Ben Andrews - MSI - ANU

Publications

[A46] Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature (with Xuzhong Chen, Hanlong Fang and James McCoy).

In this paper we consider co-compact spacelike hypersurfaces in Minkowski space evolving by functions of curvature. The results are parallel to those known for the Euclidean case, but the details are somewhat different: The solutions exist for all time, and are asymptotic for large times to an expanding hyperboloid. The results are in fact a little easier to prove than in the corresponding Euclidean case, partly because the expanding hyperboloids converge to each other after rescaling, so simply trapping the hypersurface between two hyperboloids already gets us a long way. The results are in fact quite a bit stronger than the Euclidean ones: For example we can use speeds which are arbitrary positive powers of elementary symmetric functions of principal curvatures. One of the reasons for our interest in this situation comes from the fact that the cross-curvature flow of a negatively curved Riemannian metric on a compact three-manifold coincides with the Gauss curvature flow, provided the initial metric can be locally isometrically embedded as a hypersurface in Minkowski space. Our result therefore gives an interesting large class of examples of negatively curved metrics which evolve to become hyperbolic (modulo scaling) under the cross-curvature flow. It is conjectured that this should be true for arbitrary negatively curved metrics on three-manifolds, but this is not known.

[A45] Curvature flow in hyperbolic spaces (with Xuzhong Chen).

There are many results for fully nonlinear flows of hypersurfaces in Euclidean space, but the understanding of what happens in more general background spaces is much weaker: Huisken has considered the mean curvature flow in Riemannian backgrounds, but a convexity condition must be imposed on the initial hypersurface depending on the size of the gradient of the curvature in the ambient space. In general it seems very hard to preserve convexity, though some special flows can do this (see [A2] below). In this case we focus on one of the simplest background spaces, the hyperbolic space. Some previous work has considered this setting, particularly for mean curvature flow. We provide several new results: We prove an analogue of my earlier result for Gauss curvature flow of surfaces --- but in the hyperbolic background we have to consider instead the speed K-1. We also give an improved result for mean curvature flow of convex surfaces: If the initial scalar curvature R=2(K-1) is positive, then the surface contracts to a point and becomes spherical in shape. We prove the same result for flows of surfaces by arbitrary flows of the form F=(1-1/K)G where K is the Gauss curvature, and G is any increasing homogeneous degree one symmetric function of the principal curvatures. Finally, we give an improved result for higher-dimensional flow of convex hypersurfaces in hyperbolic space: If the hypersurface initially has positive Ricci curvature, then the mean curvature flow preserves this and contracts the hypersurface to a point. This seems a much more natural condition that previous ones which amounted to something like `horospherical convexity' in which all principal curvatures are greater than 1.

[A44] Convexity estimates for hypersurfaces moving by convex curvature functions (with Mat Langford and James MCCoy).

In this paper we make a first attempt to prove the asymptotic convexity result for other flows in higher dimensions. The flows we consider here are convex functions of the principal curvatures, which is a reasonably large class but unfortunately does not include many of the most commonly studied examples such as powers of Gauss curvature, elementary symmetric functions of curvature and their ratios. However the result works out most simply in the convex case. In a third paper in the series we will consider a wider class of flows which includes more of the classical examples, but this requires more difficult and delicate estimates.

[A43] Convexity estimates for fully nonlinear surface flows (with Mat Langford and James McCoy).

One of the important results proved for mean curvature flow is the theorem of Huisken and Sinestrari which states that any compact hypersurface with positive mean curvature moving by mean curvature flow is almost convex in regions where the curvature is large --- so in particular the limits of blowups at singularities are convex. In this paper (the first in a series) we consider the same question for other flows of surfaces. The result is surprisingly strong: We show that the asymptotic convexity result holds under any fully nonlinear parabolic flow in which the speed is a homogeneous degree one function of the principal curvatures, provided the speed of motion on the initial surface is positive. To prove this we have to work quite hard to find an argument which does not rely on delicate algebraic structure of particular functions of curvatures, as was the case in the Huisken-Sinestrari argument: Any such structure is lost when combined with the arbitrary nonlinearity in the flow. The argument we produce is quite geometric, and we also use similar ideas in the follow-up papers (see [A44] above). This result of this paper is definitely two-dimensional: We combine the argument for asymptotic convexity with the careful accounting of the components of the derivatives of second fundamental form which were the essential method in [A9], [A28] and [A37] below.

[A42] Non-collapsing for hypersurface flows in the sphere and hyperbolic space (with Xiaoli Han, Haizhong Li and Yong Wei), to appear in Annali SNS.

