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.
Journal Article: On Chen's biharmonic conjecture for hypersurfaces in R5
Fu, Yu, Hong, Min-Chun and Zhan, Xin (2021). On Chen's biharmonic conjecture for hypersurfaces in R5. Advances in Mathematics, 383 107697, 1-28. doi: 10.1016/j.aim.2021.107697
Journal Article: The Oseen–Frank energy functional on manifolds
Hong, Min-Chun (2021). The Oseen–Frank energy functional on manifolds. Vietnam Journal of Mathematics, 49 (2), 597-613. doi: 10.1007/s10013-020-00468-2
Journal Article: Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system
Feng, Zhewen, Hong, Min-Chun and Mei, Yu (2020). Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system. SIAM Journal on Mathematical Analysis, 52 (1), 481-523. doi: 10.1137/18m1182887
Geometric evolution problems in nonlinear partial differential equations
(2015–2019) ARC Discovery Projects
Geometric partial differential systems and their applications
(2009–2011) ARC Discovery Projects
A new enabling technology for learning and teaching quantitative skills
(2006–2008) Carrick Competitive Grants
(2022) Doctor Philosophy
Finite time blowup of the n-harmonic flow
(2020) Doctor Philosophy
Mathematical modelling of fluid flow models
(2016) Doctor Philosophy
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.
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.
On Chen's biharmonic conjecture for hypersurfaces in R5
Fu, Yu, Hong, Min-Chun and Zhan, Xin (2021). On Chen's biharmonic conjecture for hypersurfaces in R5. Advances in Mathematics, 383 107697, 1-28. doi: 10.1016/j.aim.2021.107697
The Oseen–Frank energy functional on manifolds
Hong, Min-Chun (2021). The Oseen–Frank energy functional on manifolds. Vietnam Journal of Mathematics, 49 (2), 597-613. doi: 10.1007/s10013-020-00468-2
Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system
Feng, Zhewen, Hong, Min-Chun and Mei, Yu (2020). Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system. SIAM Journal on Mathematical Analysis, 52 (1), 481-523. doi: 10.1137/18m1182887
The energy identity for a sequence of Yang–Mills α -connections
Hong, Min-Chun and Schabrun, Lorenz (2019). The energy identity for a sequence of Yang–Mills α -connections. Calculus of Variations and Partial Differential Equations, 58 (3) 83. doi: 10.1007/s00526-019-1535-y
Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in Luloc3(R3)
Hong, Min-Chun and Mei, Yu (2019). Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in Luloc3(R3). Calculus of Variations and Partial Differential Equations, 58 (1) 3. doi: 10.1007/s00526-018-1453-4
Biharmonic hypersurfaces with constant scalar curvature in space forms
Fu, Yu and Hong, Min-Chun (2018). Biharmonic hypersurfaces with constant scalar curvature in space forms. Pacific Journal of Mathematics, 294 (2), 329-350. doi: 10.2140/pjm.2018.294.329
The rectified n-harmonic map flow with applications to homotopy classes
Hong, Min-Chun (2018). The rectified n-harmonic map flow with applications to homotopy classes. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 18 (4), 1249-1283. doi: 10.2422/2036-2145.201701_010
Finite time blowup of the n-harmonic flow on n-manifolds
Cheung, Leslie Hon-Nam and Hong, Min-Chun (2017). Finite time blowup of the n-harmonic flow on n-manifolds. Calculus of Variations and Partial Differential Equations, 57 (9) 9, 1-24. doi: 10.1007/s00526-017-1282-x
The Yang-Mills α-flow in vector bundles over four manifolds and its applications
Hong, Min-Chun, Tian, Gang and Yin, Hao (2015). The Yang-Mills α-flow in vector bundles over four manifolds and its applications. Commentarii Mathematici Helvetici, 90 (1), 75-120. doi: 10.4171/CMH/347
Blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3
Hong, Min-Chun, Li, Jinkai and Xin, Zhouping (2014). Blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3. Communications in Partial Differential Equations, 39 (7), 1284-1328. doi: 10.1080/03605302.2013.871026
Hong, Min-Chun (2014). Some results on harmonic maps. Bulletin of the Institute of Mathematics Academia Sinica New Series, 9 (2), 187-221.
