Paying Tributeto James Eells andJoseph H. Sampson:In Commemoration ofthe Fiftieth Anniversaryof Their PioneeringWork on Harmonic Maps

Yuan-Jen Chiang and Andrea Ratto

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Professor J. Eells (1926–2007).

DOI: http://dx.doi.org/10.1090/noti1225

James Eells was born in Cleveland, Ohio, in 1926and passed away in Cambridge, England, in Feb-ruary 2007. He earned his PhD from HarvardUniversity under the guidance of the topologistand analyst Hassler Whitney in 1954. He workedat the Institute for Advanced Study in Princetonand at the University of California in Berkeleyfrom 1956–1958. He then returned to the EastCoast where he accepted a position at ColumbiaUniversity (1958–1964). He also taught at ChurchillCollege, Cambridge, and Cornell University. Later,attracted by the atmosphere and potential at theUniversity of Warwick, he joined the mathematicsdepartment there and became a professor of globalanalysis and geometry in 1969. Eells organizedyear-long highly successful Warwick symposia,namely, “Global Analysis” (1971–1972) and “Ge-ometry of the Laplace Operator” (1976–1977). Hewas the first director of the Mathematical Sectionof the International Centre for Theoretical Physicsin Trieste from 1986 to 1992. As a prominentprofessor and an inventive mathematician, Eells’smathematical influence in the field of harmonic

Yuan-Jen Chiang is professor of mathematics at the Uni-versity of Mary Washington, VA. Her email address isychiang@umw.edu.

Andrea Ratto is professor of mathematics at the Universityof Cagliari, Italy. His email address is rattoa@unica.it.

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maps became internationally recognized and wide-spread. According to the Mathematics GenealogyProject, he advised thirty-eight PhDs, who, in turn,produced 198 descendants. One of Eells’s PhDsstudents was Andrea Ratto, who describes hisexperience as follows:

“When I first met Jim at Warwick in 1985,he gave me a copy of his survey article [9] (withL. Lemaire). He told me, ‘We look for a candidate forthe best map in a given homotopy class.’ This is afascinating problem, because it combines calculusof variations, differential equations, Riemanniangeometry and topology. A special instance concernsthe case of mappings between spheres, and Jimwas intrigued by this topic, because in this contextsolutions are unstable and more difficult to obtainand thus less likely to be understandable. A fewweeks after our first meeting, Jim told me that,in his opinion, the right topic for my researchactivity could be the development of the beautifulexistence theory of R. T. Smith for equivariantharmonic maps of spheres. That was the beginningof my adventure in mathematics, and all thatsprang from our conversations always remains, inmy mind, tightly tied to the extraordinary charismaand exquisite mathematical taste of J. Eells.”

James Eells retired from the University ofWarwick in 1992. Then he moved to Cambridgeand continued to work on harmonic maps for theremainder of his life.

Joseph H. Sampson was born in Philadelphia in1926 and passed away in the South of France inJuly 2003. He earned his PhD from Princeton Uni-versity under Salomon Bochner in 1951 (we oftenutilize Bochner’s techniques in harmonic maps).Sampson then worked as a Moore Instructor at MIT.He was appointed as visiting assistant professor atthe Johns Hopkins University in 1955 and then asassistant professor and was promoted to associateprofessor in 1963. He finally became full professorin 1965. He was an editor of the American Journalof Mathematics from 1978 to 1992 and the chairof the mathematics department at Johns Hopkinsfrom 1969 to 1979. Sampson retired from JohnsHopkins in 1990. He was Yuan-Jen Chiang’s advi-sor at Johns Hopkins, where she earned her PhDin 1989. For insights into her interactions withSampson, Chiang provides the following:

