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Polynomial Harmonic Morphisms Polynomial Harmonic Morphisms Martin Svensson Examensarb ete p oang Lunds Universitet Novemb er Contents Abstract Acknowledgements Chapter Intro duction Motivation History Chapter Harmonic Maps The Second Fundamental Form Harmonic Maps Harmonic Functions Chapter Harmonic Morphisms Horizontal Conformality Harmonic Morphisms The Existence Problem Polar Sets Chapter Polynomial Harmonic Morphisms Globally Dened Harmonic Morphisms The Classication of Ou Polynomial Harmonic Morphisms of Higher Degree Chapter Applications The Theorems of Eells and Yiu The Symb ol of Harmonic Morphisms Bibliography 1 Abstract The aim of this Masters thesis is to b e the rst survey of known results on p olynomial harmonic morphims b etween Euclidean spaces These were rst studied by Baird in in the early s He obtained several results on the sub ject but left op en the still unsolved classica tion of such maps In the article from Eells and Yiu classied the homogeneous p olynomial harmonic morphisms whose restrictions to spheres are again harmonic morphisms to spheres These are the well known Hopf p olynomials of degree This result revitalized the sub ject and so on thereafter Ou obtained a complete classication of the homogeneous p olynomial harmonic morphisms of degree During the preparation of this thesis a very interesting development has taken place with Abab ou Baird and Brossard writing the article proving that this is still a very active area of research In Chapter we discuss the W eierstrass representation of minimal surfaces as a motivating example and the history of general harmonic some years ago morphisms b eginning with Jacobi Chapters and are devoted to the intro duction of harmonic maps and harmonic morphisms where we also derive some of their basic prop erties In Chapter we then study p olynomial harmonic morphisms We show that every globally dened harmonic morphism b etween Eu clidean spaces of suciently high dimensions is necessarily p olynomial We give the complete classication due to Ou of those homogeneous of degree and discuss some examples of higher degree A general metho d for constructing nontrivial examples is provided and we make a conjecture on the structure of p olynomial harmonic morphisms based on known results on those of degree In Chapter we use the results of Chapter to give a new pro of of the ab ove mentioned result by Eells and Yiu regarding homogeneous p olynomial harmonic morphisms b etween Euclidean spheres Finally we show how the results derived so far can b e used to give information concerning the singularities of general harmonic morphisms has been my rm intention throughout this work to give references It to the stated results and credit to the work of others The only results I claim are mine wil l appear in chapter and and have been marked Any statement example or proof left unmarked is with an asterix considered to be too wel l known for a reference to be given 2 Acknowledgements I am grateful to all those who help ed me improve this work by use ful comments and suggestions in particular my sup ervisor Sigmundur Gudmundsson to whom I am also grateful for his patience and encour agement Martin Svensson 3 4 CHAPTER Intro duction Motivation The study of harmonic morphisms involves to a large extent the study of harmonicity and minimal submanifolds two concepts which themselves are strongly related In this section we illustrate this rela tionship with an example from classical dieren tial geometry relating 3 the mean curvature of a surface in R to the Laplacian of the co ordinate functions For details see 2 A regular parametrized surface is a C map 3 X U C R where U is op en and connected The map X is assumed to have injec 3 tive dierential so that along the image X U in R we have a normal vector eld X X u v N X X j j u v It is customary to use the following notation D E 2 2 X X X X E F G u u v v and D E D E D E 2 2 2 X X X e N f N g N 2 2 u u v v Then the mean curvature of the surface is dened as eG g E f F tracedN H 2 EG F and X is said to b e a parametrized minimal surface if the mean curva ture vanishes everywhere b e a relatively compact subset of the domain U and h Now let V 1 b e a C function on V Then X u v X u v thu v N u v t 5 is called the normal variation of X V determined by h The following very geometric result motivates the name minimal see do Carmo for a pro of 3 2 Theorem Let X U R be a regular parametrized C V surface Then X is minimal if and only if for every bounded V U and every normal variation X of X V we have t d AreaX V t dt t=0 2 3 It is