Publications


[A1] ``Contraction of convex hypersurfaces in Euclidean space'', Calc. Var. 2 (1994), 151-171.

In this paper I considered the motion of convex hypersurfaces by speeds which are homogeneous degree one functions of the principal curvatures, satisfying some natural monotonicity and concavity conditions. It turns out that for a very wide range of such flows, all smooth strictly convex initial hypersurfaces become spherical as they contract to points. The concavity condition is necessary partly to apply regularity results for fully nonlinear parabolic equations, and partly to obtain bounds on the ratios of principal curvatures as the hypersurfaces evolve.

The proof simplifies earlier work by using a simple geometric Lemma: We say the width of a convex hypersurface M in some direction is the separation of the two tangent planes of M which are orthogonal to that direction. Then the result is: If M is a convex hypersurface (of dimension at least 2) such that at every point x of M the ratio of largest and smallest principal curvatures at x is no greater than C, then the ratio of the largest and smallest widths of M is no greater than C.

Some natural questions arising from the paper are:
[A2] ``Contraction of convex hypersurfaces in Riemannian spaces'', J. Differential Geometry 39 (1994), 407-431.

In this paper the techniques of [A1] were extended to the case where the hypersurface is in a Riemannian background space satisfying some curvature condition. In particular, it is shown that a smooth, strictly convex hypersurface in a background space of non-negative principal curvature can be contracted to a point, so that it becomes spherical in the limit. If the background space has all sectional curvature greater than or equal to -1, then the same result holds for hypersurfaces which have all principal curvatures greater than 1.

This result has some nice applications: In particular it gives a fairly simple new proof of the 1/4-pinching sphere theorem of Klingenberg, Berger and Rauch, as well as a generalisation allowing some negative curvature (a `dented sphere' theorem).

The class of evolution equations used in the proof is much more restrictive than those allowed in [A1] -- it seems that the more general class does not always preserve the curvature condition on the hypersurgace as it evolves. An example of a speed which works for hypersurfaces in non-negatively curved background spaces is the harmonic mean curvature, which is the reciprocal of the sum of the reciprocals of the principal curvatures. If the background space has sectional curvatures at least -1, then a speed that works is the harmonic mean of the difference of the principal curvatures from 1 -- this is interesting because the speed is not homogeneous.

Two important questions which arise:
[A3] ``Harnack inequalities for evolving hypersurfaces'', Math. Zeitschrift 217 (1994), 179-197.

This paper proves Harnack inequalities for a range of evolution equations for convex hypersurfaces, including flows where the speed depends on the curvature and on the normal direction. The key observation is that the computations become extremely simple when the flow is written as a scalar evolution equation for the support function on the unit sphere. This explains apparent miracles in earlier calculations for special cases by Richard Hamilton (for the mean curvature flow) and Bennett Chow (for flows by powers of Gauss curvature).

[A4] ``Entropy inequalities for evolving hypersurfaces'', Communications in Analysis and Geometry 2 (1994), 53-64.

In this paper the Aleksandrov-Fenchel inequalities for convex bodies are applied to prove that certain integral quantities monotonically decrease if their speed is of a special form -- in particular, the speed can be apower of the Gauss curvature or of the harmonic mean curvature. One consequence of these estimates is that for motion by small powers of these curvature functions, it is not true that all convex bodies become spherical.

It would be very interesting to know if there are any such nice integral quantities which decrease under other apparently natural flows such as the mean curvature flow, or more generally flows in which the speed is a power of a ratio of elementary symmetric functions of curvature.
[A5] ``Contraction of convex hypersurfaces by their affine normal'', J. Differential Geometry 43 (1996), 207-230.