“In 1982, when I first met Joe, I had only a Bach-elor of Science degree. The requirements for a PhDincluded passing three oral examinations (eachwith two professors). For my three exams I chosereal analysis, differential geometry, and algebraictopology. When I took Differential Geometry, Joeand I had arguments, which caused me to liter-ally cry. Later, however, I recalled the questionsand then realized that he was exceptionally bril-liant. In 1984, and even to this day, I remember

asking Joe the question, ‘What is the goal and ob-jective of your life?’ Joe replied, ‘Life is hostile, notevil. Finance is very important, but mathematics ismost important to me.’ Then I asked, ‘Would youlike to be my PhD advisor?’ Joe then asked, ‘Doyou really want to study harmonic maps?’ ‘Yes!’I replied. Then we started a long journey whichbecame an ordeal and Joe made me upset severaltimes. He was bombastic. At first he wanted meto generalize (1) value distribution theory (VDT)for holomorphic maps of Riemann surfaces intoharmonic maps and (2) VDT for harmonic maps ofcomplex manifolds. Unfortunately, after approxi-mately two years studying S. S. Chern’s articles andinvestigating other related papers, I realized thatsuch generalizations would not work. Essentially Iwas able to show Joe that such a generalization hadan obstruction since the pull-back f∗ commuteswith ∂ and ∂ for a holomorphic map, but not fora harmonic map. I remember that his face turnedgray suddenly, and he said, ‘Mathematics is diffi-cult.’ Then we investigated another topic, whichdid not work well either. Finally, he suggested ‘Har-monic maps of V-manifolds’ for me, and we bothexperienced a happy ending. I was so fortunate tobe Joe’s only PhD student to study harmonic mapswith him directly (his other student, E.-B. Lin, wentto MIT to study geometric quantization and cameback for defense). I was very pleased to learn thebeautiful tensor techniques from Joe through Eisen-hart’s book. I admired his penetrating insight andimpeccable taste which characterized the preciousguidance that he provided over those years.”

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Professor J. H. Sampson(1926–2003).

For more than fourdecades, Eells and Samp-son were very goodfriends; both wereoutstanding mathemati-cians, brilliant teachers,and great experts inanalysis, geometry, andtopology. Sampson in-vited Eells to speak onharmonic maps at amathematics departmentcolloquium at Johns Hop-kins around 1985. Thejoint paper of Chiang andRatto [5], published inthe Bulletin of the FrenchMathematical Society in1992, was dedicated toEells and Sampson. OnJuly 20, 1992, Chiangreceived a letter fromEells: Dear Dr. Chiang,I have just received my copy of Bull. SMF 120and was most surprised and pleased to read a

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dedication by you and our excellent friend Andrea.What a nice idea! Thank you very much, and allmy best. Yours cordially, James Eells. That was thestyle of Jim: genuine, simple, and direct.

The theory of harmonic maps between Riemann-ian manifolds was first established by Eells andSampson [14]. However, the notion of harmonicmap was introduced by Sampson in the hope of ob-taining a homotopy version of the highly successfulHodge theory for cohomology in 1952. Not longafter that, his then colleague John Nash (one of thethree Nobel laureates in economics in 1994) pro-posed a quite different but equivalent definition—both of them were Moore Instructors at MIT atthat time. Fuller [15] also came upon harmonicmaps in 1954. Later, when both were working atthe Institute for Advanced Study at Princeton, Eellsand Sampson wrote their famous and outstandingpaper “Harmonic mappings of Riemannian man-ifolds” [14]. This article is considered as the pio-neering work in the theory of harmonic maps andhas provided the seeds for many further develop-ments. As Eells pointed out in the preface of [7],“Harmonic maps pervade differential geometry andmathematical physics: they include geodesics, mini-mal surfaces, harmonic functions, abelian integrals,Riemannian fibrations with minimal fibres, holo-morphic maps between Kahler manifolds, chiralmodels, and strings.”