On the sacks-uhlenbeck flow of Riemannian surfaces
Hong, Min-Chun and Yin, Hao (2013). On the sacks-uhlenbeck flow of Riemannian surfaces. Communications in Analysis and Geometry, 21 (5), 917-955. doi: 10.4310/CAG.2013.v21.n5.a3
Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2
Hong, Min-Chun and Xin, Zhouping (2012). Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2. Advances in Mathematics, 231 (3-4), 1364-1400. doi: 10.1016/j.aim.2012.06.009
Partial regularity of a minimizer of the relaxed energy for biharmonic maps
Hong, Min-Chun and Yin, Hao (2012). Partial regularity of a minimizer of the relaxed energy for biharmonic maps. Journal of Functional Analysis, 262 (2), 682-718. doi: 10.1016/j.jfa.2011.10.003
A new approximation of relaxed energies for harmonic maps and the Faddeev model
Giaquinta, Mariano, Hong, Min-Chun and Yin, Hao (2011). A new approximation of relaxed energies for harmonic maps and the Faddeev model. Calculus of Variations and Partial Differential Equations, 41 (1-2), 45-69. doi: 10.1007/s00526-010-0353-z
Global existence of solutions of the simplified Ericksen-Leslie system in dimension two
Hong, Min-Chun (2011). Global existence of solutions of the simplified Ericksen-Leslie system in dimension two. Calculus of Variations And Partial Differential Equations, 40 (1-2), 15-36. doi: 10.1007/s00526-010-0331-5
Global existence for the Seiberg–Witten flow
Hong, Min-Chun and Schabrun, Lorenz (2010). Global existence for the Seiberg–Witten flow. Communications In Analysis And Geometry, 18 (3), 433-473. doi: 10.4310/CAG.2010.v18.n3.a2
Curvature flow to the Nirenberg problem
Ma, Li and Hong, Min-Chun (2010). Curvature flow to the Nirenberg problem. Archiv der Mathematik, 94 (3), 277-289. doi: 10.1007/s00013-010-0101-9
The heat flow for H-systems on higher dimensional manifolds
Hong, Min-Chun and Hsu, Deliang (2010). The heat flow for H-systems on higher dimensional manifolds. Indiana University Mathematics Journal, 59 (3), 761-790. doi: 10.1512/iumj.2010.59.3917
Anti-self-dual connections and their related flow on 4-manifolds
Hong, M.-C. and Yu, Z. (2008). Anti-self-dual connections and their related flow on 4-manifolds. Calculus of Variations, 31 (3), 325-349. doi: 10.1007/s00526-007-0114-9
Partial regularity of weak solutions to Maxwell's equations in a quasi-static electromagnetic field
Hong, Min-Chun, Tonegawa, Yoshihiro and Yassin, Alzubaidi (2008). Partial regularity of weak solutions to Maxwell's equations in a quasi-static electromagnetic field. Methods and Applications of Analysis, 15 (2), 199-215.
Stability of the equator map for the hessian energy
Hong, M. C. and Thompson, B. (2007). Stability of the equator map for the hessian energy. Proceedings of the American Mathematical Society, 135 (10), 3163-3170. doi: 10.1090/S0002-9939-07-08950-2
Hong, M. C. (2007). Existence of infinitely many equilibrium configurations of a liquid crystal system prescribing the same nonconstant boundary value. Pacific Journal of Mathematics, 232 (1), 177-206. doi: 10.2140/pjm.2007.232.177
Partial regularity of stable p-harmonic maps into spheres
Hong, M. C. (2007). Partial regularity of stable p-harmonic maps into spheres. The Bulletin of the Australian Mathematical Society, 76 (2), 297-305. doi: 10.1017/S0004972700039678
Regularity and relaxed problems of minimizing biharmonic maps into spheres
Hong, Min-Chun and Wang, Changyou (2005). Regularity and relaxed problems of minimizing biharmonic maps into spheres. Calculus of Variations and Partial Differential Equations, 23 (4), 425-450. doi: 10.1007/s00526-004-0309-2
Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections
Hong, Min-Chun and Tian, Gang (2004). Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections. Mathematische Annalen, 330 (3), 441-472. doi: 10.1007/s00208-004-0539-9