well known that every regular C surface in R may lo cally b e parametrized by isothermal co ordinates ie co ordinates for which E G and F Let us therefore assume that X is isothermal Then we have for the Laplacian X of X D E D E D E 2 2 X X X X X X 2 2 u u u u v E D D E 2 2 X X X X 2 u u v u v 2 2 X X u u u v D E X and similarly X Hence X is normal to the surface v and e g hN X i H E E This implies the following 3 2 Theorem If X U R is a parametrized C surface with X X X X 2 2 j j j j and h i then X is minimal if and only if X is u v u v harmonic In the spirit of Theorem a minimal parametrized surface can b e dened as a map 3 X U C R satisfying D E 2 2 X X X X u v u v and X 6 If we in addition to this assume that U is simply connected it follows from elementary complex analysis that there exists a holomorphic map 3 U C such that X Re alent to That X is isothermal is then equiv 1 3 2 2 2 2 z z z Cho osing suitably leads us to the famous representation by Weier strass Theorem The Weierstrass Representation Let U be an 3 open simply connected subset of C and X U R a parametrized surface satisfying E D 2 2 X X X X u v u v and X Then there is exists pair of meromorphic functions f g in U such that 2 f and f g are holomorphic f g and Z z 2 2 X z X z Re f w g w i g w g w dw 0 z 0 for al l z U Conversely every pair f g of meromorphic functions 0 as above dene a minimal parametrized surface in this way In the next section we shall see how the results presented here demonstrate a certain duality b etween minimal conformal immersions and harmonic morphisms History The history of harmonic morphisms is generally thought to have b egun with the article of Jacobi from on the solutions of Laplaces equation in three dimensions Here Jacobi investigated nec dened on an op en essary conditions for a complex valued function 3 subset of R such that for any holomorphic function f the comp osition f is harmonic ie f A harmonic morphism though ought to b e a map that in some sense preserves a harmonic structure It was for that purp ose more than 7 a century after Jacobi that harmonic morphisms were formally intro duced by Constantinescu and Cornea in in the context of harmonic spaces in abstract p otential theory In general a harmonic space in the sense of Brelot see is a lo cally compact Hausdor space X endowed with a sheaf H assigning to each op en subset U of X a real subspace H U of the continuous functions on U such that the following conditions are satised X has an op en base for its top ology consisting of regular sets A regular set is an op en relatively compact subset V of X with nonempty b oundary V such that for every continuous function V f on V there is a unique element H H V which can b e f extended to V and equals f on V Furthermore if f then V H f If U X is op en and connected and fu g is an updirected A family in H U then either sup u is in H U or sup u A A For an op en subset U of X we call H U the harmonic functions on n U It is well known that R is a harmonic space with the harmonic functions as solutions to Laplaces equation More generally every Riemannian manifold is a harmonic space with the harmonic functions as zeros to the LaplaceBeltrami op erator These results are essentially e who showed see Chapter that the solutions due to R M Herv to a uniformly elliptic equation m m 2 X X f f b a cf ik i x x x i k i i=1 ik =1 m with co ecients a b and c lo cally Lipschitz in a domain R ik i denes a system satisfying the axioms of a harmonic space As dened by Constantinescu and Cornea in a harmonic mor phism is a continuous map X X b etween harmonic spaces X and X such that for every op en U X and harmonic function f on U the comp osition 1 U R f is harmonic Since every harmonic function on C is lo cally the real part of a holomorphic function we see that this is exactly what Jacobi was investigating The aim of Constantinescu and Cornea was to generalize results from the theory of Riemann surfaces to harmonic spaces with harmonic morphisms replacing the holomorphic maps 8 Some decade after Constantinescu and Corneas article Fuglede and Ishihara published indep endently their investigations on harmonic morphisms in Riemannian geometry see and Their results showed that in the sp ecial case when the harmonic spaces are Riemann ian manifolds the harmonic morphisms are rich in geometric features with several interesting applications and problems If we return to Jacobi for a while assume that C is a 3 harmonic morphism where R is op en By cho osing f w w for w C we see that is in fact smo oth Furthermore for a holomorphic
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