A harmonic map f : (Mm, g) → (Nn, h) froman m-dimensional Riemannian manifold into ann-dimensional Riemannian manifold is defined asa critical point of the energy functional

E(f ) = 12

∫M|df |2dv,

where dv is the volume form of M determined bythe metric g. The Euler-Lagrange equation associ-ated with the energy functional is

(1) τ(f ) = traceg(∇df) = 0 ,

where ∇df is the second fundamental form of themap and the left member of (1) is called the tensionfield of f . Equation (1) is a second-order semilinearsystem of partial differential equations: these ana-lytical features can be deduced by the expressionof (1) in local charts, i.e.,

gij(fαij − MΓ kij fαk + NΓαβγ fβi f γj ) = 0,

where the sum over repeated indices is under-stood and MΓ kij and NΓαβγ denote the Christoffelsymbols of the Levi-Civita connections on M andN, respectively.

In simple terms, the main idea of Eells and Samp-son can be described as follows. Suppose that f0wraps (in a given homotopy class) a rubber (M) intoa marble (N). If we deform f0 (the initial condition)by using heat, then its energy decreases and we

expect that the system evolves towards a configu-ration of minimum energy (i.e., a harmonic map).Note that this is the first example of a significantapplication of a nonlinear heat flow, where the non-linearity has a strong geometrical meaning becauseit depends on the curvature of N. The novelty ofthis approach is that its success is strictly relatedto the curvature itself (and so, to the topology) ofthe target manifold N. In this sense, it is importantto say that Eells-Sampson’s Theorem was the firstinstance where curvature played a basic role inanalysis, a fact which was a crucial milestone forglobal analysis. More specifically, Eells and Samp-son [14] were able to prove the following assertion:Let N have nonpositive Riemannian curvature andlet f : M → N be a continuously differentiable map.Let ft be the solution of the heat equation associ-ated to (1); i.e., τ(ft) = ∂f

∂t , which reduces to f att = 0. If ft is bounded as t →∞, then f is homotopicto a harmonic map f ′ for which E(f ′) ≤ E(f ). Inparticular, if N is compact, then every continuousmap M → N is homotopic to a harmonic map.

In order to study geometric and topologicalproblems, Hamilton introduced the Ricci flow withthe aim of attacking the Thurston geometrizationconjecture and the Poincare conjecture. At the2006 International Congress of Mathematiciansin Madrid, he said that his initial inspiration camein the late 1960s, when he attended the seminarsof Eells and Sampson on harmonic maps, who firstsuggested that one might be able to utilize evolu-tion equations to study the Poincare conjecture.Later, Hamilton used the idea of evolution equa-tions and developed the theory of the Ricci flow,which laid the foundation for G. Perelman’s workfor solving both the Poincare and the Thurstonconjectures.

Shortly after the publication of [14], the explo-sion of interest in harmonic maps started (unlessotherwise indicated, detailed bibliographical ref-erences for all the works that we mention can befound in [1], [9], and [10]: Hartman showed thatif f : M → N is harmonic with M compact andRiemN ≤ 0 at every point of f (M) and there is apoint of f (M) at which RiemN < 0, then f is uniquein its homotopy class. The previous results of Eellsand Sampson were extended to harmonic mapsbetween manifolds with boundary by Hamilton.The curvature condition RiemN ≤ 0 was modifiedby Sacks and Uhlenbeck as follows: if m = 2 andπ2(N) = 0, then given a map f0 : M → N there isa harmonic map f homotopic to f0. The conditionRiemN ≤ 0 can also be replaced by the conditionthat the image of f0 (and hence of f ) supports a uni-formly strictly convex function (by Jost). Moreover,Schoen and Uhlenbeck proved a partial regular-ity theorem which asserts that a bounded energy

