# Associate Professor Min-Chun Hong

## Personal page

Associate Professor Min-Chun Hong's personal page

## Teaching and learning

Assoc Professor Hong teaches 3rd and 4th year mathematics courses including optimisation, functional analaysis and partial differential equations.

## Researcher biography

Dr Min-Chun Hong has solved a number of open problems and conjectures on harmonic maps, liquid crystals and Yang Mills equations in the areas of nonlinear partial differential equations and geometric analysis. He has collaborated with top mathematicians such as Professor Mariano Giaquinta (SNS-Pisa), Professor Jurgen Jost (Germany), Professor Michael Struwe (Zurich), Professor Gang Tian (Princeton) and Professor Zhouping Xin (Hong Kong).

Some highlights of his research after joining UQ in 2004 are:

In the area of harmonic maps, collaborated with Giaquinta and Yin (Calc. Var. PDEs 2011), 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 by him and Hao Yin (J. Funct. Anal. 2012). Based on the well-known result of Sack and Uhlenbeck in 1981 (Uhlenbeck 2019 Abel Award Winner), with collaboration of Hao Yin in 2013, he introduced the Sack-Uhlenbeck flow to prove new existence results of the harmonic map flow in 2D and made new application to homotopy classes.

Collaborated with his PhD student L. Cheng (Calc. Var. PDEs 2018), he settled a conjecture of Hungerbuhler on the n-harmonic map flow.

Bang-Yen Chen in 1991 proposed a well-known conjecture on biharmonic submanifolds: Any biharmonic submanifold in the Euclidean space is minimal. Collaborated with Fu and Zhan (Adv. Math 2021), he confirmed Chen's conjecture for hypersurfaces in R5 with n=4.

In the area of Yang-Mills equations, with Gang Tian (Math. Ann. 2004), he 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. Collaborated with Tian and Yin (Commun. Math. Helv. 2015), he extended the Sack-Uhlenbeck program to Yang-Mills equations and introduced the Yang-Mills alpha-flow to approximate the Yang-Mills flow in 4D. More recently, collaborated with his PhD student Schabrun (Calc. Var. PDEs 2019), he proved the energy identity for a sequence of Yang-Mills α-connections.

In the area of liquid crystals, he (Calc. Var. PDEs 2011) resolved a long-standing open problem on the global existence of the simplified Ericksen-Leslie system in 2D. Collaborated with Zhouping Xin (Adv. Math. 2012), he solved the global existence problem on the Ericksen-Leslie system with unequal Frank constants in 2D. Collaborated with Li and Xin (CPDE 2014), he resolved a problem on converging of the approximate Ericksen-Leslie system in 3D.