Skip to main content

Wheeler, Glen E. Dr

Senior Lecturer

  • Faculty of Engineering and Information Sciences
  • School of Mathematics and Applied Statistics
  • Institute for Mathematics and its Applications
  • Senior Lecturer - University of Wollongong (Organization) 2019 -
  • Lecturer - UOW 2015 - 2018
  • Associate Lecturer - UOW 2014
  • Research Associate - UOW 2012 - 2013
  • Research Fellow at OvGU Magdeburg - Alexander-von-Humboldt Stiftung 2010 - 2012
  • Research Associate - UOW 2010
  • Fellow at Freie Universitaet Berlin - DAAD 2009

Overview


I am a geometric analyst.  I am interested in partial differential equations and differential geometry, in particular geometric evolution equations, such as surface diffusion flow, Helfrich flow, Willmore flow, and mean curvature flow.

Subfields that I have work in include


 - Calculus of variations and the study of functionals of submanifolds (such as the area functional, Willmore functional, Helfrich functional, ...)

 - Local and global differential geometry

 - Higher order elliptic and parabolic partial differential equations on manifolds

Top Publications


Research Overview


  • My research lies in the intersection of differential geometry and partial differential equations (PDE). I have a particular focus on nonlinear higher-order geometric PDE, in particular the class of equations known as curvature flow.

    Curvature flow are evolution equations that describe how a geometric object (we call this a manifold) changes over time. The velocity of the flow is given by internal and external forces, usually arising from physical considerations. This makes curvature flow a natural fit for applications. In recent times, curvature flow have also proved to be powerful tools for the solution of long-standing problems in mathematics. For instance, the Poincare Conjecture was solved using Ricci flow, a flow of Riemannian metrics. There are many such applications, for instance of extrinsic curvature flow to geometric inequalities, such as the isoperimetric inequality. This is a relationship between area and length; in particular, it quantifies the idea that as you pack in more volume with fixed area, the shape of the region inside becomes round. We see this for instance when blowing up a balloon.

    If my work is to have a single unifying characteristic, it is the specific study and consideration of difficult (but well-motivated) flows that require the development of new tools to treat. This is why much of my research concerns flows that are higher-order or have boundary conditions. Both of these cases prevent application of standard theory and techniques, but are absolutely critical from an applications perspective.

    Here are some specific developments:

