On the existence of complex-valued harmonic morphisms

Jonas Nordström

On the existence of complex-valued harmonic morphisms

Jonas Nordström

Forskningsoutput: Avhandling › Doktorsavhandling (sammanläggning)

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Popular Abstract in Swedish
En harmonisk morfi är en avbildning som bevarar harmoniska funktioner. De studerades först av Jacobi år 1848, han studerade dem i syfte att hitta harmoniska avbildningar från rummet till planet.

Man kan visa att harmoniska morfier är precis de avbildingar som bevarar Brownska rörelser. En Brownsk rörelse är en typ av slumpvandring där riktingen i varje steg är likafördelad och steglängden är normalfördelad.

Fuglede och Ishihara visade att en harmonisk morfi måste vara både en harmonisk avbildning och en konform avbildning, detta är för många vilkor för att garantera existens.

Om vi går från två dimensioner till två dimensioner så är harmoniska morfier precis de vinkelbevarande avbildningarna. Mellan två mångfalder av samma dimension som är högre än två är det precis de avståndsbevarande avbildningarna.

Avhandlingen behandlar existens och icke-existens resultat för harmoniska morfier. Artikel 1 och 3 handlar om existens medans artikel 2 och 4 handlar om icke-existens.

I artikel 1 går vi igenom en metod för att producera harmoniska morfier på lösbara Lie grupper. Metoden fungerar utmärkt på nästan alla homogena Hadamardrum.

Detta för oss till artikel 2 som visar att det inte finns vänster-invarianta harmoniska morfier på några av de homogena Hadamardrum där metoden från artikel 1 ej fungerade.

I artikel 3 visar vi att det går att konstruera egenfamiljer på G2. Vi använder dessa egenfamiljer för att konstruera harmoniska morfier på öppna delmängder av G2.

I den sista artikeln visar vi att harmoniska morfier med totalt geodetiska fiber måste uppfylla ett starkt geometriskt vilkor. På Einstein-mångfalder får vi ett nödvänligt vilkor som är enkelt att använda för att motbevisa existens.

Originalspråk	engelska
Kvalifikation	Doktor
Tilldelande institution	Matematik (naturvetenskapliga fakulteten)
Handledare	Gudmundsson, Sigmundur, handledare
Tilldelningsdatum	2015 sep. 9
Förlag	Centre for Mathematical Sciences, Lund University
ISBN (tryckt)	978-91-7623-291-0
Status	Published - 2015

Bibliografisk information

Defence details

Date: 2015-09-09
Time: 13:00
Place: Sölvegatan 18, Lund, sal MH:C

External reviewer(s)

Name: Montaldo, Stefano
Title: Dr.
Affiliation: University of Cagliari

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Matematik

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abstract = "This thesis consists of 4 papers, their content is described below: Paper I. We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms. Paper II. We study left-invariant complex-valued harmonic morphisms from Riemannian Lie groups. We show that in each dimension greater than $3$ there exist Riemannian Lie groups that do not have any such solutions. Paper III. We construct harmonic morphisms on the compact simple Lie group $G_{2}$ using eigenfamilies. The construction of eigenfamilies uses a representation theory scheme and the seven-dimensional cross product. Paper IV. We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the existence of complex-valued harmonic morphisms with totally geodesic fibers on Einstein manifolds.",

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N2 - This thesis consists of 4 papers, their content is described below: Paper I. We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms. Paper II. We study left-invariant complex-valued harmonic morphisms from Riemannian Lie groups. We show that in each dimension greater than $3$ there exist Riemannian Lie groups that do not have any such solutions. Paper III. We construct harmonic morphisms on the compact simple Lie group $G_{2}$ using eigenfamilies. The construction of eigenfamilies uses a representation theory scheme and the seven-dimensional cross product. Paper IV. We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the existence of complex-valued harmonic morphisms with totally geodesic fibers on Einstein manifolds.

AB - This thesis consists of 4 papers, their content is described below: Paper I. We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms. Paper II. We study left-invariant complex-valued harmonic morphisms from Riemannian Lie groups. We show that in each dimension greater than $3$ there exist Riemannian Lie groups that do not have any such solutions. Paper III. We construct harmonic morphisms on the compact simple Lie group $G_{2}$ using eigenfamilies. The construction of eigenfamilies uses a representation theory scheme and the seven-dimensional cross product. Paper IV. We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the existence of complex-valued harmonic morphisms with totally geodesic fibers on Einstein manifolds.

KW - Harmonic morphisms

KW - foliations

KW - minimal submanifolds

KW - Lie groups

M3 - Doctoral Thesis (compilation)

SN - 978-91-7623-291-0

T3 - Doctoral Theses in Mathematical Sciences

PB - Centre for Mathematical Sciences, Lund University

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On the existence of complex-valued harmonic morphisms

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