In their celebrated paper (Ann Math 1981), Sacks and Uhlenbeck introduced a family of the perturbed functional, which is now called the Sacks and Uhlenbeck functional (or alpha functional).
In this talk, I will give some survey about the Sacks and Uhlenbeck program on harmonic maps and its applications to Yang-Mills equations in vector bundle over 4- manifolds. In particular, we discuss some results on Yang-Mills alpha flow and the energy identity of Yang-Mills alpha connections in 4 dimensions.
Dr Min-Chun Hong is well known by his works on harmonic maps, liquid crystals and Yang Mills equations in the areas of nonlinear partial differential equations and geometric analysis, by solving a number of open problems and conjectures on these fields. With his collaborators, he developed a new approximation of the Dirichlet energy, yielding a new proof on partial regularity of minimizers of the relax energy for harmonic maps as well as for the Faddeev model. The method leads to solve an open problem on partial regularity in the relax energy of biharmonic maps. They introduced the Sack-Uhlenbeck flow to prove new existence results of the harmonic map flow in 2D and made new application to homotopy classes. They established asymptotic behaviour of the Yang-Mills flow to prove the existence of singular Hermitian-Yang-Mills connections, which was used to settle a well-known conjecture of Bando and Siu. In the area of liquid crystals, he resolved a long-standing open problem on the global existence of the simplified Ericksen-Leslie system in 2D. With his collaborators, they solved the global existence problem on the Ericksen-Leslie system with unequal Frank constants in 2D and resolved a problem on converging of the approximate Ericksen- Leslie system in 3D.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai