This paper is in AMS-TeX, and contains space for
six hand-drawn figures, which
are NOT included in electronic form. For a hard copy of
the figures, send a message to the authors at
dav@math.utexas.edu or sadun@math.utexas.edu. The captions
for the six figures are:
Figure 1: M\"obius invariance of $dx\, dy/|x-y|^2$.
Figure 2: Inversion of a knot in $S_x$.
Figure 3: Two interpretations of the angle $\alpha$.
Figure 4: The cylinder.
Figure 5: The dimple.
Figure 6: The helix.
BODY
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\documentstyle{amsppt}
\magnification=\magstep1
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\pagewidth{5.3in}
\pageheight{7.0in}
\overfullrule=0pt
\CenteredTagsOnSplits
\TagsOnRight
\define\Mob{M\"obius }
\define\tx{\tilde x}
\define\ty{\tilde y}
\define\half{{1 \over 2}}
\define\real{{\Bbb R}}
\define\complex{{\Bbb C}}
\define\ep{\epsilon}
\topmatter
\title A Family of M\"obius Invariant 2-Knot Energies
\endtitle
\author David Auckly and Lorenzo Sadun \endauthor
\address{Department of Mathematics, University of Texas, Austin, Texas 78712}
\endaddress
\email sadun\@math.utexas.edu, dav\@math.utexas.edu \endemail
\thanks {This work is partially supported by an NSF Mathematical
Sciences Postdoctoral Fellowship}
\endthanks
\abstract
After reviewing energy functionals for 1-dimensional knots and links,
we define a family of \Mob invariant energy functionals $E_s$
for surfaces embedded in $\real^n$. These functionals are all finite
for smoothly embedded compact surfaces and infinite for self-intersecting
immersed surfaces. They treat disconnected surfaces and connected sums of
surfaces correctly. For sufficiently negative $s$, $E_s$ is not bounded
from below. For sufficiently positive $s$, the evidence to date suggests
that $E_s$ {\it is} bounded from below, but we have not yet found a proof.
We also discuss alternate methods of defining surface energy.
\endabstract
\endtopmatter
Several years ago Jun O'Hara [O1, O2, O3] defined a functional for
embeddings of
circles in $\real^n$ that he called the energy of a knot.
Bryson, Freedman, He and Wang [BFHW, FHW] proved that this
energy is \Mob invariant,
and bounded the complexity of knots by their energy. Since then
others, notably Doyle and Schramm [D,S], have modified and reformulated
O'Hara's energy. The formulations
are all essentially equivalent, but each exhibits a different
important property of the energy. In this paper
we generalize some of these ideas to 2 dimensions
and define a family of energy functionals for embedded surfaces.
These functionals are different from, and postdate, the surface energy defined
by Rob Kusner and John Sullivan, which is discussed elsewhere in these
proceedings [KS].
Section 1 is a review of the 1-dimensional energy. We begin with the
unregularized energy formula and discuss its properties. We then list
several ideas for regularization and show how they are essentially
equivalent. Each of these methods suggests a regularization in 2 dimensions,
most of which turn out {\it not} to be equivalent. The material in this
section is not original, but rather reflects past work by many authors
[O1, O2, O3, BFHW, FHW, D, S].
In section 2 we generalize O'Hara's original regularization to
2 dimensions. There is some freedom in the choice of regularization,
yielding a 1-parameter family of \Mob invariant surface energies $E_s$
that differ by multiples of the Willmore functional [W]. We compute
the energies of several examples, and establish that these energy
functionals have all but one of the desirable properties shared by
the 1-dimensional energy. The one uncertain property is boundedness,
which is discussed further in section 4.
In section 3 we consider generalizations of the other 1-dimensional
regularizations to 2 dimensions. Some approaches fail,
due to a log divergence that
has no analog in 1 dimension. One successful approach
gives the Kusner-Sullivan energy. Another gives an energy
similar to ours. We are presently exploring, with
Kusner and Sullivan, this last approach.
In section 4 we consider boundedness.
The primary difficulty with our energy functionals $E_s$
is that they are not
manifestly bounded from below. Indeed, for $s$ sufficiently
negative, $E_s$ is {\it not} bounded from below. However, for
$s$ sufficiently positive, it appears likely that $E_s$ is positive
semi-definite, although we have not yet found a proof.
We thank Peter Doyle, Rob Kusner, Dennis Roseman, Oded Schramm
and John Sullivan for freely sharing their ideas with us and
updating us on work in progress. We thank Rob Kusner,
Karen Uhlenbeck and Michael Freedman for useful face-to-face
discussions, and thank Boris Apanasov for suggesting this problem
to us. L.S. is partially supported by an NSF Mathematical
Sciences Postdoctoral Fellowship.
\medskip
\centerline{\bf 1. The 1-dimensional energy }
\medskip
Let $\gamma_{1,2}$ be 1-manifolds in $\real^n$, and consider the functional
$$\int\int_{\gamma_1 \times \gamma_2} {dx \; dy \over |x-y|^2},
\eqno (1.1) $$
where $dx$ and $dy$ are the arclength measures
on $\gamma_{1,2}$. This functional is finite if $\gamma_1$ and
$\gamma_2$ are compact and do not intersect, it diverges if they
do intersect, and it is invariant under \Mob transformations.
\vskip 1 in
%Insert Figure 1: \Mob invariance of $dx\,dy/|x-y|^2$.
\vskip 1.5 in
\vfill\eject
\Mob invariance is seen in figure 1. Recall that the \Mob group
is generated by inversions in spheres. Let $\tx$ and $\ty$ be the
images of $x$ and $y$ under inversion in the sphere S. Since
$|x|/|y| = |\ty|/|\tx|$, the triangles $Oxy$ and $O\ty\tx$ are similar,
so $|\tx - \ty|^2 /|\tx||\ty| = |x-y|^2/|x||y|$.
Since $|d\tx| = (|\tx|/|x|) |dx|$, the invariance of the integral follows.
Next we consider the case where $K=\gamma_1=\gamma_2$ is a smooth knot
or link, with one component possibly extending to the point at infinity.
The unregularized ``energy of the knot (link)'' is
$$E_0(K) = \int\int_{K \times K} |x-y|^{-2} dx \; dy.
\eqno(1.2) $$
This obviously diverges for any nonempty $K$. The problem is to find
a regularization of the functional, finite for smoothly embedded $K$,
that preserves the desirable formal properties of the unregularized
energy. In particular, the regularized energy should:
\item {1.} Be finite for sufficiently smooth knots and links,
but infinte for self-intersecting immersions.
\item {2.} Be \Mob invariant.
\item {3.} Be bounded from below.
\item {4.} Satisfy a {\it connected sum rule}. If $K_1$ and $K_2$
are open knots, then, in the limit of large separation, the energy
of $K_1 \# K_2$ should approach the sum of the energies of $K_1$ and $K_2$.
