[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:
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?
[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 diffeomorphicversion 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.
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:
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?
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.