Glen Wheeler

• Ph.D.
• Lecturer at University of Wollongong

In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \use...

This article investigates a mathematical model for bushfire propagation, focusing on the existence and properties of translating solutions. We obtain quantitative bounds on the environmental diffusion coefficient and ignition kernels, identifying conditions under which fires either propagate across the entire region or naturally extinguish. Our ana...

We introduce a model to describe the interplay between prescribed burning and bushfires based on a system of ordinary differential equations. We show that the system possesses a unique steady-state, whose stability depends on the policy governing prescribed burning. Specifically, a Reactive Policy in which prescribed burning activities are increase...

In this paper we introduce the target flow -- a specific curve shortening flow with an ambient forcing term -- that, given an embedded (not necessarily convex) target curve, will attempt to evolve a given source curve to that target. The motivation for this flow is to address a question of Yau. Our main result is that the target flow with uniformly...

In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the $L^{\infty}$ sense. We prove that closed embedded curves that have a minimum of curvature sufficiently large shrink to round points. The method of proof follows...

In this paper we consider the steepest descent L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-gradient flow of the entropy functional. The flow e...

In this article we completely classify solitons (equilibria, self-similar solutions and travelling waves) for the surface diffusion flow of entire graphs of function over the real line.

In this article, we consider the length functional defined on the space of immersed planar curves. The $$L^2(ds^\gamma )$$ L 2 ( d s γ ) Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the vanishing of the $$L^2(ds^\gamma )$$ L 2 ( d s γ ) Riemannian distance, we consider the gra...

We study a class of fourth-order geometric problems modeling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, and bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.

In [25], Smoczyk showed that expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we consider L2(dθ)-gradient flows for a class of higher-order cur...

We show stabilisation of solutions to the sixth-order convective Cahn-Hilliard equation. {The problem} has the structure of a gradient flow perturbed by a quadratic destabilising term with coefficient $\delta>0$. Through application of an abstract result by Carvalho-Langa-Robinson we show that for small $\delta$ the equation has the structure of gr...

A recent article [1] considered the so-called ‘ideal curve flow’, a sixth-order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the L2 sense. It was critical in the analysis in that article that there was a length bound on the evolving curves. It is natural to suspect therefore...

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and...

In this paper, we consider the $L^2$-gradient flow for the modified $p$-elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prove the existence of global-in-time weak solutions with $p \ge 2$ for initial curves in the energy class via minimizing movements. Moreover, we prove the existence of uniqu...

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and...

In this article we consider the length functional defined on the space of immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the triviality of the metric topology in this space, we consider the gradient flow of the length functional with respect...

We consider the parabolic polyharmonic diffusion and the L 2 {L^{2}} -gradient flow for the square integral of the m -th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 {L^{2}} , then the e...

In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a convex domain...

In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture).

A recent article by the first two authors together with B Andrews and V-M Wheeler considered the so-called `ideal curve flow', a sixth order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the $L^2$ sense. Critical in the analysis there was a length bound on the evolving curves....

We study the evolution of closed inextensible planar curves under a second order flow that decreases the p-elastic energy. A short time existence result for p∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setleng...

In this article we study Chen's flow of curves from theoreical and numerical perspectives. We investigate two settings: that of closed immersed $\omega$-circles, and immersed lines satisfying a cocompactness condition. In each of the settings our goal is to find geometric conditions that allow us to understand the global behaviour of the flow: for...

In this note we establish exponentially fast smooth convergence for global curve diffusion flows that satisfy an isoperimetric inequality, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture).

We are interested in surfaces with boundary satisfying a sixth order non-linear elliptic partial differential equation associated with extremal surfaces of the L2-norm of the gradient of the mean curvature. We show that such surfaces satis-fying so-called ‘flat boundary conditions’ and small L2-norm of the second fundamental form are necessarily pl...

In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.

We consider surfaces with boundary satisfying a sixth-order nonlinear elliptic partial differential equation corresponding to extremising the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...

We consider the parabolic polyharmonic diffusion and $L^2$-gradient flows of the $m$-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in $L^2$, then the evolving curve converges exponentially in...

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{U}$ (where $\mathcal{U}\subset...

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow...

We show that small energy curves under a particular sixth order curvature flow with generalised Neumann boundary conditions between parallel lines converge exponentially in the C∞ topology in infinite time to straight line segments.

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm of the second fundamental form and satisfying so-called `flat boundary conditions' are necessarily planar.

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow...

We study a class of fourth-order geometric problems modelling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.

We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieved.

In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply-covered circle. Moreover,...

The paper studies a curvature flow linked to the physical phenomenon of wound closure. Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity, showing that the singularity profile after continuous rescaling is that of a circle. We additionally give a max...

The paper studies a curvature flow linked to the physical phenomenon of wound closure. Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity, showing that the singularity profile after continuous rescaling is that of a circle. We additionally give a max...