\item {5.} Have the {\it additive link property} (ALP). The regularization
should not affect the force between non-intersecting neighborhoods.
The energy of a link should equal the sum of the energy of the
components plus the {\it unregularized} cross terms, and the force
between two neighborhoods should be independent of whether they are
in the same link component or not.
\Mob invariance, boundedness, and the connected sum rule together
imply that the energy is minimized by straight lines, or equivalently
by circles. By scale invariance, a line must have energy zero.
If any open knot had energy less than a line, the energy of the
connected sum of that knot with itself $N$ times could be made
arbitrarily negative, violating boundedness. Therefore all knots
must have positive or zero energy.
The 1-dimensional regularizations of O'Hara, Schramm, and
Doyle and Schramm have all the above properties, up to unimportant
additive constants, and are all equivalent. However, in
generalizing to two dimensions it is easy to lose one or more of
these properties. The Kusner-Sullivan energy has the first four
properties, but not the ALP. Our energy,
described in section 2 below, has the ALP, but we have not yet
managed to prove that it is bounded.
\vfill\eject
\noindent {\bf O'Hara's counterterm regularization}
O'Hara [O1] was the first to define a sensible regularized energy.
Let
$$ V(x) = \lim_{\ep\to 0} \left (\int_ {| x-y |>\ep}
{dy \over | x-y |^2} \qquad -{2\over\ep}\right ).
\eqno (1.3)
$$
The {\it counterterm} $-2/\ep$ exactly cancels the divergence of the
integral, so the limiting potential is well-defined and finite.
We then define the energy of a knot to be
$$ E^{(c)}(K) = \int V(x) dx. \eqno(1.4) $$
Although O'Hara only defined this energy for closed knots,
the same definition works for open knots and for links.
Since the counterterm is local, the ALP is immediate. It is also
an easy calculation to see that the connected sum rule holds, with
$$E^{(c)}(K_1 \# K_2) = E^{(c)}(K_1) + E^{(c)}(K_2) + O(D^{-2}), \eqno (1.5)
$$
where $D$ is the separation between $K_1$ and $K_2$ in the connected sum.
\Mob invariance is, however, more subtle, and was first proven
by Bryson et al [BFHW] using a different regularization.
\noindent {\bf The arclength regularization}
O'Hara [O2, O3] also defined an energy for knots by subtracting a term
involving arclength,
$$E^{(a)}(K) = \int\int_{K \times K} \left (\| x-y \|^{-2} - D(x,y)^{-2}
\right )dx dy , \eqno(1.6) $$
where $D(x,y)$ is the arclength between $x$ and $y$. The beauty of this
regularization is that it is manifestly positive semi-definite, and
equals zero only for $K$ being a straight line.
For closed knots, this arclength-regularized energy
turns out to be 4 greater than the
counterterm energy, while for open knots it equals the counterterm energy.
To see this, hold $x$ fixed and integrate $dy/D(x,y)^{2}$ over
the region $|x-y|>\ep$.
$$ \int_{|x-y|>\ep} {D(x,y)^{-2}} dy = {2 \over \ep} - {4 \over l} + O(\ep),
\eqno (1.7) $$
where $l$ is the length of the knot. The $O(\ep)$ term comes
from the difference between cutting off the integral at
Euclidean distance $\ep$ and arclength $\ep$.
Taking the $\ep \to 0$ limit we get that
$$ V(x) = -4/l + \int \left (\| x-y \|^{-2} - D(x,y)^{-2}
\right )dy . \eqno (1.8)$$
Integrating over $x$ we find that $-4/l$ integrates to $-4$
if $l$ is finite and zero if $l$ is infinite.
Bryson et al [BFHW] proved that $E^{(a)}$ is \Mob invariant
up to this difference of 4. If 2 knots, $K_1$ and $K_2$, are
related by a \Mob transformation, then $E^{(a)}(K_1)=E^{(a)}(K_2)$
if both knots are open or both are closed, while
$E^{(a)}(K_1)=E^{(a)}(K_2) - 4$ if $K_1$ is open and $K_2$ is closed.
Or to put it another way, $E^{(c)}(K_1)=E^{(c)}(K_2)$ in {\it all} cases.
For links, Freedman, He and Wang define the link energy to
be the sum of the energies of the components plus the
unregularized cross-terms. For a link $L$ with $n_c$ closed
components, $E^{(a)}(L) = E^{(c)}(L) + 4 n_c$. So even though
the $E^{(a)}$ energy formula makes a distinction between points
on the same or different link components, $E^{(a)}$ does have the ALP.
\noindent {\bf Schramm's wasted length regularization}
Oded Schramm [S] discovered an interesting geometrical interpretation
of O'Hara's local potential $V(x)$. Fix $x$, let $S_x$ be the unit
sphere centered at $x$, let $I_x$ be the inversion in $S_x$,
and let $\ty=I_x(y)$ be the image of $y$.
Then $|dy|/|x-y|^2 = |I_x^*d\ty|$, so $\int dy/|x-y|^2$
is just the length of the inverted curve.
This is shown in figure 2. The points on $K$ a distance $\ep$
from $x$ are $p_1$ and $p_2$. The coordinates of $\tilde p_{1,2}$
are $x \pm (1/\ep){\vec t} + k {\vec n}/2 + O(\ep)$, where
${\vec t}$, ${\vec n}$, and $k$ are the unit tangent, principal normal
and curvature of the knot at $x$. The integral in (1.3) gives the
length of the inverted curve from $\tilde p_1$ to $\tilde p_2$,
while $2/\ep$ is, to
within $O(\ep)$, the straight-line distance from
$\tilde p_1$ to $\tilde p_2$. So, to within $O(\ep)$, $V_\ep(x)$
gives the slack, or ``wasted length'', in the inverted knot between
$\tilde p_1$ and $\tilde p_2$. $V(x)$ gives the total slack in the
inverted knot.
\vskip 1 in
%Insert Figure 2: Inversion of a knot in $S_x$.
\vskip 1.2 in
\noindent {\bf Doyle and Schramm's \Mob invariant approach}
Suppose for simplicity that the knot $K$ is tangent to the $x_1$ axis
at $x$. Then the slack in the inverted knot $\tilde y$ can be written as
$ \int |d\ty| - d\ty_1$, or equivalently as $\int |d\ty|(1-\cos(\alpha))$,
where $\alpha$ is the angle between the tangent vector at $\tilde y$ and
the $x_1$ axis. Since $|d\ty|$ can be expressed in terms of $|dy|$, we
can rewrite the entire energy functional as
$$E^{(m)}(K) = \int\int {(1-\cos(\alpha)) \over |x-y|^2}dx dy . \eqno(1.9)$$
The angle $\alpha$ can be understood in terms of the original knot $K$,
without recourse to inversions. Let $S_1$ be the unique circle,
tangent to $K$ at $x$, that passes through $y$.
