univ. St. Paul.31 (1982), no. 2, 223–230;MR0701010(84k:58062)].

Reviewed bySeiichi Udagawa

References

1. P. Baird,Harmonic morphisms onto Riemann surfaces and generalized analytic functions,Ann. Inst. Fourier (Grenoble)37 (1987), 135–173.MR0894564 (88h:31009)

2. P. Baird and J. Eells,A conservation law for harmonic maps, Geometry Symposium, eds. E.

Looijenga, D. Siersma and F. Takens, Utrecht 1980, pp. 1–25. Lecture Notes in Math. 894,Springer-Verlag, Berlin, Heidelberg, New York, 1981.MR0655417 (83i:58031)

3. J. Y. Chen,Stable harmonic maps intoS2, Geometry and Global Analysis, eds. T. Kotake, S.Nishikawa and R. Schoen, pp. 431–436. Tohoku Univ., Sendai, 1993.MR1361209 (96k:58054)

4. J. Eells and L. Lemaire,A report on harmonic maps, Bull. London Math. Soc.10(1978), 1–68.MR0495450 (82b:58033)

5. J. Eells and L. Lemaire,Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math.Vol. 50. Amer. Math. Soc., Providence, R.I., 1983.MR0703510 (85g:58030)

6. J. Eells and L. Lemaire,Another report on harmonic maps, Bull. London Math. Soc.20(1988),385–524.MR0956352 (89i:58027)

7. B. Fuglede,Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Greno-ble)28 (1978), 107–144.MR0499588 (80h:58023)

8. S. Gudmundsson,On the existence of harmonic morphisms from symmetric spaces of rank one,preprint, 1996.cf. MR 98f:58063

9. S. Gudmundsson, The bibliography of harmonic morphisms, http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/bibliography.html.

10. S. Gudmundsson, E. Loubeau, S. Montaldo and T. Mustafa,The atlas of harmonic morphisms,http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/atlas.html.

11. T. Ishihara,A mapping of Riemannian manifolds which preserves harmonic functions, J. Math.Kyoto Univ. 19 (1979), 215–229.MR0545705 (80k:58045)

12. H. B. Lawson,Lectures on minimal submanifolds, Mathematics Lecture Series9, Publish orPerish, 1980.MR0576752 (82d:53035b)

13. R. Lipschitz,Ausdehnung der theorie der minimalflachen, J. Reine Angew. Math.78 (1874),1–45.

14. S. Montaldo,On the stability of harmonic morphisms, preprint, 1997.cf. MR 2000j:5308815. Y. Ohnita and S. Udagawa,Stability, complex-analyticity, and constancy of pluriharmonic maps

from compact Kahler manifolds, Math. Z.205(1990), 624–644.MR1082880 (91k:58108)16. H. Takeuchi,A stable harmonic map of a manifold with boundary, Comment. Math. Univ. St.

Paul.31 (1982), 223–230.MR0701010 (84k:58062)17. H. Urakawa,Stability of harmonic maps and eigenvalues of the Laplacian, Trans. Amer. Math.

Soc.301(1987), 557–589.MR0882704 (88g:58046)18. J. C. Wood,Harmonic maps and morphisms in4 dimensions, Proc. first Brazilian—USA

Workshop on Geometry, Topology and Physics, to appear.cf. MR 99b:5806719. Y. L. Xin, Some results on stable harmonic maps, Duke Math. J.47 (1980), 609–613.

MR0587168 (81j:58041)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

c© Copyright American Mathematical Society 2000, 2008

Article

Citations

From References: 2From Reviews: 2

MR1420937 (97k:58055)58E20Montaldo, Stefano(4-LEED-SM)p-harmonic maps and stability of Riemannian submersions. (Italian summary)Boll. Un. Mat. Ital. A (7)10 (1996),no. 3,537–550.

As usual we say that a mapϕ: (Mm, g)→ (Nn, h) is a Riemannian submersion ifMx =Hx⊕Vxfor everyx ∈M , where the orthogonal decomposition is determined by the metricg. Then, foreacht ∈R+, let gt be the unique Riemannian metric onM such thatgt(X,Y ) = g(X,Y ) for allX,Y ∈ Hx; Hx andVx are orthogonal to each other with respect to eachgt andgt(X,Y ) = t2 ·g(X,Y ) for all X,Y ∈ Vx and for allx ∈M . Then, for allt ∈R+, the Riemannian submersionwith totally geodesic fibersϕ: (M, gt)→ (N,h) is called the canonical variation.

A mapu: (Mm, g)→ (Nn, h) is said to bep-harmonic if it is a critical point of thep-energy func-tionalEp = 1

2

∫M ‖dϕ(x)‖pvg. The author shows in this article, for example, that ifϕ: (M, g)→

(N,h) is a Riemannian submersion with totally geodesic fibres and if the identity map ofN is p-stable for somep≥ 2 then there existsε > 0 such that the canonical variationϕ: (M, g)→ (N,h)is p-stable for somep and for everyt < ε. For related material see for example the work of H.Urakawa [Trans. Amer. Math. Soc.301(1987), no. 2, 557–589;MR0882704 (88g:58046)].

Reviewed byCaio J. C. Negreiros

c© Copyright American Mathematical Society 1997, 2008