Harmonic Maps Between Surfaces and Teichmüller Theory - Lecture 1

Explore harmonic maps between surfaces and Teichmüller theory in this lecture by Michael Wolf, part of the Geometry, Groups and Dynamics program at the International Centre for Theoretical Sciences. Delve into the fundamentals of harmonic maps, energy functionals, and their applications in surface geometry. Examine the Hopf differential and its relationship to harmonic maps, and gain insights into the structure of Teichmüller space. Learn how these concepts connect to partial differential equations and geometric analysis. Engage with examples, remarks, and a comprehensive overview of the topic, concluding with a perspective on Teichmüller space and a Q&A session.

Start
Harmonic Maps between surfaces and
Outline
Harmonic maps
A map u0 which is critical for the energy functional Eu
Example
Equation
Example
Remark
How to think about harmonicity?
Applications
Application of chain rule
Example: if M is a surface, N=R
Conclusion
II. Harmonic maps between surfaces
How to compare
Hopf differential
How does H relate to ϕ?
Remarks: This is a friendly PDE
What can we learn about 0H?
Note
Picture of Teichmuller space Teich S
From the perspective of teichmuller space
Q&A