In this paper we introduce a remarkable affine-invariant evolution equation. In affine differential geometry, this corresponds to motion with unit speed in the direction of the affine normal vector. In terms of Euclidean-geometric invariants, this is equivalent to motion in the unit normal direction with speed equal to the (n+2)nd root of the Gauss curvature, where n is the dimension of the hypersurface. This is affine invariant in the following sense: Suppose we start with a smooth, strictly convex hypersurface M(0), and evolve under this evolution equation to obtain a hypersurface M(t) for each time t. Now take any volume-preserving affine transformation T of (n+1) dimensional Euclidean space, and consider the evolution equation applied to the initial hypersurface T(M(0)), giving the family of hypersurfaces (TM)(t). Then (TM)(t) = T(M(t)) for each t. Who would have guessed? In particular it follows that any ellipsoid evolves by contracting to its centre without changing shape (in particular these do not become spherical in the limit).

The main result of the paper is that any smooth, strictly convex initial hypersurface evolves to become ellipsoidal in shape as it contracts to a point. The key estimate is to control the cubic ground form (an affine-invariant tensor which depends on first derivatives of the Euclidean second fundamental form). The paper also gives a new proof of the affine isoperimetric inequality, which follows from the convergence result together with the Entropy estimate of [A4].

This led me to conjecture that flow by powers of Gauss curvature should give spherical limiting shapes if the power is greater than 1/(n+2), but usually not if the power is smaller.

[A6] ``Monotone quantities and unique limits for evolving convex hypersurfaces'', Int. Math. Res. Not. 20 (1997), 1001-1031.

This paper extends the results of [A4] by finding a family of monotone integral quantities for special curvature evolution equations. These are applied to prove that convergence to limiting shapes is always smooth, rather than just subsequential.

[A7] ``Evolving convex curves'', Calc. Var. 7 (1998), 315-371.

This paper gives a comprehensive discussion of the behaviour of curves evolving by functions of curvature and normal direction. It covers contraction and expansion flows, isotropic and anisotropic flows, homogeneous and nonhomogeneous flows. It gives existence results for singular initial data, optimal regularity estimates, and detailed convergence results.

There are still some interesting questions:
[A8] ``The affine curve-lengthening flow'', J. reine angew. Math. 506 (1999), 43-83.

This paper studies the affine-geometric analogue of the curve-shortening flow, which is not the affine normal flow mentioned above, but a fourth-order flow. In affine-geometric terms in corresponds to moving a convex curve in the direction of its affine normal with speed equal to its affine curvature.

The main result is that any embedded convex curve evolves to infinite size, becoming elliptical in shape as it does so. Maximum principle arguments cannot be used since the flow is of fourth order, so instead an isoperimetric-type inequality is used to obtain geometric control (in particular showing that the evolving curves remain convex and that any limit must expand homothetically). Then some hard work is done to establish regularity estimates and to show that the only curves which expand homothetically are ellipses.

The higher dimensional case looks interesting but more difficult, partly because I can't prove such a nice isoperimetric estimate. The volume-preserving affine mean curvature flow might be easier, however.

Other invariance groups also give rise to interesting higher order invariant evolution equations. It would be nice to be able to deal with some of these as well.
[A9] ``Gauss curvature flow: The fate of the rolling stones'', Invent. Math. 138 (1999), 151-161.

This paper proves a 1974 conjecture of Firey that convex surfaces evolving by their Gauss curvature become spherical. This flow was introduced by Firey as a model of the way that stones change in shape as they tumble around.

The argument is simple but surprising: I prove (using the maximum principle) that the maximum difference between the two principal curvature over the surface does not increase in time. It follows right away that the surfce rapidly becomes spherical in shape as it shrinks.

The methods used here make it possible to deal with a very large family of flows, particularly in the two-dimensional case.
[A10] ``Motion of hypersurface by Gauss curvature'', Pacific J. Math. 195 (2000), pp. 1-34.

This paper concerns evolution of hypersurfaces by Gauss curvature to a power no bigger than 1/n, possibly also with some dependence on the normal direction. The main result is that solutions immediately become smooth and strictly convex, and converge in shape to that of a homothetically contracting solution. The paper also gives a proof of the affine isoperimetric inequality without any smoothness assumption, by showing that solutions of the affine normal flow can be found for any convex initial hypersurface (without any smoothness assumption). Another application is given to prove the existence of non-spherical homothetic solutions for isotropic flow by Gauss curvature to a small power. Examples are given for flow by powers of Gauss curvature bigger than 1/n where the hypersurfaces do not immediately become smooth or strictly convex.
[A11] ``Volume-preserving anisotropic mean curvature flow'', to appear in Indiana Math. J.