In this paper we made a first attempt to understand the effects of background geometry on the non-collapsing argument which appeared in [A35] and [A36], by looking at the relatively nice case of hypersurfaces in constant curvature spaces (spheres and hyperbolic spaces). To a large extent the results are as expected: The argument essentially goes through as in the Euclidean case, but in the presence of some negative background curvature the collapsing ratio can degenerate as time approaches infinity. For the purpose of understanding singularities this is not an issue, so we still obtain a useful result. It seems a reasonable conjecture that some kind of non-collapsing should hold in arbitrary backgrounds if we assume bounds on curvature and its first derivative, and a lower bound on the injectivity radius.

[A41] Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue (with Julie Clutterbuck).

We derive sharp estimates on the modulus of continuity of solutions of `isotropic' quasilinear heat equations on Riemannian manifolds, in terms of initial modulus of continuity, the diameter of the manifold, and a Ricci curvature lower bound. Applying this in the case of the heat equation we derive sharp modulus of continuity estimates for solutions of `isotropic' parabolic equations on manifolds with a lower Ricci curvature bound. The eigenvalue result itself is not new, but our proof is much easier than the gradient estimate methods used in previous proofs. The main idea is to extend the methods of [A20] and [A21] to the manifold setting (see also [A31] and [A38]).

[A40] F-stability for self-shrinking solutions to mean curvature flow (with Haizhong Li and Yong Wei).

The main work of this paper is to extend ideas of stablility for solutions of mean curvature flow to higher codimension. Along the way we prove that the only self-similar solution which is a minimal surface in a sphere (an important class of examples) which is F-stable is the shrinking sphere (to prove this we modify an old argument of Jim Simons for minimal submanifolds in spheres).

[A39] Embedded constant mean curvature tori in the three-sphere (with Haizhong Li).

Simon Brendle found a beautiful proof of the Hsiang-Lawson conjecture (that the only embedded minimal torus in the three dimensional sphere is the Clifford torus) by applying the non-collapsing estimate I proved for mean curvature flow [A35] to a minimal torus in the three-sphere (here it does not make sense to use the mean curvature to measure the sizes of touching spheres, since this vanishes for a minimal surface, and Brendle shows that the mean curvature can be replaced with the largest principal curvature). In this paper we find a way to apply a similar argument to embedded constant mean curvature tori, and prove a 1989 conjecture Pinkall and Sterling: Any such torus is a surface of rotation. We further classify such surfaces of rotation, making use of earlier work by Perdomo and combining this with a new proof of monotonicity of an associated period function. Curiously, if the mean curvature is either zero or has square equal to 1/3, then the only such torus is the Clifford (product) torus, while for any other values of the mean curvature there are other examples which are not products. The mean curvature zero rigidity is the Hsiang-Lawson conjecture proved by Brendle, but the other is rather surprising: It turns out that the mean curvature having square equal to 1/3 is just too small to allow a surface made of three identical pieces glued together, and is just too large to allow a surface made by gluing two identical pieces together.

[A38] Eigenvalue comparison on Bakry-Emery manifolds (with Lei Ni, to appear in Communications in PDE).

This paper applies the methods developed by Julie Clutterbuck and myself (see [A20], [A21]) on controlling the modulus of continuity of heat equations, to prove sharp lower bounds on the first eigenvalue for the natural Laplacian on a Bakry-Emery space. We show that the estimate is sharp by constructing model spaces on which equality holds (at least in some limit). The result implies a curious lower diameter bound for non-Einstein gradient Ricci solitons.

[A37] Surfaces moving by powers of Gauss curvature (with Xuzhong Chen), Pure and Applied Mathematics Quarterly 8 (2012) no. 4, 825-834

This paper adapts the methods I used to prove the Firey conjecture in [A9] to other flows. We obtain sharp pinching estimates for flows by powers of Gauss curvature between 1/2 and 1, and also for powers of mean curvature and several other flows. Perhaps the most important point is the observation that the `reaction' part of the evolution equation for curvature can always be completely solved, and this gives a natural candidate for a pinching quantity. We also announce some results to appear in a future paper, perhaps most interesting is that flow of surfaces by sums of pth powers of principal curvatures always gives spherical limits for p at least 1.

[A36] Non-collapsing in fully nonlinear curvature flows (with Mat Langford and James McCoy), Annales de l'Institut Henri Poincare / Analyse non lineaire, 30 (2013), no. 1, 23--32.