$I_x(S_1)$ is the line through $\tilde y$ parallel
to the $x_1$ axis, which intersects $I_x(K)$ at angle $\alpha$.
Since \Mob transformations are conformal, the angle between $S_1$ and $K$
at $y$ is also $\alpha$. Alternatively, let $S_2$ be the circle,
tangent to $K$ at $x$ that passes through $y$. The angle between
$S_1$ and $S_2$, measured at either $x$ or $y$, is again $\alpha$.
\vskip 1 in
%Insert Figure 3: Two interpretations of the angle $\alpha$.
\vskip 1.2 in
The functional $E^{(m)}$ is manifestly \Mob invariant, positive
semi-definite, and minimized by circles and lines. The ALP is
less obvious, but it follows from the equality of $E^{(m)}$ and
$E^{(c)}$.
There are a few special cases where the equivalence of $E^{(m)}$
and $E^{(c)}$ breaks down. The two functionals are finite and
equal when the knot (or link) is embedded. They are both infinite
when there are self-intersections at isolated points. However, they
disagree on curves that wrap around themselves several times.
For example, let $L$ be two superimposed copies of the knot $K$.
If the two copies are given the same orientation,
$E^{(m)}(L)=4 E^{(m)}(K)$, while if the orientations disagree,
$E^{(m)}(L)=\infty$. By constrast, $E^{(c)}(L)=\infty$ in both cases.
The reason for the discrepancy is that, at points of self-intersection, the
effect of the $\cos(\alpha)$ term is to subtract $2(1 + \cos(\beta))/\ep$,
rather than $2/\ep$, where $\beta$ is the angle between the two
intersecting strands.
When self-intersection points are absent, or form a set of measure zero,
this does not matter, but for multiple copies of the same curve it makes
a great difference.
Doyle and Schramm [D,S] have proposed a family of \Mob invariant knot
energies. Given any function $f(\alpha)$ that goes to zero sufficiently
fast at $\alpha=0$, they define the functional
$$E^{(f)}(K)= \int \int {f(\alpha) |x-y|^{-2}} dx dy . \eqno(1.10)$$
This is manifestly \Mob invariant, and is positive semi-definite if $f$ is.
For $f \ne 1 - \cos(\alpha)$ this is NOT equivalent to the other energies,
and typically does not have the ALP.
\bigskip
\centerline{\bf 2. Energy of surfaces --- counterterm approach}
\bigskip
We are now ready to generalize the definition of energy to surfaces
embedded in $\real^n$. In this section we present our generalization
of O'Hara's counterterm approach, and define a
1-parameter family of surface energies. Generalizations of the other
1-dimensional regularizations will be considered in section 3.
The naive unregularized energy of a $k$-manifold $F$ is
$$\int_{F \times F}|x-y|^{-2k} d^kx d^ky, \eqno(2.1)$$
where $d^kx$ is the $k$-dimensional volume form on $F$.
Since \Mob transformations are conformal, $d^kx d^ky$ transforms
like the $k$-th power of $dx dy$, so (2.1) is formally \Mob invariant.
However, it diverges
like $\ep^{-k}$ at short distance. To get a sensible definition we
must remove this divergence.
In 1 dimension the divergence was $2/\ep$, regardless of the local
geometry, and was easy to subtract off. In 2 dimensions there is
a $\pi/\ep^2$ divergence that doesn't depend on local geometry, plus
a $\ln(\ep)$ divergence that does depend on local geometry. Natural
generalizations of some 1-dimensional approaches fail because they do
not cancel the log divergence. Kusner and Sullivan [KS] have defined
an energy that avoids these problems, but which sacrifices the ALP.
The challenge is to remove both divergences while preserving the ALP.
We begin by computing the local divergence more precisely. Let
$$V(\ep,x) = \int_{|x-y|>\ep} |x-y|^{-4} d^2y. \eqno(2.2)$$
For simplicity, take $x$ to be at the origin, and let the tangent plane
$T_xF$ be the $x_1$-$x_2$ plane. Let
$$ L = \left ( \matrix e & f \cr f & g \endmatrix
\right ) \eqno(2.3) $$
be the second fundamental form at $x$, where $e$, $f$, and $g$ are
vectors normal to the $x_1$-$x_2$ plane. Defining
$$\align A(\theta) = & e \cos^2(\theta)/2 + f \cos(\theta)\sin(\theta)
+ g \sin^2(\theta)/2 \\
B(\theta) = & |e|^2 \cos^2(\theta) + |f|^2
+ |g|^2 \sin^2(\theta) + 2f \cdot (e + g)\sin(\theta)\cos(\theta)
\tag 2.4 \endalign
$$
we see that, near $x$, the surface $F$ takes the form
$$ z = A(\theta) r^2 + O(r^3), \eqno(2.5)$$
where $z=(y_3, y_4, \ldots, y_n)$ and $(r,\theta)$ are polar
coordinates for $T_xF$. The area form is
$$ d^2y = \sqrt{1 + |\nabla z|^2} \; r dr d\theta =
r + B(\theta)r^3/2 + O(r^4) \; dr d\theta,
\eqno(2.6)$$
while the distance from the origin is
$$ \sqrt{r^2 + |z|^2} = r + |A(\theta)|^2 r^3/2 + O(r^4) \eqno(2.7) $$
and the distance $\ep$ cutoff is at
$$ r = r_\ep(\theta) = \ep - |A(\theta)|^2 \ep^3/2 + O(\ep^4). \eqno(2.8) $$
Combining (2.4) through (2.8) we get that
$$\align V(\ep,x) = & \int_{r>r_\ep} [r^2 + |A(\theta)|^2r^4 + O(r^5)]^{-2}
[r + B(\theta)r^3/2 + O(r^4)] dr d\theta \\
= & \int_{r>r_\ep} [r^{-3} + (B(\theta)/2 - 2|A(\theta)|^2)/r
+ O(1)] dr d\theta \\
= & \int_0^{2\pi} {1\over 2} r_\ep^{-2} -
(B(\theta)/2 - 2|A(\theta)|^2)\ln(r_\ep) + O(1) d\theta \\
= & \pi \ep^{-2} - (\pi/8) \Delta(x) \ln(\ep) + O(1), \tag 2.9 \endalign
$$
where
$$ \Delta(x) = (e + g)^2 - 4(e \cdot g - |f|^2) = (e-g)^2 + 4 |f|^2. \eqno(2.10) $$
$\Delta$ is a measure of how far a point is from being umbilical.
If the surface is in $\real^3$, with principal curvatures
$k_1$ and $k_2$, then $\Delta = (k_1-k_2)^2$.
\noindent{\bf The counterterm regularization}
To get a suitable regularization, we must find a local expression that has
the form (2.9) and transforms correctly under \Mob transformations,
and subtract it from $V(\ep,x)$. Note that $\ln(\ep)$ does {\it not}
transform correctly, since $\ep$ is a dimensionful quantity.