The Helfrich functional, denoted by Hc0 , is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise Hc0 among all possible configurations. The functional integrates a spontaneous curvature parameter c0 toge...

We show that small energy curves under a particular sixth order curvature flow with generalised Neumann boundary conditions between parallel lines converge exponentially in the smooth topology in infinite time to straight lines.

We show that small energy curves under a particular sixth order curvature flow with generalised Neumann boundary conditions between parallel lines converge exponentially in the smooth topology in infinite time to straight lines.

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-con...

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-con...

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{U}$ (where $\mathcal{U}\subset...

Chen's flow is a fourth-order curvature flow motivated by the spectral decomposition of immersions, a program classically pushed by B.-Y. Chen since the 1970s. In curvature flow terms the flow sits at the critical level of scaling together with the most popular extrinsic fourth-order curvature flow, the Willmore and surface diffusion flows. Unlike...

In this paper, we study families of immersed curves $\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported on parallel lines $\{\eta_1, \eta_2\}:\mathbb{R}\rightarrow\mathbb{R}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux co...

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces $F:M^n\rightarrow\mathbb{R}^{n+1}$ with free boundary on the standard unit sphere. First we show that if $F$ is graphical with respect to any Killing field, then $F(M^n)$ is a flat disk. Second, if $M^n = \mathbb{D}^n$ is a disk, we sh...

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces $F:M^n\rightarrow\mathbb{R}^{n+1}$ with free boundary on the standard unit sphere. First we show that if $F$ is graphical with respect to any Killing field, then $F(M^n)$ is a flat disk. This result is independent of the topology or nu...

In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Minkowski integral formula to pro...

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Minkowski integral formula to pro...

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of the total amount of time durin...

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of the total amount of time durin...

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (...

We consider closed immersed surfaces in R^3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L^2. We further use an {\epsilon}-regularity type...

The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a spontaneous curvature parameter...

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (...

We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining speed and curvature pinching estimates for the flows by the approximating spe...

The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preservation of graphicality for the...

We consider smooth solutions to the biharmonic heat equation on Euclidean space for which the square of the Laplacian at time t is globally bounded from above by k/t for some k in R, for all t in [0,T]. We prove local, in space and time, estimates for such solutions. We explain how these estimates imply uniqueness of smooth solutions in this class.

The merging of two lines of fire is a relatively common occurrence in landscape fire events. For example, it can arise through the coalescence of two wildfires or when a prescribed fire meets a wildfire as part of suppression efforts. When two fires approach one another, the effects of convective and radiative heat transfer are compounded and high...

In this paper we propose a novel mathematical model for describing the evolution of a fire front. Specifically, for a homogeneous fuel bed of varying height with constant ignition temperature T ig we model the isosurface corresponding to T ig . The intersection of this isosurface with the fuel bed defines the evolving front. There are three natural...

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the cat...

In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature and area at initial time. The es...

Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Eu...

In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples include Willmore surfaces, stationary solitons for the surface diffusion flow, and biharmonic immersed surfaces i...

In this paper we consider the evolution of regular closed elastic curves $\gamma$ immersed in $\R^n$. Equipping the ambient Euclidean space with a vector field $\ca:\R^n\rightarrow\R^n$ and a function $f:\R^n\rightarrow\R$, we assume the energy of $\gamma$ is smallest when the curvature $\k$ of $\gamma$ is parallel to $\c = (\ca \circ \gamma) + (f...

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (\Delta H \equiv 0) hypersurface in \R^3 or \R^4 with restricted growth of the curvature at infinity and small total tracefree curvature must be an embe...

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH ≡ 0) hypersurface in \({\mathbb{R}^3}\) or \({\mathbb{R}^4}\) with restricted growth of the curvature at infinity and small total tracefree curvatur...

In this paper, we consider the steepest descent H −1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. W...

We consider closed immersed hypersurfaces in $\R^3$ and $\R^4$ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exis...

In this paper we study the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all smooth...

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our f...

We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for which a smooth solution exists, and a small upper bound on a power of the total curvature during this time. By ph...

In this thesis the chief object of study are hypersurface flows of fourth order, with the speed of the flow varying from the Laplacian of the mean curvature, to the more general constrained flows which include a function of time in the speed, and satisfy various conditions. Our aim is to instigate a study of the regularity of these flows, answering...

The binary reflected Gray code function b is defined as follows: If m is a nonnegative integer, then b(m) is the integer obtained when initial zeros are omitted from the binary reflected Gray code of m.This paper examines this Gray code function and its inverse and gives simple algorithms to generate both. It also simplifies Conder's result that th...

We introduce the notion of weighted watermarking for proof-of-ownership watermark protection of multimedia works that are the product of more than one author and where each author is considered to be of different importance relative to the other authors. We specifically examine weighted segmented watermarking for still images and generalise previou...