Global existence of the m-equivariant Yang-Mills flow in four dimensional spaces
Hong, Min-Chun and Tian, Gang (2004). Global existence of the m-equivariant Yang-Mills flow in four dimensional spaces. Communications In Analysis And Geometry, 12 (1), 183-211. doi: 10.4310/CAG.2004.v12.n1.a10
Partial regularity of minimizers of a functional involving forms and maps
Giaquinta, M and Hong, MC (2004). Partial regularity of minimizers of a functional involving forms and maps. Nodea-nonlinear Differential Equations And Applications, 11 (4), 469-490. doi: 10.1007/s00030-0004-2015-3
Partial regularity of weak solutions of the liquid crystal equilibrium system
Hong, Min-Chun (2004). Partial regularity of weak solutions of the liquid crystal equilibrium system. Indiana University Mathematics Journal, 53 (5), 1401-1414. doi: 10.1512/iumj.2004.53.2459
Minimizing harmonic maps into ellipsoids and harmonic diffeomorphisms
Hong, MC (2002). Minimizing harmonic maps into ellipsoids and harmonic diffeomorphisms. Mathematische Zeitschrift, 241 (2), 313-327. doi: 10.1007/s002090200416
On the minimality of the p-harmonic map x/|x| : Bn → Sn-1
Hong, MC (2001). On the minimality of the p-harmonic map x/|x| : Bn → Sn-1. Calculus of Variations And Partial Differential Equations, 13 (4), 459-468.
Heat flow for Yang-Mills-Higgs fields, part II
Yi,Fang and Hong, MC (2001). Heat flow for Yang-Mills-Higgs fields, part II. Chinese Annals of Mathematics Series B, 22 (2), 211-222. doi: 10.1142/S0252959901000206
Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric
Hong, MC (2001). Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric. Annals of Global Analysis And Geometry, 20 (1), 23-46. doi: 10.1023/A:1010688223177
Heat flow for Yang-Mills-Higgs fields, part I
Yi, Fang and Hong, MC (2000). Heat flow for Yang-Mills-Higgs fields, part I. Chinese Annals of Mathematics Series B, 21 (4), 453-472. doi: 10.1142/S0252959900000455
On the Jager-Kaul theorem concerning harmonic maps
Hong, MC (2000). On the Jager-Kaul theorem concerning harmonic maps. Annales De L Institut Henri Poincare-analyse Non Lineaire, 17 (1), 35-46. doi: 10.1016/S0294-1449(99)00103-1
On the singular set of stable-stationary harmonic maps
Hong, MC and Wang, CY (1999). On the singular set of stable-stationary harmonic maps. Calculus of Variations And Partial Differential Equations, 9 (2), 141-156. doi: 10.1007/s005260050135
On the hausdorff dimension of the singular set of stable-stationary harmonic maps
Hong, MC (1999). On the hausdorff dimension of the singular set of stable-stationary harmonic maps. Communications In Partial Differential Equations, 24 (11-12), 1967-1985. doi: 10.1080/03605309908821490
Some new examples for nonuniqueness of the evolution problem of harmonic maps
Hong, MC (1998). Some new examples for nonuniqueness of the evolution problem of harmonic maps. Communications In Analysis And Geometry, 6 (4), 809-818. doi: 10.4310/CAG.1998.v6.n4.a7
Asymptotic limits of a self-dual Ginzburg-Landau functional in bounded domains
Hong, MC (1998). Asymptotic limits of a self-dual Ginzburg-Landau functional in bounded domains. Manuscripta Mathematica, 97 (2), 251-267. doi: 10.1007/s002290050100
Two estimates concerning asymptotics of the minimizations of a Ginzburg-Landau functional
Hong, MC (1997). Two estimates concerning asymptotics of the minimizations of a Ginzburg-Landau functional. Journal of The Australian Mathematical Society Series A-pure Mathematics And Statistics, 62 (1), 128-140. doi: 10.1017/S1446788700000598
Hong M.-C. (1996). Asymptotic behavior for minimizers of a Ginzburg-Landau-type functional in higher dimensions associated with n-harmonic maps. Advances in Differential Equations, 1 (4), 611-634.