    • In a series of five papers I developed a monotonicity at low associated energies method for the (constrained and unconstrained) surface diffusion flow. This is now a standard technique in the area, and has been one of my most important contributions to the field of higher-order curvature flow.
    • Surface diffusion flow is fourth-order; the method has been extended to the case of sixth-order, and recently arbitrarily high order for curves. This is the first time global results for such flows have been obtained.
    • I considered, together with V.-M. Wheeler (UOW), the mean curvature flow with mixed boundary conditions for the first time: a new geometry where one component is fixed and another free. We give partial preservation of graphicality, singularity criteria and stability analysis.
    • I introduced the normalised oscillation of curvature for curve diffusion flow, which is later used in both my work and the work of others to study rescaling limits for the curve diffusion flow and reformulate Giga's conjecture.
    • In a series of papers I established general rigidity statements (this relies on a new technique involving test functions) for a variety of functionals and higher-order operators.
    • I proposed an elementary cut-off function technique that implies a local decay estimate can give new information on the Chen conjecture. Note: This idea was discovered independently by Nakauchi-Urakawa-Gudmundsson. Their article appeared after mine, and although the B.-Y. Chen of the Chen conjecture cites my article together with theirs, many subsequent works using this same idea only cite Nakauchi-Urakawa-Gudmundsson. Nevertheless the fact remains that this idea has been fundamental to much of the new exciting progress on Chen's conjecture.
    • Together with A. Dall'Acqua and K. Deckelnick, we solved an open question of Grunau and settled the existence of unstable Willmore surfaces. We did this by developing several novel new observations, including that unstable surfaces of revolution have an associated derived function that solves a Dirichlet problem.
    • Together with B. Andrews, J. McCoy, V.-M. Wheeler and A. Holder we tackled for the first time curvature flow that are not only fully nonlinear, but whose speed is a non- smooth function of the Weingarten map. To handle this we mollified the speed of the flow (and not the flow itself), and showed how uniform estimates independent of the mollification parameter can be derived. A limiting process allows us to solve the original problem. Convergence is extremely delicate, and required geometrisation of a weak Harnack inequality combined with the strong maximum principle on subintervals.
    • Together with J. McCoy, M. Edwards, A. Gerhard-Bourke and V.-M. Wheeler we classified solitons for the curve diffusion flow, and showed that the Lemniscate of Bernoulli is a shrinker. This allows us to settle an old question of Polden.
    • I studied the evolution of space curves evolving by the L^2-gradient of a functional incorporating curvature and length, with anisotropy. This is interesting from an applied perspective, since the Helfrich functional is of this form. Several new techniques needed to be developed, notably convergence to an equilibrium required a new application of the Llojasiewicz-Simon gradient inequality. This idea resulted from discussions with Ben Andrews. It proved to be fundamental and now many papers in the field are using the same method.
    • Together with J. McCoy we approached the problem of finite-time singularities for a constrained Willmore functional. This allowed us to demonstrate for the first time the existence of finite-time singularities for these flows. Previously, either singularities were known in infinite time only or for very special highly singular flows. Unfortunately the family does not include the Willmore flow (or the surface diffusion flow), and existence of finite-time singularities for these flows remains an important open question.
    • Together with M. Simon we proved a new uniqueness criterion for the biharmonic heat equation on n-dimensional space that uses a new geometric blowup and decay argument. The method has been developed to be particularly relevant to fourth and higher order curvature flow.
    • Together with H. Lee and G. Drugan I reduced the existence problem for solitons of the inverse mean curvature flow to an ODE. By proving various qualitative properties of the ODE we found a semi-explicit expression for the desired solitons. This is a completely new family of solutions to the flow, hinting at a much deeper and more extensive structure than had been previously known. We also give some rigidity results.
    • Together with Y. Bernard and V.-M. Wheeler I considered problems directly motivated by the blood disease spherocytosis. We take the Helfrich model and ask ourselves: for which parameter values does one see a sphere? For which parameter values are spheres `stable'? We give various answers to this question, using new techniques in the variational theory for higher-order functionals. One of the theorems generalises a well-known result of myself and J. McCoy.
    • Together with J. McCoy, J. Sharples, I. Towers and V.-M. Wheeler I produced a series of refereed conference papers that explore various applications of curvature flow to the modeling of fire fronts, especially those that arise in bushfires.
    • B, Lamichhane, J. Droniou, M. Ilyas and myself studied numerically a PDE of sixth order with various boundary conditions, with a view to eventually considering numerically a sixth-order curvature flow I introduced earlier.
    • I considered a variety of higher-order flows with boundary conditions. This is a collection of papers joint with J. McCoy and Y. Wu. We give rigidity results as well as global analysis.
    • In the elliptic setting, together with V.-M. Wheeler I considered a rigidity question for minimal hypersurfaces. When this article was placed on the global e-print archive as a preprint, it generated prompted almost immediately e-mails from eminent researchers at ETH Z\"urich and Stanford University to congratulate us on our work.
    • Together with Y. Bernard and V.-M. Wheeler we have given the first systematic analysis of Chen's flow, a higher-order curvature flow tailor-made to give new insight on Chen's conjecture. This is followed up by further work joint with M. Cooper that includes numerical work and results in the low-dimensional case.