To get a dimensionless quantity, we must divide $\ep$ by a locally
determined length scale. The natural candidate is $\Delta^{-1/2}$,
though other choices (such as $2 \Delta^{-1/2}$) are equally valid.
We define the local potential energy
$$ V_0(x) = \lim_{\ep \to 0} V(\ep,x) - \pi\ep^{-2} + {\pi \Delta(x)\over 16}
\ln(\Delta(x)\ep^2) + {\pi K(x) \over 4}, \eqno(2.11) $$
where $K(x)= e \cdot g - |f|^2$ is the Gauss curvature at $x$, and define the
energy of a surface to be
$$ E_0(F) = \int_F V_0(x) d^2x. \eqno(2.12) $$
This is not the only reasonable choice. Had we taken the length scale
to be $2 \Delta^{-1/2}$ rather than $\Delta^{-1/2}$, we would have
gotten a potential that differs from $V$ by $\pi \ln(2) \Delta/8$.
Other length scales give potentials that differ by other factors of $\Delta$.
It therefore makes sense to consider the entire
1-parameter family of potentials
$ V_s(x) = V_0(x) + s \Delta(x)$
and the corresponding energies
$$E_s(F) = \int V_s(x) d^2 x = E_0(F) + s \int \Delta(x) d^2x \eqno(2.13) $$
The integral of $\Delta$ is closely related to the Willmore functional
$W(F)$, a well-known conformal invariant [W]. Specifically,
$$W(F) = \int_F H^2 d^2x = \int_F \left ( {1\over 4}\Delta + K
\right) d^2x, \eqno(2.14) $$
where $H=(e+g)/2$ is the mean curvature. The 2-form $\Delta \; d^2x$ is in
fact invariant under all conformal changes of the ambient metric, not
just under \Mob transformations, while $\int K$ is a topological invariant.
$W(F)$ is invariant under \Mob transformations that do not change
the Euler characteristic, but decreases by $4\pi$ when a point is sent
to infinity.
The $\Delta\ln(\Delta)$ term in the regularization is an object better
known to measure theorists than to topologists. Minus $\int\Delta
\ln(\Delta)d^2x$ is the {\it entropy} of the \Mob invariant measure $\Delta
d^2x$ relative to the area measure $d^2x$.
The integral (2.2), while well-defined, can be difficult to compute.
To aid computation, we define an equivalent energy using a
smooth cutoff function.
Let $\rho(r)$ be a function that approaches 1 as $r \to \infty$
and approaches zero rapidly as $r \to 0$. Define the regularized
potential
$$ V^\rho(x) = \lim_{\ep \to 0} \left ( \int {\rho(|x-y|/\ep) d^2y \over
|x-y|^4} - c(\rho) \pi\ep^{-2} + {\pi \Delta \over 16}
\ln(\Delta\ep^2) + {\pi \over 4}K\right ) , \eqno(2.15) $$
where $c(\rho) = \int_0^\infty r^{-2} \rho'(r) dr$. When $\rho$ is the
characteristic function of $[1,\infty]$, $V^\rho$ is the same as $V_0$,
while if $\rho=\chi_{[R,\infty]}$, $V^\rho$ differs from $V_0$ by $\pi
\Delta \ln(R)/8$. For a general $\rho$, $V^\rho=V_s$, where
$s = - {\pi\over 8} \int_0^\infty \rho'(r) \ln(r) dr$.
Varying the cutoff function $\rho$ is equivalent to varying $s$, and
we get exactly the same family of energies as before.
A particularly useful cutoff function is $\rho(r) = r^4/(1+r^2)^2$,
which reduces the integral in (2.15) to
$\int (\ep^2 + |x-y|^2)^{-2} d^2y$. This particular cutoff gives
$V_s$ with $s=-\pi/16$.
\noindent{\bf Examples}
\noindent{\it The plane.}
Before exploring the general properties of the energies $E_s$, we compute
a few examples. Our first and simplest example is the
plane. Here $K=\Delta=0$, and
$$V(\ep,x) - \pi/\ep^2 = -\pi/\ep^2 + \int_0^{2\pi}
\int_\ep^\infty r^{-4} r dr d\theta = 0.
\eqno(2.16) $$
Our energy is zero, as it must be by scale invariance.
\noindent{\it The sphere.}
The next example is the unit sphere, where $\Delta=0$ and $K=1$.
The potential $V(x)$ is constant by symmetry, and we compute it at
the north pole.
$$V(\ep,N) = 2\pi \int_{2\sin^{-1}(\ep)}^\pi
{\sin(\theta) d\theta \over (2 \sin(\theta/2))^4} = \pi/\ep^2
-\pi/4. \eqno(2.17) $$
Subtracting the counterterms we get $V_s(N)=0$, and hence $E_s(S^2)=0$,
reflecting the \Mob equivalence of the
sphere to the plane. Note the effect of the topological ${\pi K/4}$
counterterm in the definition of $V_s$. Had this term not been included,
the sphere would have had energy $-\pi^2$.
\noindent{\it The cylinder.}
Our third example is the right circular cylinder of radius 1.
In this example the two principal curvatures are unequal, and
$\Delta =1$. All points have the same potential,
and this is what we compute. Multiplying this potential by
$2\pi$ gives the energy per unit length of the cylinder.
\vskip 2 in
\phantom{Insert Figure 4: The cylinder}.
\vskip 0.9 in
The natural coordinates on the cylinder are $(\alpha,z)$,
where $\alpha$ is the angle around the circle and $z$ is height,
and we take our reference point $x$ to be $(0,0)$. It is also
useful to convert from $\alpha$ to $t=2\sin(\alpha/2)$, the
straight-line distance across the circle, as shown in figure 4.
By symmetry, it is sufficient to integrate over the shaded regions
in figure 4, and then to multiply the total by 4.
The integral over the lightly shaded region is
$$\align & \int_\ep^2 \int_0^\infty {2 dz dt \over \sqrt{4-t^2} (t^2+z^2)^2}
= {\pi \over 2} \int_\ep^2 {dt \over t^3 \sqrt{4-t^2}} \tag 2.18 \\
& \qquad = {\pi \over 32}
\left ( {2 \sqrt{4-\ep^2} \over \ep^2} + \ln\left ({2 + \sqrt{4-\ep^2}
\over \ep}\right )
\right ) = {\pi \over 32} \left ( 4\ep^{-2} \! + \! 2 \ln(2) \! - \! {1\over 2}
\! - \! \ln(\ep)
\! + \! O(\ep^2) \right ). \endalign $$
To integrate over the heavily shaded region we expand $2/\sqrt{4-t^2}
= 1 + t^2/8 + O(t^4)$. We also switch to polar coordinates
$r^2=t^2+z^2$, $\tan(\theta)=z/t$. Our integral is then, to within $O(\ep^2)$,
$$ \align \int_{-\pi/2}^0 \int_\ep^{\ep\sec(\theta)} \! \! \!
r^{-4} (1 + r^2 \cos^2(\theta)/8) r dr d\theta = &
\int_{-\pi/2}^0 \!\!\ep^{-2} \sin^2(\theta)/2 - \cos^2(\theta)
\ln(\cos(\theta))/8 d\theta \\
= & \pi \ep^{-2}/8 + \pi(2\ln(2) - 1)/64. \tag 2.19 \endalign $$
Adding these two expressions, multiplying by 4, subtracting the counterterms and taking the $\ep \to 0$ limit gives
$$V_s(x) = s + {\pi \over 8} (3 \ln(2) - 1). \eqno(2.20) $$
Note that, for $s$ sufficiently negative, the cylinder has negative
energy per unit length. By taking a very long cylinder with smooth
caps at the ends, we can create a smooth surface with arbitrarily
negative energy. As a result, for $s < -(3\ln(2) - 1)\pi/8$, the
energy functional $E_s$ is unbounded from below.