This paper follows on from [35] below, and both gives a new interpretation of the results for mean curvature flow and an extension of them to flows where the speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures.

[A35] Non-collapsing in mean-convex mean curvature flow, Geometry & Topology 16 (2012), 1413 1418.

This short paper provides a direct proof (using the maximum principle) that compact embedded mean-convex solutions of the mean curvature flow do not `collapse'. Precisely, suppose that at each point on the initial hypersurface there touches a sphere in the enclosed region with radius equal to a constant divided by the mean curvature. Then this remains true, with the same constant, for all future times. Results of this kind were deduced by Weimin Sheng and Xujia Wang using a much more involved argument, requiring quite detailed analysis of the singular structure. The argument here is rather straightforward, involving an application of the maximum principle on a function of pairs of points on the evolving hypersurface.

[A34] Contracting convex hypersurfaces by curvature (with James McCoy and Zheng Yu), to appear in Calc. Var. PDE

Our main goal in writing this paper was to understand which flows of hypersurfaces can be used to take an arbitrary (possibly non-smooth or non-strictly convex) convex hypersurface as initial data, and deform it to immediately become smooth and uniformly convex. This behaviour is well known for the mean curvature flow, where it follows from a combination of regularity estimates and the strong maximum principle, and depends crucially on the fact that the mean curvature flow is uniformly parabolic. Our motivation for understanding this problem comes from the setting of convex hypersurfaces in Riemannian backgrounds, where the mean curvature flow does not in general preserve convexity, but where one would still like to be able to deform weakly convex hypersurfaces to uniformly convex ones under reasonable conditions. My previous work on Gauss curvature flows ([A10] below) gave some examples of flows which are degenerate parabolic but still deform weakly convex or non-smooth hypersurfaces to become smooth and strictly convex. In the process of investigating this we discovered quite a lot of interesting behaviour: We found examples of flows (even with speeds which are concave functions of the principal curvatures) where smooth, uniformly convex hypersurfaces deform to become non-convex, and we found surprisingly many flows in which hypersurfaces remain convex, but develop curvature singularities even while the inradius remains strictly positive; in both cases we gave essentially necessary and sufficient conditions for these phenomena to occur. We completely characterized the flows in which planar pieces in the intiial hypersurface persist for some time, and deduced that for the flows where this does not occur the speed immediately becomes strictly positive. And we gave examples where the speed is strictly positive, but the hypersurface does not immediately become uniformly convex. Surprisingly, in the latter examples the situation is so bad that the evolving hypersurfaces do not even contract to points, but instead collapse onto line segments of positive lengths, or higher-dimensional discs of positive radius. This happens even for the apparently well-behaved flow by speed equal to the length of the second fundamental form, for which it is known that any uniformly convex smooth initial hypersurface does contract to a point and become spherical in shape. At the end of this process of understanding various kinds of counterexample, we arrive at a reasonably good (though not really complete) understanding of which flows have the desired property of immediately producing smooth, uniformly convex hypersurfaces. We also provide some new contributions to the `pinching estimates' required to prove that limiting shapes are spherical, proving a new pinching estimate for (homogeneous degree one) speeds which are concave in the principal radii of curvature. Next we hope to look at the question in Riemannian backgrounds, where there are quite different analytical difficulties to overcome in addition to those we had to handle here.

[A33] Gradient and oscillation estimates and their applications in geometric PDE, AMS/IP Studies in Advanced Mathematics Vol. 51 (2012), 3--19.

This is a survey written for the proceedings of the 5th International Congress of Chinese Mathematicians, Beijing 2010.

[A32] A comparison theorem for the isoperimetric profile under curve shortening flow (with Paul Bryan), Comm. Analysis and Geometry 19 (2011), 503-530.

This paper is related to my two previous papers with Paul Bryan, in that it derives curvature bounds for geometric evolution equations from control on isoperimetric quantities. In this case we control the isoperimetric profile of the enclosed region for a solution of the normalized curve shortening flow (normalized to have fixed enclosed area), by comparing it with the isoperimetric profile of a model region (the `paperclip' solution of curve shortening flow found by Sigurd Angenent). This gives an upper bound on curvature, and a similar comparison for the isoperimetric profile of the exterior region gives a lower bound on curvature. Along the way we determine the isoperimetric subregions for any convex region which is symmetric under relection in the coordinate axes, and has exactly four critical points of curvature.

[A31] Proof of the Fundamental Gap Conjecture (with Julie Clutterbuck), Journal of the American Mathematical Society 24 (2011), 899-916.