In this paper it is shown that the gradient descent flows of anisotropic area functionals with a volume constraint always deform convex hypersurfaces smoothly to the corresponding isoperimetrix.

It seems much more difficult to handle anisotropic mean curvature flows without the fixed volume constraint. It is easy enough to show that the hypersurfaces stay convex (and even become smooth and strictly convex for small positive times), but the asymptotic behaviour is difficult. The solutions converge to points in finite time, but perhaps their isoperimetric ratios could blow up; even if the isoperimetric ratio stays bounded, I can't deduce much about the limit (I would like to say it becomes homothetic, but this seems to require some kind of improving integral - it is probably not true that ratios of anisotropic principal curvature are decreasing in time as in the isotropic case, and there are no known monotonicity formulae for these flows).
[A12] ``Non-convergence and instability in the limiting behaviour of curves evolving by curvature'', to appear in Communications in Analysis and Geometry.

This paper completes the story for evolving convex curves by curvature, by investigating the case of flow by small powers of curvature. [A7] proved that powers bigger than 1/3 of the curvature always give convergence to a homothetic limit. Here it is shown that for powers less than 1/3 (or equal to 1/3 with some anisotropy), generic initial conditions do not give convergence to any nice limit - instead the isoperimetric ratios blow up as the final time is approached.
[A13] ``Nonlocal geometric expansion of convex plane curves'' (with Mikhail Feldman, University of Wisconsin), to appear in Journal of Differential Equations.

We consider a family of non-local expansion flows for convex sets in the plane, in which the speed depends on the curvature but also on the `ridge function' - that is, the radius of the largest ball contained in the set which touches at a given point of the boundary. Such equations arise in models of collapsing sand piles and compression molding, and in population models.

We construct convex viscosity solutions for these flows, and prove results about the asymptotic behaviour.
[A14] ``Convergence of the iterates of descent methods for analytic cost functions'' (with Robert Mahony and Pierre-Antoince Absil), SIAM J, Optim, 16 (2005), 531-547.

This paper concerns discrete approximations to gradient descent algorithms for analytic functions. The main result is that these always converge to a critical point for large times (this is sometimes not true for functions which are not analytic).
[A15] ``Singularities in crystalline curvature flows'' Mathematics Research Report MRR 01.011, 2001, School of Mathematical Sciences, ANU.

This paper considers polygonal curves moving by discrete analogues of the curve-shortening equations, and shows that these can display some quite different behaviour to the smooth case: In the speed of motion (as a function of `crystalline curvature') does not grow fast enough, then there are convex polygonal curves which do not shrink to points, but collapse to line segments; also, there are crystalline curve-shortening flows which have no homothetic solutions, in contrast to the smooth case. Both of these results are in contradiction to conjectures made in the literature.
[A16] ``Notes on the isometric embedding problem and the Nash-Moser implicit function theorem'', Surveys in analysis and operator theory (Canberra, 2001), 157--208, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002.

These are notes I put together for a working seminar we ran here at ANU on the Nash-Moser implicit function theorem. There's nothing new here, but a variety of existing results and techniques are presented.
[A17] Classification of limiting shapes for isotropic curve flows", J. Amer. Math. Soc. 16 (2003), 443--459.

In this paper a complete classification is given for the homothetic solutions for flows of curves in the plane by powers of curvature. In particular this paper contains the first proof of the classification of homothetic solutions for the curve shortening flow (due to Abresch and Langer) which is not computer assisted.

[A18] ``Pinching estimates and motion of hypersurfaces by curvature functions'', J. Reine Angew. Math. 608 (2007), 17-33.