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minimizing map f : M → N between two Riemann-ian manifolds is regular (in the interior) except fora closed singular set S of Hausdorff dimensionat most (m− 3). In the meantime, Giaquinta andGiusti and also Hildebrandt-Kaul-Widman proved apartial regularity theorem in the case that f (M) iscontained in a single chart of N. In the 1970s and1980s, utilizing the ideas and methods of complexand algebraic geometries, Eells and Wood, Chernand Wolfson, Burstall and Wood, Eells and Salamon,etc., constructed and characterized harmonic mapsfrom Riemann surfaces into projective spaces orcomplex Grassmannians. All these important de-velopments were made easy to access, thanks tothe precious surveys of Eells and Lemaire ([9], [10],and [11]). In the 1980s, Siu and Sampson [19], [21],and [22] made breakthroughs on harmonic maps ofKahler manifolds. Carlson and Toledo also studiedharmonic maps of Kahler manifolds into locallysymmetric spaces. In 1989, Uhlenbeck exploredharmonic maps into Lie groups and obtained ele-gant results. Afterwards, two books on harmonicmaps, integrable systems, conservation laws, andmoving frames were published by Hélein [17], etc.In summary, a multitude of mathematicians, withprofoundly different backgrounds, turned theirresearch interest to harmonic maps.

Harmonic maps are intimately connected withthe classical study of minimal and constant meancurvature submanifolds: that is not simply becausea Riemannian immersion is minimal (i.e., a criti-cal point of the volume functional) if and only ifit is harmonic, but also because of the commonmethods in analysis. In this spirit, Eells and Rattocollaborated on a book [12] (published by Prince-ton University Press) about harmonic maps andminimal immersions such that the presence of suit-able symmetries (due to invariance with respectto suitable group actions or isoparametric func-tions) reduces the analytical problem to the studyof a nonlinear ODE system. Writing this mono-graph was an amazing experience for Ratto sincehe could really enjoy a deep and daily interactionwith Jim. In particular, he could share Jim’s viewabout the value of simple, key starting ideas andexamples and, in a sort of ideally complementaryvision, about the fact that all branches of mathe-matics (and the sciences) are just one thing. Jimcould imagine, in the future, mathematical meth-ods providing increasing support for all the otherscientific disciplines, and that was a strong mo-tivation for his research and for the enthusiasticguidance with which he encouraged and motivatedhis great number of students and collaborators. Inthat period, we also had the opportunity to worktogether on some extensions of the equivariant the-ory of spheres to the case of Euclidean ellipsoids[13]: Jim made me proud by telling me that he was

extremely happy with our construction of a signifi-cant family of harmonic morphisms from suitable3-dimensional ellipsoids to S2. He considered theseamong the geometrically more interesting exam-ples within the theory of harmonic morphisms, afield which now, thanks to the work of world-classmathematicians such as Wood and Baird (both for-mer PhD students of Jim at Warwick), has attractedthe interest of a very ample new generation ofmathematicians (see [2] and references therein).

The notion of a harmonic map interestinglyextends to the cases that manifolds are notsmooth, but singular spaces. In this direction,Chiang studied “Harmonic maps of V-manifolds”[4] in 1990. The main difficulties arise from thecomplicated behavior near the singular locus ofV-manifolds. Therefore, a new method is requiredto study spectral geometry of V-manifolds byapplying the techniques in Baily’s paper andSampson’s book manuscripts [20]. Then we usetriangulations and simplices to deal with harmonicmaps of V-manifolds. Afterwards, Chiang andRatto [5] studied harmonic maps of spaces withconical singularities from a different approachin 1992 since their structures are different fromthose of V-manifolds. In the meantime, Gromovand Schoen [16] also investigated harmonic mapsinto singular spaces and p-adic superrigidity forlattices into groups of rank one. All these worksmotivated Eells’s final monograph, Harmonic Mapsbetween Riemannian Polyhedra [8], coauthoredwith Fuglede, on harmonic theory with singulardomains and targets, which was published in2001.

Wave maps are harmonic maps on Minkowskispaces and have been studied since the late 1980s.In this quarter, there have been many new de-velopments achieved by a number of well-knownmathematicians. Yang-Mills fields (cf. Donaldson[6]) are the critical points of the Yang-Mills func-tionals of connections whose curvature tensors areharmonic. They were first explored by a few physi-cists in the 1950s, and since then there have beenmany striking developments in this topic. Chiang’srecent book Developments of harmonic maps, wavemaps and Yang-Mills fields into biharmonic maps,biwave maps and bi-Yang-Mills fields [3] was pub-lished by Birkhauser-Springer in 2013. Due to herfeelings and personal experience, it has been anhonor to conceive this book in the memory of Pro-fessors Eells and Sampson as follows: The namesof these two pioneers of the theory of harmonicmaps will be engraved in the minds of all mathe-maticians who work on harmonic maps, wave mapsand Yang-Mills fields, for their great and everlast-ing contributions. Chiang met 2010 Fields MedalistC. Villani (director of the Institut Henri Poincarein Paris) at the Second Pacific Rim Mathematics