Available as Research Supervisor

Selected Publications


Impact Story


Available as Research Supervisor

Potential Supervision Topics


  • Analysis of higher order PDE
  • Curvature Flow
  • Differential geometry
  • Rigidity for PDE and submanifolds

Advisees


  • Graduate Advising Relationship

    Degree Research Title Advisee
    Doctor of Philosophy Non-Compact Higher-Order Curvature Flow
    Doctor of Philosophy Gradient Flow of the Dirichlet Energy for the Mean Curvature of Closed Surfaces Fang, Yanqin
    Doctor of Philosophy Gradient Flow of the Dirichlet Energy for the Curvature of Plane Curves Wu, Yuhan

Keywords


  • Analysis
  • Mathematics
  • Pure Mathematics

Web Of Science Researcher Id


  • G-1735-2014

Top Publications


Research Overview


  • My research lies in the intersection of differential geometry and partial differential equations (PDE). I have a particular focus on nonlinear higher-order geometric PDE, in particular the class of equations known as curvature flow.

    Curvature flow are evolution equations that describe how a geometric object (we call this a manifold) changes over time. The velocity of the flow is given by internal and external forces, usually arising from physical considerations. This makes curvature flow a natural fit for applications. In recent times, curvature flow have also proved to be powerful tools for the solution of long-standing problems in mathematics. For instance, the Poincare Conjecture was solved using Ricci flow, a flow of Riemannian metrics. There are many such applications, for instance of extrinsic curvature flow to geometric inequalities, such as the isoperimetric inequality. This is a relationship between area and length; in particular, it quantifies the idea that as you pack in more volume with fixed area, the shape of the region inside becomes round. We see this for instance when blowing up a balloon.

    If my work is to have a single unifying characteristic, it is the specific study and consideration of difficult (but well-motivated) flows that require the development of new tools to treat. This is why much of my research concerns flows that are higher-order or have boundary conditions. Both of these cases prevent application of standard theory and techniques, but are absolutely critical from an applications perspective.

    Here are some specific developments:

