Conformally Invariant Variational Problems

TL;DR

This paper explores conformally invariant variational problems in analysis, focusing on nonlinear PDEs from conformally invariant 2D variational problems like harmonic maps and Willmore surfaces, highlighting their significance in geometry and physics.

Contribution

It provides an in-depth study of nonlinear PDEs arising from conformally invariant variational problems, connecting geometric analysis with physical theories.

Findings

Analysis of conformally invariant PDEs in 2D

Connections between geometry and physics in conformal invariance

Lecture notes from international courses on the topic

Abstract

Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant two dimensional variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Willmore and Constrained conformal surfaces, isothermic surfaces). The present manuscript are lecture notes of courses given by the author at several places including UBC Vancouver, SNS Pisa, IHP Paris, ICTP Trieste.