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Association Congress in Shanghai, China, in June2013. He delivered a great and very impressiveopening address, and he was one of six advisoryeditorial board members of her book. The citationfor Villani’s Fields Medal was “For his proofs ofnonlinear Landau damping and convergence toequilibrium for the Boltzmann equation.” Chiangalso presented “Some properties of biwave Maps”at ICM 2010 in Hyderabad, India.

In the 1990s (see [3] for detailed references),Klainerman and Machedon and Klainerman andSelberg investigated the general Cauchy problemfor wave maps, in any dimension greater than orequal to two, and obtained the almost optimallocal well-posedness for regular data. In the dif-ficult case of dimension two, Christodoulou andTahvildar-Zadeh studied the regularity of spheri-cally symmetric wave maps by imposing a convex-ity condition for the target manifold. Shatah andTahvildar-Zadeh also studied the optimal regular-ity of equivariant wave maps into 2-dimensionalrotationally symmetric and geodesically convexRiemannian manifolds. The study of the generalwave maps problem incorporated methods thatexploited the null-form structure of the wave mapsystem, as in the work of Grillakis, as well as thegeometric structure of the equations as done byStruwe. Keel and Tao studied the 1-(spatial) dimen-sional case. Tataru, following Tao, has used newtechniques which allow one to treat the Cauchyproblem with critical data. Their methods rely onharmonic analysis, such as adapted frequency andgauge-theoretic geometric techniques. Tao estab-lished the global regularity for wave maps fromR×Rm into the sphere Sn for high (I) and low (II) di-mensionsm. Similar results were obtained by Klain-erman and Rodnianski for target manifolds thatadmit a bounded parallelizable structure. Nahmod,Stefanov, and Uhlenbeck studied the Cauchy prob-lem for wave maps from R × Rm into a (compact)Lie group (or Riemannian symmetric spaces) whenm ≥ 4 and established global existence and unique-ness, provided the Cauchy initial data are smallin the critical norm. Shatah and Struwe obtainedsimilar results simultaneously, also in the case thatthe target is any complete Riemannian manifoldwith bounded curvature. Recently, Kenig, Merle,and Duyckaerts investigated global well-posedness,scattering, and finite time blow-up. Furthermore,it is incredible that Tao wrote a book review [23]in the Bulletin of the AMS in October 2013, and helisted his five preprints about global regularity ofwave maps III, IV, V, VI, and VII.

Other variant topics of harmonic maps are theobjects of significant present research: for instance,biharmonic maps were first studied by Jiang [18]in 1986. Also, biwave maps (i.e., biharmonic maps

on Minkowski spaces), bi-Yang-Mills fields, expo-nentially harmonic maps, exponential wave maps,and exponential Yang-Mills connections are topicsof growing interests, (see [3]). Teichmuller theory,minimal surfaces, loop groups, integrable systems,etc., are among them as well.

Professors Eells and Sampson are gone, and theexemplars are in the past. As Jim pointed out thatall branches of mathematics (and the sciences) arejust one thing, he could have imagined that mathe-matical methods would provide increasing supportfor all the other scientific disciplines in the future.To commemorate the fiftieth anniversary of theEells-Sampson pioneering work in harmonic mapsin 2014, we sincerely hope that our generation ofmathematicians will light the torch and carry onthe Eells-Sampson legacy and spirit to the next gen-eration of young mathematicians and scientists. Itis not easy to predict now which developments,among those that we cited in this article, will bemore fruitful in the future. But, for sure, we cansay that the study of nonlinear evolution equa-tions will continue to be a fundamental tool forthe understanding of the central open problems ingeometric analysis, partial differential equations,topology, and mathematical physics.