    • In a series of five papers I developed a monotonicity at low associated energies method for the (constrained and unconstrained) surface diffusion flow. This is now a standard technique in the area, and has been one of my most important contributions to the field of higher-order curvature flow.
    • Surface diffusion flow is fourth-order; the method has been extended to the case of sixth-order, and recently arbitrarily high order for curves. This is the first time global results for such flows have been obtained.
    • I considered, together with V.-M. Wheeler (UOW), the mean curvature flow with mixed boundary conditions for the first time: a new geometry where one component is fixed and another free. We give partial preservation of graphicality, singularity criteria and stability analysis.
    • I introduced the normalised oscillation of curvature for curve diffusion flow, which is later used in both my work and the work of others to study rescaling limits for the curve diffusion flow and reformulate Giga's conjecture.
    • In a series of papers I established general rigidity statements (this relies on a new technique involving test functions) for a variety of functionals and higher-order operators.
    • I proposed an elementary cut-off function technique that implies a local decay estimate can give new information on the Chen conjecture. Note: This idea was discovered independently by Nakauchi-Urakawa-Gudmundsson. Their article appeared after mine, and although the B.-Y. Chen of the Chen conjecture cites my article together with theirs, many subsequent works using this same idea only cite Nakauchi-Urakawa-Gudmundsson. Nevertheless the fact remains that this idea has been fundamental to much of the new exciting progress on Chen's conjecture.
    • Together with A. Dall'Acqua and K. Deckelnick, we solved an open question of Grunau and settled the existence of unstable Willmore surfaces. We did this by developing several novel new observations, including that unstable surfaces of revolution have an associated derived function that solves a Dirichlet problem.
    • Together with B. Andrews, J. McCoy, V.-M. Wheeler and A. Holder we tackled for the first time curvature flow that are not only fully nonlinear, but whose speed is a non- smooth function of the Weingarten map. To handle this we mollified the speed of the flow (and not the flow itself), and showed how uniform estimates independent of the mollification parameter can be derived. A limiting process allows us to solve the original problem. Convergence is extremely delicate, and required geometrisation of a weak Harnack inequality combined with the strong maximum principle on subintervals.
    • Together with J. McCoy, M. Edwards, A. Gerhard-Bourke and V.-M. Wheeler we classified solitons for the curve diffusion flow, and showed that the Lemniscate of Bernoulli is a shrinker. This allows us to settle an old question of Polden.
    • I studied the evolution of space curves evolving by the L^2-gradient of a functional incorporating curvature and length, with anisotropy. This is interesting from an applied perspective, since the Helfrich functional is of this form. Several new techniques needed to be developed, notably convergence to an equilibrium required a new application of the Llojasiewicz-Simon gradient inequality. This idea resulted from discussions with Ben Andrews. It proved to be fundamental and now many papers in the field are using the same method.
    • Together with J. McCoy we approached the problem of finite-time singularities for a constrained Willmore functional. This allowed us to demonstrate for the first time the existence of finite-time singularities for these flows. Previously, either singularities were known in infinite time only or for very special highly singular flows. Unfortunately the family does not include the Willmore flow (or the surface diffusion flow), and existence of finite-time singularities for these flows remains an important open question.
    • Together with M. Simon we proved a new uniqueness criterion for the biharmonic heat equation on n-dimensional space that uses a new geometric blowup and decay argument. The method has been developed to be particularly relevant to fourth and higher order curvature flow.
    • Together with H. Lee and G. Drugan I reduced the existence problem for solitons of the inverse mean curvature flow to an ODE. By proving various qualitative properties of the ODE we found a semi-explicit expression for the desired solitons. This is a completely new family of solutions to the flow, hinting at a much deeper and more extensive structure than had been previously known. We also give some rigidity results.
    • Together with Y. Bernard and V.-M. Wheeler I considered problems directly motivated by the blood disease spherocytosis. We take the Helfrich model and ask ourselves: for which parameter values does one see a sphere? For which parameter values are spheres `stable'? We give various answers to this question, using new techniques in the variational theory for higher-order functionals. One of the theorems generalises a well-known result of myself and J. McCoy.
    • Together with J. McCoy, J. Sharples, I. Towers and V.-M. Wheeler I produced a series of refereed conference papers that explore various applications of curvature flow to the modeling of fire fronts, especially those that arise in bushfires.
    • B, Lamichhane, J. Droniou, M. Ilyas and myself studied numerically a PDE of sixth order with various boundary conditions, with a view to eventually considering numerically a sixth-order curvature flow I introduced earlier.
    • I considered a variety of higher-order flows with boundary conditions. This is a collection of papers joint with J. McCoy and Y. Wu. We give rigidity results as well as global analysis.
    • In the elliptic setting, together with V.-M. Wheeler I considered a rigidity question for minimal hypersurfaces. When this article was placed on the global e-print archive as a preprint, it generated prompted almost immediately e-mails from eminent researchers at ETH Z\"urich and Stanford University to congratulate us on our work.
    • Together with Y. Bernard and V.-M. Wheeler we have given the first systematic analysis of Chen's flow, a higher-order curvature flow tailor-made to give new insight on Chen's conjecture. This is followed up by further work joint with M. Cooper that includes numerical work and results in the low-dimensional case.

Selected Publications


Impact Story


Potential Supervision Topics


  • Analysis of higher order PDE
  • Curvature Flow
  • Differential geometry
  • Rigidity for PDE and submanifolds

Advisees


  • Graduate Advising Relationship

    Degree Research Title Advisee
    Doctor of Philosophy Non-Compact Higher-Order Curvature Flow
    Doctor of Philosophy Gradient Flow of the Dirichlet Energy for the Mean Curvature of Closed Surfaces Fang, Yanqin
    Doctor of Philosophy Gradient Flow of the Dirichlet Energy for the Curvature of Plane Curves Wu, Yuhan

Keywords


  • Analysis
  • Mathematics
  • Pure Mathematics

Web Of Science Researcher Id


  • G-1735-2014
uri icon

Geographic Focus