\noindent{\it The ``dimple''.}
Our final example shows that, for any $s$, there exist surfaces with points
where $V_s$ is negative. Let $d$ be a small parameter. Consider the surface
in $\complex^2$ given by
$$ z_1=r \exp(i\theta), \qquad \qquad
z_2(r,\theta) = \cases r^2 \exp(2i\theta) & r \le d \\
d^2 \exp(2i\theta) & r \ge d. \endcases
\eqno( 2.21) $$
See figure 5.
This, of course, is not a smooth surface, but it can be arbitrarily
well approximated by smooth surfaces. We compute the potential
$V_s(x)$ at $x=0$ for $d <\!< 1$.
The second fundamental form at 0 is
$\pmatrix 2 & 2i \\ 2i & -2 \endpmatrix$
and $\Delta(0)=32$.
\vfill\eject
\phantom{This is a blank line}
\vskip 1.5 in
The integral for $V(\ep,x)$ breaks into two pieces, one for $r>d$ and one for
$rd$ we have
$$\int_0^{2\pi} \int_d^\infty \sqrt{1 + 4 d^4/r^2}
(r^2+d^4)^{-2} r dr d\theta = \pi/d^2 + O(d^2). \eqno(2.22) $$
For $r\ep>|y-x|} - \int_{|y-x|>\ep>|I(y)-x|} \right )
|x-y|^{-4} d^2y \eqno(2.25) $$
plus $\pi/4$ times the change in $K$. By a power-series expansion,
this is seen to equal zero in all cases.
Boundedness is problematic. The example of the cylinder shows that
$E_s$ is {\it not\/} bounded from below for $s<-(3\ln(2) - 1)\pi/8$.
Furthermore, the example of the dimple shows that $V_s$ cannot be
bounded by local information, so a bound on $E_s$ cannot be achieved
by integrating
a bound on $V_s$. However, experimentation suggests that, for large $s$,
$E_s$ {\it is\/} positive semi-definite and is minimized by spheres.
This conjecture is discussed further in section 4.
The connected sum rule follows from \Mob invariance. First suppose
that $K_1$ and $K_2$ each contain neighborhoods that are exactly spherical.
By inverting about a sphere centered in this neighborhood we can assume
that $K_1$ and $K_2$ are exactly planar outside of compact regions $M_1$
and $M_2$, respectively, with finite areas $A_1$ and $A_2$. We take
$K_1 \# K_2$ to be a plane outside of two regions, isometric to $M_1$
and $M_2$, that are separated by a distance $D$. An easy calculation
then shows that
$$|E_s(K_1 \# K_2) - E_s(K_1) - E_s(K_2)| \le A_1 A_2/D^4. \eqno(2.26) $$
Taking $D\to\infty$ gives the connected sum rule for surfaces with
spherical neighborhoods. But since {\it any} surface can be given a small
spherical neighborhood at an arbitrarily small cost in energy,
the connected sum rule holds for all surfaces. (This is essentially
the same argument as found in [K], where a connected sum rule for
the Willmore functional is proven).
Finally, $E_s$ manifestly has the ALP, for the same reason that O'Hara's
$E^{(c)}$ does. Namely, the regularization at $x$ is
independent of whether additional link components are present or not.
\vfill\eject
\centerline{\bf 3. Alternative approaches to surface energy}
\bigskip
In this section we consider generalizations of other 1-dimensional
regularizations. Several natural ideas simply do not work, thanks
to the $\ln(\ep)$ divergence in $V(\ep,x)$. One approach gives
functionals similar to the counterterm energies of section 2.
Another approach gives the Kusner-Sullivan ``first-order'' energy,
which is discussed elsewhere in these proceedings [KS].
\noindent{\bf Arclength regularization}
There are several possible ways to generalize the 1-dimensional
arclength formula (1.7). The most obvious is to define
$$ E(F) = \int_{F\times F} (|x-y|^{-4} - D(x,y)^{-4}) d^2x d^2y, \eqno(3.1)
$$
but this expression is divergent. At short distance $|x-y|^{-4}$
equals $r^{-4} - 2|A(\theta)|^2 r^{-2} + O(r^{-1})$, while
$D(x,y)^{-4}= r^{-4} - 8 |A(\theta)|^2 r^{-2}/3 + O(r^{-1})$. Unless
the second fundamental form is identically zero, the difference between
these two expressions is $O(r^{-2})$, and is not integrable.
A different generalization is
$$ E(F) = \int_{F\times F} (|x-y|^{-2} - D(x,y)^{-2})^2 d^2x d^2y. \eqno(3.2)
$$
This is finite, as $|x-y|^{-2}-D(x,y)^{-2}=O(1)$, but it is not
\Mob invariant. Also, as a practical matter, it is nearly impossible to
compute, since just finding the shortest path between two distant points
is a difficult endeavor.
In one dimension, arclength was an easy quantity to work with, in that there
were only two paths between any two points to choose from, and in
that arclength served as a canonical parametrization for any curve.
In two dimensions, arclength is neither calculable nor canonical, and
does not appear to provide a useful regularization.
The closest thing to a canonical parametrization in 2 dimensions is
a map that preserves conformal structure. Kusner has suggested an energy
based on this structure. If $f:\real^k \to \real^n$ is
a conformal embedding, he defines
$$E(f) = \int_{\real^k \times \real^k} {|f'(z)|^k|f'(w)|^k \over
|f(z)-f(w)|^{2k}} - {1\over |z-w|^{2k}} d^2z d^2w. \eqno(3.3) $$
This energy is an invariant of the surface, since if $g:\real^k \to \real^n$
is a conformal map with the same image as $f$, then $E(g)=E(f)$.
In one dimension this expression is well-defined, and gives yet
another description of O'Hara's energy $E^{(c)}$. Kusner and
Sullivan have applied this approach to graphs, giving a relative
energy for distinct embeddings of the same graph.