AcknowledgmentThe authors appreciate Drs. Steve Lipscomb andSoon H. Leong for their comments.

References[1] The Harmonic Maps Bibliogra-

phy, http://people.bath.ac.uk/masfeb/harmonic.html.

[2] P. Bard and J. C. Wood, Harmonic Morphisms betweenRiemannian Manifolds, L. M. S. Monographs 29, OxfordSc. Publ., 2003.

[3] Y.-J. Chiang, Developments of Harmonic Maps, WaveMaps and Yang-Mills Fields into Biharmonic Maps,Biwave Maps and Bi-Yang-Mills Fields, Frontiers inMathematics, Birkhäuser, Springer, Basel, xxi+399,2013.

[4] , Harmonic maps of V-manifolds, Ann. GlobalAnal. Geom. 8 (3) (1990), 315–344.

[5] Y. J. Chiang and A. Ratto, Harmonic maps on spaceswith conical singularities, Bull. Soc. Math. France 120(2) (1992), 251–262.

[6] S. K. Donaldson, Mathematical uses of gauge theory,in: The Encyclopedia of Mathematical Physics, Eds. J.-P.Francoise, G. Naber, and Tsou Sheung Tsun, Elsevier,2006.

[7] J. Eells, Harmonic Maps, Selected Papers of JamesEells and Collaborators, World Scientific PublishingCo., Inc., River Edge, NJ, 1992.

[8] J. Eells and B. Fuglede, Harmonic Maps between Rie-mannian Polyhedra, Cambridge Tracts in Math., 42,Cambridge University Press, Cambridge, 2001.

[9] J. Eells and L. Lemaire, A report on harmonic maps,London Math. Soc. 10 (1978), 1–68.

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[10] , Another report on harmonic maps, LondonMath. Soc. 20 (1988), 385–524.

[11] , Selected topics in harmonic maps, CBMSRegional Conf. Series 50, Amer. Math. Soc., 1983.

[12] J. Eells and A. Ratto, Harmonic Maps and MinimalImmersions with Symmetries. Methods of OrdinaryDifferential Equations Applied to Elliptic VariationalProblems, Annals of Math. Studies, no. 130, PrincetonUniv. Press, Princeton, NJ, 1993.

[13] , Harmonic maps between spheres andellipsoids, Int. J. Math. 1 (1990), 1–27.

[14] J. Eells and J. H. Sampson, Harmonic maps ofRiemannian manifolds, Amer. J. Math. 86 (1964),109–164.

[15] F. B. Fuller, Harmonic mappings, Proc. Natl. Acad. Sci.USA 40 (1954), 987–991.

[16] M. Gromov and R. Schoen, Harmonic maps into sin-gular spaces and p-adic superrigidity for lattices ingroup of rank one, Publ. IHES N◦ 76 (1992).

[17] F. Hélein, Harmonic Maps, Conservation Laws andMoving Frames, Cambridge University Press, 2002.

[18] G. Y. Jiang, 2-harmonic maps and their first and sec-ond variations between Riemannian manifolds, Chin.Ann. Math. A 7 (4) (1986), 130–144.

[19] Y.-T. Siu, The complex-analyticity of harmonic mapsand the strong rigidity of compact Kähler manifolds,Ann. of Math. 112 (1) (1980), 73–111.

[20] J. H. Sampson, Cours de Topologie Algébraic,Départment of Mathématique, Strasbourg, 1969.

[21] , Harmonic mappings and minimal immersions,C. I.M.E. Conf. Lecture Notes in Math, 1161, Springer,Berlin, 1984, pp. 193–205.

[22] , Applications of harmonic maps to Kählergeometry, Contemp. Math 49 (1986), 125–134.

[23] T. Tao, Concentration compactness for critical wavemaps by J. Krieger and W. Schlag, Bull. AMS 50 (4)(2013), 655–662.

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