In this paper we extend the methods developed in our two earlier joint papers [A20],[A21] to prove a sharp log-concavity estimate for the first Dirichlet eigenfunction for Schr\"odinger operators on convex domains, and use this to prove the fundamental gap conjecture: If the potential is convex, then the gap between the first two eigenvalues is at least as large as that for the zero potential operator on an interval with the same diameter.

[A30] ``The Ricci Flow in Riemannian Geometry'' (with Chris Hopper), Lecture Notes in Mathematics vol. 2011 (2011).

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.

[A29] ``All roads lead to Newton: Feasible second-order methods for equality-constrained optimization'', preprint (with Pierre-Antoine Absil, Robert Mahony and Jochen Trumpf).

[A28] ``Moving surfaces by non-concave curvature functions'', Calculus of Variations and PDE 39 (2010), 649--657.

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.

[A27] ``Fully nonlinear parabolic equations in two space variables'', preprint.

A short paper dealing with regularity theory for fully nonlinear parabolic equations in two space variables (with a section also devoted to what the same methods say about higher dimensions). The key estimate (with a view to applications in geometric evolution equations) is a Holder continuity estimate for second spatial derivatives, which is the key step to higher regularity.

[A26] ``Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature'' (with James McCoy), Trans. Amer. Math. Soc. Vol. 364, No. 7 (2012), 3427--3447.

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 hypersurfaces are close 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.

[A25] ``Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere'', (with Paul Bryan), Calc. Var. PDE. 39 (2010), 419--428.

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.

[A24] ``Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem'' J. Reine Angew. Math. 653 (2011), 179-187 (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.

[A23] ``Mean curvature flow of pinched submanifolds to spheres'', (with Charles Baker), J. Differential Geometry 85 (3) 2010, 357--396.

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.

[A22] ``Four-manifolds with 1/4-pinched Flag Curvatures'', Asian Journal of Mathematics 13 (2009), p. 251-270 (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).

[A21] ``Time-interior gradient estimates for quasilinear parabolic equations'', Indiana Univ. Math. J. 58 (2009), 351--380 (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.

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

[A19] ``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.

[A18] ``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.

[A17] ``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!

[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.

[A15] ``Singularities in crystalline curvature flows'' Asian J. Math. 6 (2002), 101-121.

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.

[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).

[A13] ``Nonlocal geometric expansion of convex plane curves'' (with Mikhail Feldman, University of Wisconsin), Journal of Differential Equations 182, Issue 2, Pages 298-343.

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.

[A12] ``Non-convergence and instability in the limiting behaviour of curves evolving by curvature'', Comm. Anal. Geom. 10 (2002) 409-449.

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.

[A11] ``Volume-preserving anisotropic mean curvature flow'', Indiana Univ. Math. J. 50 (2) (2001) 783-827

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).

[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.

[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.

[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.

[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:

In the isotropic case, how can the possible limiting shapes be classified?
Can the results on these smooth flows be extended to cases where the anisotropy is not smooth? The extreme cases of crystalline flows are quite well studied, but there is a big range in between these two extremes.
In extensions to non-convex curves, these flows usually become either singularly parabolic or degenerate parabolic. The singular case has been studied somewhat (in the particular case of the affine normal flow). What happens in the degenerate parabolic case? Does a Grayson-type theorem hold? That is, do all embedded closed curves eventually become convex?

[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.

[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.

[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.

[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).

[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:

Can this method be used to obtain a diffeomorphism version of the sphere theorem, rather than just a homeomorphism version? This would probably require some smart choice of parametrisation of the hypersurfaces as they contract.
An intriguing feature of the proof is the following: In order to show that convexity is preserved as the hypersurfaces evolve, and to control the ratio of the principal curvatures at each point, I am forced to use a speed which becomes degenerate when any principal curvature approaches zero. This means a lot of work would be needed to get the result without assuming strict convexity of the hypersurface. Is there some flow which will allow us to assume merely that the principal curvature are everywhere non-negative, and that there is some point where they are all positive? This would have a number of useful geometric applications.

[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:

How important is the homogeneity of the speed? It seems that one could take higher powers of curvature and still have a reasonable parabolic equation, but the methods used in this paper just don't seem to give good answers.
How important is the concavity? Although the regularity theory seems to require some kind of concavity/convexity condition, the maximum principle arguments used in this paper can be used if the speed is either convex or concave, so it doesn't seem to matter either way! Maybe the concavity isn't needed in this step at all. Could we also avoid the concavity requirement in the regularity theory?