Multiple Solutions of the Static Landau-Lifshitz Equation From B-2 Into S-2
Hong, MC and Lemaire, L (1995). Multiple Solutions of the Static Landau-Lifshitz Equation From B-2 Into S-2. Mathematische Zeitschrift, 220 (2), 295-306. doi: 10.1007/BF02572616
Hong, MC (1995). The Landau-Lifshitz Equation with the External-Field - a New Extension for Harmonic Maps with Values in S-2. Mathematische Zeitschrift, 220 (2), 171-188. doi: 10.1007/BF02572608
On a Problem of Bethuel, Brezis and Helein Concerning the Ginzburg-Landau Functional
Hong, MC (1995). On a Problem of Bethuel, Brezis and Helein Concerning the Ginzburg-Landau Functional. Comptes Rendus De L Academie Des Sciences Serie I-Mathematique, 320 (6), 679-684.
Heat-Flow of P-Harmonic Maps with Values Into Spheres
Chen, YM, Hong, MC and Hungerbuhler, N (1994). Heat-Flow of P-Harmonic Maps with Values Into Spheres. Mathematische Zeitschrift, 215 (1), 25-35. doi: 10.1007/BF02571698
The Landau-Lifshitz Equation of the Ferromagnetic Spin Chain and Harmonic Maps
Guo, BL and Hong, MC (1993). The Landau-Lifshitz Equation of the Ferromagnetic Spin Chain and Harmonic Maps. Calculus of Variations and Partial Differential Equations, 1 (3), 311-334. doi: 10.1007/BF01191298
Liouville Theorems for Exponentially Harmonic-Functions On Riemannian-Manifolds
Hong, MC (1992). Liouville Theorems for Exponentially Harmonic-Functions On Riemannian-Manifolds. Manuscripta Mathematica, 77 (1), 41-46. doi: 10.1007/BF02567042
On the Conformal Equivalence of Harmonic Maps and Exponentially Harmonic Maps
Hong, MC (1992). On the Conformal Equivalence of Harmonic Maps and Exponentially Harmonic Maps. Bulletin of the London Mathematical Society, 24 (5), 488-492. doi: 10.1112/blms/24.5.488
The Equator Map and the Negative Exponential Functional
Hong, MC (1992). The Equator Map and the Negative Exponential Functional. Manuscripta Mathematica, 75 (1), 49-63. doi: 10.1007/BF02567071
Some Remarks On the Minimizers of Variational Integrals with Nonstandard Growth-Conditions
Hong, MC (1992). Some Remarks On the Minimizers of Variational Integrals with Nonstandard Growth-Conditions. Bollettino Della Unione Matematica Italiana, 6A (1), 91-101.
Partial Regularities of Minimizers of Certain Quadratic Functionals with Unbounded Obstacles
Hong, MC (1992). Partial Regularities of Minimizers of Certain Quadratic Functionals with Unbounded Obstacles. Annali Di Matematica Pura Ed Applicata, 161 (1), 113-138. doi: 10.1007/BF01759634
Existence and Partial Regularity in the Calculus of Variations
Hong, MC (1987). Existence and Partial Regularity in the Calculus of Variations. Annali Di Matematica Pura Ed Applicata, 149 (1), 311-328. doi: 10.1007/BF01773940
Geometric evolution problems in nonlinear partial differential equations
(2015–2019) ARC Discovery Projects
Geometric partial differential systems and their applications
(2009–2011) ARC Discovery Projects
A new enabling technology for learning and teaching quantitative skills
(2006–2008) Carrick Competitive Grants
Variational methods in partial differential equations
(2006–2008) ARC Linkage International
Geometric variational problems and nonlinear partial differential systems
(2004–2006) ARC Discovery Projects
Analytic Problems of Bi-harmonic Maps
(2004–2005) UQ New Staff Research Start-Up Fund
(2022) Doctor Philosophy — Principal Advisor
Other advisors:
Finite time blowup of the n-harmonic flow
(2020) Doctor Philosophy — Principal Advisor
Mathematical modelling of fluid flow models
(2016) Doctor Philosophy — Principal Advisor
(2011) Doctor Philosophy — Principal Advisor
Other advisors:
(2010) Doctor Philosophy — Principal Advisor
Other advisors:
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.