Unfortunately, in 2 dimensions this energy has a log divergence
at non-umbilical points.
For example, consider the map $f: \complex \to \complex^2$, $f(z)=(z,z^2)$.
Holding $w=0$ fixed and integrating over $z$, we see that
$|f'(z)|^2|f(z)-f(w)|^{-4}= |z|^{-4} + 2|z|^{-2} + O(1)$. The second term
in the integrand cancels the $|z|^{-4}$, but the remaining $2|z|^{-2}$ term
is not integrable.
\noindent{\bf Wasted area regularization}
As in the 1-dimensional case, let $S_x$ be the unit sphere centered at $x$,
let $I_x$ be inversion in this sphere, and let $\tilde y=I_x(y)$. Then
the 2-form $|x-y|^{-4} d^2y$ is the pullback, under $I_x$, of the area form
$d^2\tilde y$. As before, the area of $I_x(F)$ is infinite, as $I_x(F)$
extends to infinity. We try to regularize by taking the
excess area of $I_x(F)$ relative to some standard surface. The
question is, which surface?
Before considering reference surfaces, we investigate what $I_x(F)$ looks like
near infinity. Since $F$ takes the form (2.5) near zero, $I_x(F)$ has the
form
$$ z = A(\theta) + O(1/r), \qquad \partial z/\partial r =
O(1/r^2), \qquad \partial z/\partial \theta = d A/d\theta
+ O(1/r). \eqno(3.4)
$$
The area form is
$$ \sqrt{1 + |\nabla z|^2} \; r dr d\theta = (r + |A'(\theta)|^2/2r + O(r^{-2}))
dr d\theta, \eqno(3.5) $$
where $A'$ denotes $dA(\theta)/d\theta$. Integrating out to $r=R$ gives
$\pi R^2 + \pi\Delta\ln(R)/8 + O(1)$. These are precisely the
divergences we have already seen in (2.9), with $R=1/\ep$.
Our reference surface must have the same asymptotic area as $I_x(F)$. This
rules out the natural choice of the plane, whose area is merely
$\pi R^2$. Another possibility is to take the curve
$z=A(\theta), r=R$, fill it in
with a minimal surface, and take the area of that. Unfortunately,
the area of such a minimal surface is $\pi R^2 + O(1)$,
with no $\ln(R)$ term, so this approach also fails.
A reference surface that {\it does\/} work is the fan shape
$z(r,\theta) = A(\theta)$, which we denote $M_x$. For $r>d$, this is
the same as the ``dimple'' of figure 5.
The area form for $M_x$ is exactly
$\sqrt{1 + |A'|^2/r^2} \; r dr d\theta$, which has the form (3.5)
for large $r$. We can therefore define
$$ \align V^{(w)}(x)= & \hbox{Excess area of $I_x(F)$ relative to $M_x$} \\
E^{(w)}= & \int_F V^{(w)}(x) d^2x \tag 3.6 \endalign
$$
This definition is the product, and the subject, of a continuing
collaboration with Kusner and Sullivan.
Since $M_x$ is not a minimal surface, it is possible for $I_x(F)$ to
have less area than $M_x$, and hence for $V^{(w)}(x)$ to be negative.
Indeed, if $I_x(F)$ is a minimal surface for $rD$, then $V^{(w)}(x)= -\pi\Delta\ln(D)/8 + O(1)$.
(The ``dimple'' of section 2 is a surface of precisely this type,
with $D=1/d$). This shows that $V^{(w)}$, like $V_s$, cannot be
pointwise bounded from below.
$E^{(w)}$ is finite for smooth embeddings and infinite for immersions with
self-intersections. It is \Mob invariant, obeys a connected sum rule and
has the ALP. We will prove \Mob invariance directly, and then show that
$E^{(w)}-E_s$ is the integral of a local and \Mob invariant quantity.
This then establishes the other properties for
$E^{(w)}$, and shows that $E_s$ is \Mob invariant.
We examine what a \Mob transformation does to $I_x(F)$ and $M_x$. $\tilde x
= \infty$ is fixed, so the transformation can only act on $I_x(F)$ by rotation,
reflection, translation or scale change. We show that in all cases
$V^{(w)}(x) d^2x$ is unchanged.
A rotation (or reflection) of
$I_x(F)$ corresponds to a rotation (reflection) of $F$ around $x$, hence
a rotation (reflection) of the second fundamental form at $x$, hence a
rotation (reflection) of $M_x$. $I_x(F)$ and $M_x$ still match
asymptotically, and the difference in their areas is unchanged.
A translation of $I_x(F)$ corresponds to a \Mob transformation whose derivative
is the identity at $x$. Such a transformation can change the mean curvature
$H=(e+g)/2$, but cannot change $e-g$ or $f$.
A translation of $I_x(F)$ normal to the $x_1$-$x_2$ plane corresponds to a change in $H$, which in turn translates
$M_x$ the same amount as $I_x(F)$, leaving $V^{(w)}$ unchanged. A translation
in the $x_1$-$x_2$ plane does not change $H$, and so is {\it not\/} matched
by a translation of $M_x$. However, the difference in area form
between $M_x$ and its translate is absolutely integrable,
and integrates to zero, so the shift of $I_x(F)$ relative to $M_x$ does
not change $V^{(w)}(x)$.
Finally, rescaling $I_x(F)$ by a factor of $\lambda$ corresponds to
rescaling $F$ by $\lambda^{-1}$, rescaling the second fundamental form
at $x$ by $\lambda$, and so rescaling $M_x$ by $\lambda$. $V^{(w)}(x)$
gets rescaled by $\lambda^2$, while $d^2x$ gets rescaled by $\lambda^{-1}$,
leaving $V^{(w)}(x) d^2x$ unchanged. This completes the proof that
$V^{(w)} d^2x$, and therefore $E^{(w)}$,
is \Mob invariant.
Now we compare $E^{(w)}$ to $E_s$. We can compute $V^{(w)}$ by taking the area
of that part of $I_x(F)$ within distance $R$ of the origin, subtracting the
area of the corresponding part of $M_x$, and taking the limit as $R \to\infty$.
The points on $I_x(F)$ of distance less than $R$ from the origin correspond
to points on $F$ of distance greater than $\ep=1/R$ from $x$. This procedure
is therefore equivalent to a counterterm regularization, where the counterterm
is the area of $M_x$ within distance $R=1/\ep$ of the origin.
The distance cutoff is at $r=r_R = R - A(\theta)^2/(2R) + O(R^{-2})$.
