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 widthof a convex hypersurface Min some direction is the separation of the two tangent planes of Mwhich are orthogonal to that direction. Then the result is: If Mis a convex hypersurface (of dimension at least 2) such that at every point xof Mthe ratio of largest and smallest principal curvatures at xis no greater than C,then the ratio of the largest and smallest widths of Mis 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 nottrue 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 nis 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 Tof (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 should be relatively easy, 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/nwhere 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 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 discrete and continuous gradient descent algorithms for analytic cost functions'' (with Robert Mahony, Engineering Department, ANU), preprint.

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.

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Last updated: 4 Oct, 2001