This paper proves curvature pinching estimate for a class of curvature flows including powers of ratios of elementary symmetric functions (homogeneous of degree one). In particular the paper resolves a question arsing from Ben Chow's work on flow by the square root of the scalar curvature: In that work he had to assume that S/H^2 on the initial hypersurface was larger than the value on the cylinder S^{n-1} x R. Then preserving a lower bound on this ratio implies a bound on ratios of principal curvatures, and the hypersurface contracts to a `round point'. But what if the initial hypersurface is just uniformly convex? This paper provides the answer. Most interesting perhaps is not the particular application but the method used to obtain it: It involves understanding in detail the nature of the gradient terms which arise in the evolution equation for the second fundamental form.

[A19] ``Time-interior gradient estimates for quasilinear parabolic equations'', Indiana Univ. Math. J. 58 (2009), 351--380 (with Julie Clutterbuck).

This is the first of a series of papers together with Julie Clutterbuck arising from the investigations in her PhD thesis. In this paper we look in detail at quasilinear parabolic equations in one spatial variable, where the coefficients depend on the gradient of the solution. We get sharp criteria for when arbitrary continuous initial data produce solutions with bounded gradient for positive times. It turns out the curve shortening flow is rather close to critical for this kind of behaviour. We also get sharp bounds on the gradient for positive times, and produce information about the dependence of these bounds on the initial modulus of continuity.
The estimates are based on the neat trick of Khruzhkov of doubling the number of variables to change an interior estimate into a boundary estimate, so that barrier techniques can be used. We make use of explicit (translating) solutions to prove the sharp criterion for the existence of gradient bounds.

[A20] ``Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable'', J. Differential Equations 246 (2009), 4268--4283 (with Julie Clutterbuck).

Following on from the previous paper, we attack the higher-dimensional problem. We find a useful criterion for when solutions have bounds on their gradient in terms of initial oscillation and elapsed time. Using this we can treat various geometrically interesting problems such as graphical mean curvature flow and its ansiotropic analogues, under various natural boundary conditions, and under minimal assumptions on the initial data (in many cases just continuity). An interesting question we don't deal with is the corresponding interior estimates. Such gradient estimates are known for mean curvature flow, but not for anisotropic mean curvature flows except under rather restrictive assumptions on the anisotropy.

[A21] ``Four-manifolds with 1/4-pinched Flag Curvatures'', to appear in Asian Journal of Mathematics (with Huy Nguyen).

This paper, joint with my former PhD student Huy Nguyen, introduces a new curvature pinching notion: Pinching of the flag curvature. Given any unit vector v in the tangent space to a manifold M at x, the flag curvature R(v) in that direction is a symmetric bilinear form which acts on the orthogonal complement, so that R(v) applied to an orthogonal unit vector u gives the sectional curvature of the plane generated by u and v. We prove that compact four-manifolds for which each of the flag curvatures R(v) has ratio of eigenvalues less than 4 evolves under Ricci flow to a constant curvature limit, thus proving a version of the sphere theorem for flag curvature pinching.
A more recent paper of Ni and Wilking proves that 1/4 pinching of the flag curvatures implies positive complex sectional curvature, so the result itself is rather superseded by subsequent work opf Brendle and Schoen. However this work was where we came up with the technique which Nguyen later used to prove that positive curvature on totally isotropic two-planes is preserved by the Ricci flow (this was done independently by Brendle and Schoen and was a key step in their proof of the differentiable 1/4-pinching sphere theorem).

[A22] ``Mean curvature flow of pinched submanifolds to spheres'', preprint (with Charles Baker).

This is joint work with my PhD student Charles Baker. In it we prove an analogue of the old result of Huisken on contraction of convex hypersurfaces to spheres, but for higher codimension submanifolds. Instead of convexity we assume that the ratio of the length of the second fundamental form to the length of the mean curvature vector is bounded (by some explicit constant depending on dimension but not codimension). The hard work is in handling the algebra of the second fundamental form in high codimension.

[A23] ``Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem'', to appear in J. Reine Angew. Math. (with Paul Bryan).