The area is
$$\align
\int_0^{2\pi} & \int_0^{r_R} \sqrt{1 + |A'(\theta)|^2/r^2} r dr d\theta \\
& = \int_0^{2\pi} {1\over 2}\left ( r_R \sqrt{r_R^2 + |A'|^2}
+ |A'|^2\ln(r_R + \sqrt{r_R^2 + |A'|^2}) -
{1\over 2}|A'|^2\ln(|A'|^2) \right )d\theta \\
& = \int_0^{2\pi}{1\over 2}\left (r_R^2 + |A'|^2/2
+ |A'|^2\ln(2R) - {1\over 2}|A'|^2\ln(|A'|^2)
+ O(R^{-2}) \right )d\theta \\
& = \int_0^{2\pi} R^2/2 + |A'|^2/4 - |A(\theta)|^2/2
- |A'|^2\ln(|A'|^2/4R^2)/4 + O(R^{-2}) d\theta \\
& = \pi R^2 -\pi\Delta\ln(|\Delta/R^2)/16 -\pi K/4
+ \pi\Delta(10\ln(2)-1)/32 \\
& \qquad - {\Delta\over 32} \int_0^{2\pi}{8 |A'|^2 \over \Delta}\ln\left(
{8 |A'|^2 \over \Delta} \right ) d\theta +O(R^{-2}) \tag 3.7
\endalign $$
The first 3 terms match the $V_0$ counterterms. $V^{(w)}$ is therefore
equal to $V_s$,
with $s=-\pi(10 \ln(2)-1)/32$, plus $\Delta/32$ times the final
integral, which is the entropy of the $S^1$ measure
$|A'(\theta)|^2 d\theta$ relative to its average, $(\Delta/8) d\theta$.
This entropy is a locally defined \Mob invariant quantity that depends
on the relative size, and the angle between, $e-g$ and $f$. In $\real^3$
we can always choose coordinates such that $f=0$, and the integral is
$2 \pi(1-\ln(2))$. In 4 or more dimensions, the integral can take on any
value from $0$ to $2 \pi(1-\ln(2))$.
The fan shape $M_x$ is not the only reference surface that works. Given
any cutoff function $\rho(r)$, we can take the surface
$$ z(r,\theta) = A(\theta) \rho(r^2/A_0(\theta)^2), \eqno(3.8) $$
where
$A_0(\theta)=(e-g)\cos(2\theta)/4 + f\sin(2\theta)/2$ is the oscillatory part
of $A(\theta)$. This surface works just as well as $M_x$, and gives
an energy functional that differs from $E^{(w)}$ by local
terms such as $\int\Delta$ and the integral of the entropy of $8|A'|^2/\Delta$.
\noindent{\bf The manifestly \Mob invariant approach}
Earlier this year, Kusner and Sullivan [KS] generalized Doyle and Schramms's explicitly \Mob invariant approach to higher dimensions. Let $S_x$ be the sphere, tangent
to $F$ at $x$, that passes through $y$, and let $S_y$ be the sphere tangent
to $F$ at $y$ that passes through $x$. Let $\alpha(x,y)$ be the angle between
$S_x$ and $S_y$. More precisely, $\cos(\alpha)$ is the inner product of
the area form of $S_x$ and the area form of $F$ at $y$. Then, given
any reasonable function $f(\alpha)$ with $f(0)=0$ they write
$$E^{(f)}(F) = \int_{F\times F} f(\alpha) |x-y|^{-4}. \eqno(3.9) $$
This is manifestly \Mob invariant, and is finite for embedded surfaces as
long as $f \to 0$ sufficiently quickly as $\theta \to 0$. Kusner and
Sullivan concentrate in particular on $f(\alpha)=(1-\cos(\alpha))^2$
(or, in $k$ dimensions, $(1-\cos(\alpha))^k$).
This approach has several advantages over the local regularizations $E_s$, and
one major disadvantage. The advantages are that the energy is manifestly bounded from below, that it requires knowledge only of the tangent planes
at $x$ and $y$, not of the curvature, and that it easily generalizes to
higher dimensions.
The disadvantage is that it does not have the additive
link property. The force between distinct neighborhoods depends not only on
their separation but also on their orientation in space.
As such, it does not preserve
the long-distance behavior of the unregularized funtional (2.1).
To preserve the additive link property, one would have to use
$f(\alpha)=1-\cos(\alpha)$. This is equivalent to computing the wasted
area relative to a plane, or to using the local counterterm $\pi/\ep^2$, and
is log divergent at all non-umbilical points. To avoid log divergences, one
must either incorporate information about curvature into the regularization
(as with $E_s$) or sacrifice the additive link property (as in [KS]).
It is possible to incorporate information about the second derivative into
this approach. The integral
$$\int_{F\times F} |x-y|^{-4}\Big ( 1 - \cos(\alpha)
\sqrt{1 + |A'(\theta)|^2(r^2+|z|^2)^2/r^2}\Big ) d^2x d^2y \eqno(3.10)$$
is equivalent to $E^{(w)}$. Here $(r,\theta)$ are coordinates in the
tangent plane $T_xF$ and $z$ is the component of $y-x$ normal to this plane,
and $A'$ is related to the second fundamental form at $x$ as before.
The complicated second term is just the pullback, under $I_x$, of the
area form of $M_x$. There is an apparant singularity at points where
$r=0$, $z \ne 0$, but this singularity is integrable. Of course, by
introducing the second derivative we lose manifest boundedness, as the
integrand may be negative.
\medskip
\centerline{\bf 4. The question of boundedness}
\medskip
In this section we discuss the boundedness properties of the surface
energy functionals $E_s$ defined in section 2. We show that the
question may be reduced to evaluating a single number, which we call $s_c$.
We then tell what little is known about this number, and argue that $s_c$ is most likely finite.
\noindent {\bf The critical parameter}
Since the energy of a sphere is zero, for any given $s$ the functional $E_s$ is either
\item {a)} Unbounded from below,
\item {b)} Bounded from below, but not uniquely minimized by round spheres, or
\item {c)} Positive semi-definite, and uniquely minimized by round spheres.
We will prove that there is a (possibly infinite) critical value of $s$, which we call $s_c$, such that condition (a) holds for all $ss_c$. When $s=s_c$, the energy is positive semi-definite,
but it is not known whether $E_{s_c}(F)=0$ implies that $F$ is a round sphere (or a plane, which we consider a sphere of infinite radius).
First consider $W(F)=\int_F \Delta(x) d^2x$. Since
$\Delta = |e-g|^2 + 4|f|^2 \ge 0$,
$W(F)$ is never negative, and is zero only when $F$ is umbilical everywhere, i.e., a sphere. As a result, if $s_2>s_1$, then $E_{s_2}(F) \ge E_{s_1}(F)$,
with equality if and only if $F$ is a sphere.
Define
$$ s_c = \sup \{ s \; | \; E_s \hbox{ is unbounded from below.}\} \eqno(4.1) $$
If $ss_c$. This implies that $E_s$ is bounded from below. By
the connected sum rule, if $E_s(F)$ were negative for some $F$, we could get
$E_s(F\# F \# \ldots \# F)$
arbitrarily negative, and $E_s$ would be unbounded. Thus $E_s$ must be positive semi-definite. Similarly, $E_{(s+s_c)/2}$ is positive semi-definite.
