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Associate Professor Min-Chun Hong

Associate Professor

School of Mathematics and Physics
Faculty of Science
hong@maths.uq.edu.au
+61 7 334 69036

Overview

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.

Publications

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Grants

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Supervision

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Available Projects

  • Variational and evolution problems in liquid crystals

    Pierre-Gilles de Gennes was awarded the Nobel prize in physics in 1991 for his discovery of the Landau-de Gennes theory in liquid crystals and polymers. The Landau-de Gennes theory generalises the well-known Oseen-Frank theory on nematic liquid crystals and is one of the successful theories in modeling both uniaxial and biaxial nematic liquid crystals. In mathematics, the Landau-de Gennes theory is described as a variational problem of minimising the Landau-de Gennes energy functional. in this project, we aim to give a rigorous proof to verify that the biaxial Q -tensor Landau-de Gennes system can approach the uniaxial Q-tensor Oseen-Frank system.

  • Smooth approximations of the Yang-Mills flow in higher dimensional manifolds

    The Yang-Mills flow, introduced by Atiyah-Bott, has played an important role in Yang-Mills theory for classifying vector bundles over manifolds. In holomorphic vector bundles over K\"ahler manifolds, Donaldson used the Yang-Mills flow to establish that an irreducible holomorphic vector bundle on K\"ahler manifolds admits a unique Hermitian-Einstein connection if and only if the vector bundle is stable. In unstable holomorphic vector bundles over compact K\"ahler manifolds, Bando-Siu conjectured an equivalent relation between the Harder-Narashimhan filtration of holomorphic vector bundles on K\"ahler manifolds and the limiting bundle of the Yang-Mills flow. In collaboration with Gang Tian from Princeton, we established asymptotic behaviour of the Yang-Mills flow. As applications of our work, Sibley and Jacob to settle the conjecture of Bando and Siu in holomorphic vector bundles over K\"ahler manifolds. In this project, we aim to introduce a smooth Yang-Mills approximation flow in higher dimensions to establish a realted Bando-Siu conjecture.

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Publications

Book Chapter

  • Some analytic aspects of liquid crystal configurations

    Hong, Min-Chun (2010). Some analytic aspects of liquid crystal configurations. Trends in partial differential equations. (pp. 193-211) edited by Baojun Bian, Shenghong Li and Xu-Jia Wang. Beijing-Boston: Higher Education Press and International Press.

  • Asymptotic limits of a Ginzburg-Landau type functional

    Hong, Min-Chun , Jost, Jurgen and Struwe, Michael (1996). Asymptotic limits of a Ginzburg-Landau type functional. Geometric analysis and the calculus of variations. (pp. 99-124) edited by Jürgen Jost. Boston, MA, United States: International Press.

Journal Article

Grants (Administered at UQ)

PhD and MPhil Supervision

Current Supervision

  • Convergence and its applications of the Ginzburg-Landau approximation for a generalized Ericksen-Leslie system

    Doctor Philosophy — Principal Advisor

    Other advisors:

Completed Supervision

Possible Research Projects

Note for students: The possible research projects listed on this page may not be comprehensive or up to date. Always feel free to contact the staff for more information, and also with your own research ideas.

  • Variational and evolution problems in liquid crystals

    Pierre-Gilles de Gennes was awarded the Nobel prize in physics in 1991 for his discovery of the Landau-de Gennes theory in liquid crystals and polymers. The Landau-de Gennes theory generalises the well-known Oseen-Frank theory on nematic liquid crystals and is one of the successful theories in modeling both uniaxial and biaxial nematic liquid crystals. In mathematics, the Landau-de Gennes theory is described as a variational problem of minimising the Landau-de Gennes energy functional. in this project, we aim to give a rigorous proof to verify that the biaxial Q -tensor Landau-de Gennes system can approach the uniaxial Q-tensor Oseen-Frank system.

  • Smooth approximations of the Yang-Mills flow in higher dimensional manifolds

    The Yang-Mills flow, introduced by Atiyah-Bott, has played an important role in Yang-Mills theory for classifying vector bundles over manifolds. In holomorphic vector bundles over K\"ahler manifolds, Donaldson used the Yang-Mills flow to establish that an irreducible holomorphic vector bundle on K\"ahler manifolds admits a unique Hermitian-Einstein connection if and only if the vector bundle is stable. In unstable holomorphic vector bundles over compact K\"ahler manifolds, Bando-Siu conjectured an equivalent relation between the Harder-Narashimhan filtration of holomorphic vector bundles on K\"ahler manifolds and the limiting bundle of the Yang-Mills flow. In collaboration with Gang Tian from Princeton, we established asymptotic behaviour of the Yang-Mills flow. As applications of our work, Sibley and Jacob to settle the conjecture of Bando and Siu in holomorphic vector bundles over K\"ahler manifolds. In this project, we aim to introduce a smooth Yang-Mills approximation flow in higher dimensions to establish a realted Bando-Siu conjecture.

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ResearcherID: E-7095-2010

ORCID: 0000-0003-0607-7562

Scopus ID: 7402688135

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