This is work which arose partly from discussions around Paul Bryan's honours thesis. In it he presented Huisken's distance comparison principle for the curve shortening flow, which rules out `type 2' singularities and so provides an alternative proof of Grayson's theorem using a blow-up argument (modulus some machinery of blowup, and classification results for type 1 and type 2 singularities). We wondered whether the distance comparison argument could be `bootstrapped' to get higher regularity and so bypass the blowup argument. After spending a great deal of time trying to control curvatures in terms of chord-arc ratios, and various other possibilities, we finally realised that a good enough control on straight-line distance as a function of arc length automatically gives a curvature bound. From there it took some creative guesswork to produce a suitable distance comparison estimate. The result gives remarkably good control, including an explicit rate of decay of the curvature towards one for the normalized curve shortening flow.

[A24] ``Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere'', preprint (with Paul Bryan).

This follows on from the previous work in some ways: The idea is that good enough control on the isoperimetric profile implies control on the curvature. In this case the result is actually much cleaner (and less mysterious) than the case for the curve shortening flow, and is a kind of comparison theorem: If the isoperimetric profile (the function which gives the smallest length of boundary for a region containing a given area) of a given initial metric on the two-sphere is bounded by the isoperimetric profile of a positively curved axially symmetric metric on the two-sphere (of the same area), then this remains true at later times under normalized Ricci flow. This is beautiful because we have a lovely explicit positively curved axially symmetric solution of Ricci flow on the two sphere, namely the Rosenau solution. Comparison with this gives that the maximum curvature decays exponentially to 1 under the normalized Ricci flow, and the convergence to a constant curvature metric follows very easily.

[A25] ``Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature'', preprint (with James McCoy).

The main contribution here is an estimate of the following kind: Suppose f is any continuous function which approaches 1 at infinity. Then there exists a function h, which approaches zero at 1, such that any convex hypersurface with ratio of principal curvatures bounded by f(H) (where H is the mean curvature) and diameter less than h(c) has ratio of circumradius to inradius less than c. This is proved simply by combining various classical estimates involving quermassintegrals.
We then apply this to the evolution of convex hypersurfaces by speeds which are homogeneous of degree greater than 1 in the principal curvatures. It is quite well known that pinching estimates on principal curvatures can be proved using the maximum principle for flows of this kind, and quite a few papers have now appeared which treat special cases. The difficulty has been in proving that solutions converge, since there is no good regularity theory for the kinds of equations which arise. Thus results have previously only been proved for flows which have some kind of divergence structure. The geometric estimate gets rid of this obstacle, by proving directly that the hypersurface are clsoe to spheres when they are small enough. Once they are close enough to spheres, lower bounds on the speed can be deduced using barriers, and the difficulties with the regularity theory disappear.

[A26] ``Fully nonlinear parabolic equations in two space variables'', preprint.



[A27] ``Moving surfaces by non-concave curvature functions'', preprint.

A short paper in which I prove that arbitrary (strictly parabolic) homogeneous degree one speeds deform (smooth, strictly convex) hypersurfaces in three-space to round points. This ties in with questions posed earlier concerning whether concavity or convexity were necessary. I rely on the special regularity results for parabolic equations in two space variables proved in the previous paper.

[A28] ``All roads lead to Newton: Feasible second-order methods for equality-constrained optimization'', preprint (with Pierre-Antoine Absil, Robert Mahony and Jochen Trumpf).



[A29] ``The Ricci Flow in Riemannian Geometry'', book preprint (with Chris Hopper).

This expository account of Ricci flow grew from the honours thesis of Chris Hopper. In it we provide an introduction to Ricci flow, leading up to the recent proof of the differentiable 1/4-pinching sphere theorem of Brendle and Schoen.

[A30] ``Positively curved surfaces in the three-sphere''

My talk at the 2002 ICM in Beijing. It discussed some ideas on constructing flows to suit the needs of the problem, with a particular example discussed in detail: Deforming positively curved surfaces immersed in the three-sphere to totally umbillic spheres. Similar techniques apply in higher dimensions, and also give results on surfaces in hyperbolic manifolds. I'm embarrassed to say that I still haven't written up these results!

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Last updated: 4 Oct, 2001