If $F$ is not a round sphere, $E_s(F) > E_{(s+s_c)/2}(F) \ge 0=
E_s(S^2)$, so $E_s$ is
uniquely minimized by round spheres.
Finally, we consider $E_{s_c}$. For any smooth $F$, $W(F)$ is finite,
so $E_{s_c}(F) = \lim\limits_{s \to s_c^+} E_s(F) \ge 0$. This shows
that $E_{s_c}$ is positive semi-definite.
However, it is not clear whether $E_{s_c}(F)=0$ implies that $F$ is a
sphere.
The remaining task is to determine whether $s_c$ is finite or infinite, and
if finite, to estimate its value.
It is easy to see that $s_c > -\infty$. Let $F$ be any non-spherical surface.
Then $W(F)>0$ and, for any $s < - E_0(F)/W(F)$, $E_s(F)<0$. This shows that
$s_c \ge - E_0(F)/W(F)$. In particular, by looking at the cylinder we see
that $s_c \ge (1-3\ln(2))\pi/8$.
It is not known whether $s_c$ is finite or positive infinite. The evidence
collected to date suggests that $s_c$ is finite. However, no proof has been found, and the evidence is far from conclusive.
\noindent{\bf Evidence for Boundedness}
Suppose that $E_s$ is unbounded for all $s$. Then it should be possible
to find examples whose energies are extremely negative even for large values of $s$. However, all the wild examples we have constructed have had extremely
{\it positive\/} energies, as might be expected from a well-behaved and bounded functional. While this is hardly proof of boundedness for large $s$, it is certainly suggestive.
We present some of these examples here. We do so not so much to convince to
reader of boundedness as to build intuition. We hope that these examples will suggest methods by which boundedness might be proved, or by which true counterexamples might be constructed.
Our first example is the ellipsoid $x^2/a^2 + y^2/b^2 + z^2/c^2=1$ in
$\real^3$. If $a/b$ and $a/c$ are both close to 1, then both $W$ and
$E_0$ will be second order in the eccentricity, and
for reasonable values of $s$ our energy will be positive. To get candidates for negative energy we must take extreme values of $(a,b,c)$.
Suppose $a >\!> b,c$, so that our ellipsoid looks like a cigar. In that case most points will lie along the length of the cigar, which is roughly cylindrical. But we have already calculated the energy of the cylinder, and found it to be positive for $s$ positive. Increasing $a$ only increases the
length of the cigar and increases the energy.
Now suppose $a <\! < b,c$, so that our ellipsoid looks like a pancake. Most
points lie along a flat face of the pancake, with very high energy due to
the vicinity of the other face. Even along the rim, one curvature greatly exceeds the other, making the rim cylinder-like, again with positive energy.
Finally, if $a >\!> b>\!>c$, we have a cylinder whose cross-section is a highly
eccentric ellipse, again with extremely high energy.
A similar analysis applies to the Clifford torus $S^1 \times S^1 \in \complex^2$, with two
different radii. If the two radii are close, then $W$ and $E_0$ are comparable, and even for reasonably small $s$, $E_s$ is positive. If one of
the radii is much greater than the other, the neighborhood of each point resembles part of a long cylinder, and $E_0$ is very large and positive.
Next we consider perturbations of the plane. If such a perturbation has a large natural length scale, it can be rescaled to have length scale 1 and a very small amplitude. Both $W$ and $E_0$ would be quadratic in this amplitude, and so $E_s$ would be positive for $s$ bigger than the ratio of the leading-order coefficients. To get $E_s$ negative for large $s$ one has to take perturbations with very small length scales, or equivalently large amplitudes.
\vskip 1 in
%Insert Figure 6: The helix.
\vskip 1.5 in
One such example is the plane wave $(u,v, \cos(ku), \sin(ku)) \in \real^4$ for
large $k$. This is the helix $(u, \cos(ku),\sin(ku)) \in \real^3$ crossed with
$\real$. See figure 6. As $k$ increases, the loops of the helix get closer and closer. In the $k \to \infty$ limit, the contributions to $V(p)$ from the
single loop highlighted in figure 6, minus the counterterms, approach the (finite) energy of the cylinder, while the other loops give more and more positive energy. As a result, $\lim_{k\to\infty} V_s(p) = + \infty$ for all $s$.
Perturbations with point peaks do no better. Consider the surface
$z=c\exp(-(x^2+y^2)) \in \real^3$. For large $c$, the sides of the ``mountain'' get very steep and resemble the sides of a cylinder, whose potential we know to be positive. The base of the mountain (the region of
slope $\sim 1$) may have negative potential, but its area grows only as $\ln(c)$, while the area of the side grows as $c$. So again, as $c \to
\infty$, the energy of the surface goes to $+ \infty$.
This by no means exhausts the supply of potential counterexamples.
However, it shows that extreme behavior tends to generate extreme
positive energy, not negative energy. In particular, when features get
stretched they tend to locally resemble planes or cylinders. Planes
get positive energy from the presence of other nearly planes (as
in the pancake example), while cylinders have positive energy for
$s >\!> (1-3\ln(2))\pi/8$. This limiting behavior suggests
that $s_c$ is not only finite, but is fairly small.
There are two more classes of examples that should be noted, which we have not yet computed. One is where the perturbation is peaked at small rings, rather than points or lines. An example of this is the surface
$z_2 = c z_1^2 \exp(-|z_1|^2) \in \complex^2$. The other class is surfaces with many length scales.
\bigskip
\centerline{\bf 5. Conclusions}
\bigskip
We have defined a family of \Mob invariant energy functionals $E_s$
for surfaces embedded in $\real^n$. These functionals are all finite
for smoothly embedded compact surfaces and infinite for self-intersecting
immersed surfaces. They treat disconnected surfaces and connected sums of
surfaces correctly.
For a functional to be useful, it must be bounded from
below. For $s< (1-3\ln(2))\pi/8$, $E_s$ is not bounded
from below. However, for $s$ sufficiently
large and positive, the evidence to date suggests
that $E_s$ {\it is} bounded from below, although we have not yet found a proof.
\vfill\eject
\refstyle{A}
\widestnumber\key{BFHW}
\Refs
\ref\key BFHW \by S. Bryson, M. Freedman, Z.-X. He and Z. Wang
\paper M\"obius Invariance of Knot Energy
\jour Bull. Amer. Math. Soc \vol 28 \pages 99--103 \yr 1993 \endref
\ref\key D \by P. Doyle \finalinfo private communication
\endref
\ref\key FHW \by M. Freedman, Z.-X. He and Z. Wang
\paper M\"obius Energy of Knots and Unknots
\jour Ann. Math. \finalinfo in press\endref
\ref\key K \by R. Kusner \paper
Comparison Surfaces for the Willmore Problem
\jour Pacific J. Math
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\enddocument
\bye