HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Quaternionic Holomorphic Geometry

Katrin Leschke

University of Leicester

”Submanifold Geometry and Lie Group Theory”29th October 2009

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Overview

1 Introduction

2 Complex holomorphic functions

3 Conformal maps

4 2–spheres and the conformal Gauss map

5 Holomorphic structure

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Overview

1 Introduction

2 Complex holomorphic functions

3 Conformal maps

4 2–spheres and the conformal Gauss map

5 Holomorphic structure

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Overview

1 Introduction

2 Complex holomorphic functions

3 Conformal maps

4 2–spheres and the conformal Gauss map

5 Holomorphic structure

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Overview

1 Introduction

2 Complex holomorphic functions

3 Conformal maps

4 2–spheres and the conformal Gauss map

5 Holomorphic structure

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Overview

1 Introduction

2 Complex holomorphic functions

3 Conformal maps

4 2–spheres and the conformal Gauss map

5 Holomorphic structure

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Surface theory

local results around the turn of the 19th century(Darboux, Bianchi, Blaschke)

renewed interest in global results in the second half of20th century

special surface classes given by curvature properties andvariational equations (e.g., minimal, constant meancurvature (CMC), and Willmore surfaces)

small catalogue of explicit examples: Gauss–Codazziequations are nonlinear, third order systems

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Surface theory

local results around the turn of the 19th century(Darboux, Bianchi, Blaschke)

renewed interest in global results in the second half of20th century

special surface classes given by curvature properties andvariational equations (e.g., minimal, constant meancurvature (CMC), and Willmore surfaces)

small catalogue of explicit examples: Gauss–Codazziequations are nonlinear, third order systems

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Surface theory

local results around the turn of the 19th century(Darboux, Bianchi, Blaschke)

renewed interest in global results in the second half of20th century

special surface classes given by curvature properties andvariational equations (e.g., minimal, constant meancurvature (CMC), and Willmore surfaces)

small catalogue of explicit examples: Gauss–Codazziequations are nonlinear, third order systems

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Surface theory

local results around the turn of the 19th century(Darboux, Bianchi, Blaschke)

renewed interest in global results in the second half of20th century

special surface classes given by curvature properties andvariational equations (e.g., minimal, constant meancurvature (CMC), and Willmore surfaces)

small catalogue of explicit examples: Gauss–Codazziequations are nonlinear, third order systems

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Delaunay surface

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Delaunay Bubbleton

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Delaunay Bubbleton

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Delaunay Bubbleton

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Willmore Cylinder

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Willmore Torus

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Castro Urbano Tori

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Castro Urbano Tori

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Complex Analysis

theory of complex curves

lots of nontrivial examples

powerful methods, global results

linear, first order Cauchy–Riemann equation: possible towrite down explicit formulae for meromorphic functions onany Riemann surface

Idea: Combine surface theory with complex analysis:

Quaternionic Holomorphic Geometry

References: F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4and Quaternions , Lecture Notes in Mathematics 1772, 86 pages, Springer, Berlin, Heidelberg, 2002.

K. Leschke, Transformations of Willmore surfaces, Habilitationschrift, 2006 (on my webpage, UoL)

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Complex Analysis

theory of complex curves

lots of nontrivial examples

powerful methods, global results

linear, first order Cauchy–Riemann equation: possible towrite down explicit formulae for meromorphic functions onany Riemann surface

Idea: Combine surface theory with complex analysis:

Quaternionic Holomorphic Geometry

References: F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4and Quaternions , Lecture Notes in Mathematics 1772, 86 pages, Springer, Berlin, Heidelberg, 2002.

K. Leschke, Transformations of Willmore surfaces, Habilitationschrift, 2006 (on my webpage, UoL)

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Complex Analysis

theory of complex curves

lots of nontrivial examples

powerful methods, global results

linear, first order Cauchy–Riemann equation: possible towrite down explicit formulae for meromorphic functions onany Riemann surface

Idea: Combine surface theory with complex analysis:

Quaternionic Holomorphic Geometry

References: F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4and Quaternions , Lecture Notes in Mathematics 1772, 86 pages, Springer, Berlin, Heidelberg, 2002.

K. Leschke, Transformations of Willmore surfaces, Habilitationschrift, 2006 (on my webpage, UoL)

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Complex Analysis

theory of complex curves

lots of nontrivial examples

powerful methods, global results

linear, first order Cauchy–Riemann equation: possible towrite down explicit formulae for meromorphic functions onany Riemann surface

Idea: Combine surface theory with complex analysis:

Quaternionic Holomorphic Geometry

References: F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4and Quaternions , Lecture Notes in Mathematics 1772, 86 pages, Springer, Berlin, Heidelberg, 2002.

K. Leschke, Transformations of Willmore surfaces, Habilitationschrift, 2006 (on my webpage, UoL)

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Complex Analysis

theory of complex curves

lots of nontrivial examples

powerful methods, global results

linear, first order Cauchy–Riemann equation: possible towrite down explicit formulae for meromorphic functions onany Riemann surface

Idea: Combine surface theory with complex analysis

:

Quaternionic Holomorphic Geometry

References: F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4and Quaternions , Lecture Notes in Mathematics 1772, 86 pages, Springer, Berlin, Heidelberg, 2002.

K. Leschke, Transformations of Willmore surfaces, Habilitationschrift, 2006 (on my webpage, UoL)

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Introduction: Complex Analysis

theory of complex curves

lots of nontrivial examples

powerful methods, global results

linear, first order Cauchy–Riemann equation: possible towrite down explicit formulae for meromorphic functions onany Riemann surface

Idea: Combine surface theory with complex analysis:

Quaternionic Holomorphic Geometry

References: F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4and Quaternions , Lecture Notes in Mathematics 1772, 86 pages, Springer, Berlin, Heidelberg, 2002.

K. Leschke, Transformations of Willmore surfaces, Habilitationschrift, 2006 (on my webpage, UoL)

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Complex holomorphic functions

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Conformal maps

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

2–spheres and complex structures

Katrin Leschke HSL tori

HSL tori

KatrinLeschke

Overview

Intro

C

H

S2

∂̄

Holomorphic structure

Katrin Leschke HSL tori

OverviewIntroductionComplex holomorphic functionsConformal maps2--spheres and the conformal Gauss mapHolomorphic structure