Posted by: matheuscmss | August 15, 2012

## Energy of links and Freedman-He-Wang’s conjecture after Ian Agol, Fernando Coda Marques and Andre Neves

The solution of a long-standing conjecture normally provokes a lot of excitement in the mathematical community because it usually brings a considerable amount of new ideas together with the answer of the previously open problem.

As it turns out, the solution of the Willmore conjecture by Fernando Marques and André Neves adds one more example to the general “philosophical” statement above. More precisely, by using the new ideas (min-max theory) used in the solution of Willmore’s conjecture, Ian Agol, Fernando Marques and André Neves were able to solve a conjecture of M. Freedman, Z-.X. He and Z. Wang (made in this article here) that the so-called Möbius energy of (non-trivial) links in ${\mathbb{R}^3}$ is minimized by the (stereographical projection of the) Hopf link.

Evidently, the title of this post hints that we’ll say (below the fold) a few words on the article of I. Agol, André and Fernando in what follows, but before entering into let me just take the opportunity to congratulate Fernando for the Ramanujan prize 2012 and UMALCA prize 2012 in recognition for his several contributions to Differential Geometry.

1. Energy of a link in ${\mathbb{R}^3}$ and Freedman-He-Wang’s conjecture

A link ${(\gamma_1,\gamma_2)}$ (or, more precisely, a 2-component link) is a pair ${\gamma_i:S^1\rightarrow\mathbb{R}^3}$, ${i=1,2}$, of closed simple rectifiable curves such that ${\gamma_1(S^1)\cap \gamma_2(S^1)=\emptyset}$.

The simplest example of a link is a trivial link, that is, any link isotopic to the link consisting of the pair of circles ${C_0:=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1, z=0\}}$ and ${C_1:=\{(x,y,z)\in\mathbb{R}^3:(x-3)^2+y^2=1, z=0\}}$ shown in the picture below (extracted from the corresponding Wikipedia article):

The second simplest example of a link is the Hopf link ${(h_1,h_2)}$ obtained by stereographic projection to ${\mathbb{R}^3}$ of the standard Hopf link

$\displaystyle \gamma_1(s)=(\cos s, \sin s, 0, 0)\in S^3\subset\mathbb{R}^4 \quad \textrm{and} \quad \gamma_2(t)=(0,0,\cos t,\sin t)\in S^3\subset\mathbb{R}^4$

Pictorially, the Hopf link has the following aspect:

From this figure (extracted from the blog Sketches of Topology) it is intuitively “clear” that the Hopf link is not a trivial link. Of course, there are several ways of proving this fact and a very efficient one uses the so-called linking number, an integer-valued invariant ${l(\gamma_1,\gamma_2)}$ of the isotopy class of a link ${(\gamma_1,\gamma_2)}$ measuring how much ${\gamma_1}$ and ${\gamma_2}$ “wind” around each other. Indeed, by direct computation, one can show that the linking number of the trivial knot is ${0}$ while the linking number of the Hopf link is ${\pm1}$ (depending on how one orients ${(h_1,h_2)}$), so that they can’t live in the same isotopy class. For more informations about the linking number, we recommend the Wikipedia page quoted above, where it is mentioned that Gauss’ original definition of the linking number was in terms of the following integral:

$\displaystyle l(\gamma_1,\gamma_2):=\frac{1}{4\pi} \int_{(s,t)\in S^1\times S^1} \frac{\det(\gamma_1'(s),\gamma_2'(t),\gamma_1(s)-\gamma_2(t))}{|\gamma_1(s)-\gamma_2(t)|^3} \,ds\,dt$

In connection with the physical study of dynamics and extremal configurations of charged loops, Jun O’Hara introduced a family of “energies” for knots and links, and, among them, a particularly interesting member of the family is the Möbius energy ${E(\gamma_1,\gamma_2)}$ of a link ${(\gamma_1,\gamma_2)}$:

$\displaystyle E(\gamma_1,\gamma_2):=\int_{(s,t)\in S^1\times S^1}\frac{|\gamma_1'(s)|\cdot|\gamma_2'(t)|}{|\gamma_1(s)-\gamma_2(t)|^2}\,ds\,dt.$

It was discovered by M. Freedman, Z.-X. He and Z. Wang that the Möbius energy has the remarkable property of invariance under conformal transformations of ${\mathbb{R}^3}$.

Note that the Möbius energy relates to (Gauss’ integral definition of) the linking number by the following inequality:

$\displaystyle E(\gamma_1,\gamma_2)\geq 4\pi |l(\gamma_1,\gamma_2)|$

By considering the pair of circles, ${C_0:=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1, z=0\}}$ and ${D_n=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1, z=n\}}$ as ${n\rightarrow\infty}$, we see that

$\displaystyle E(C_0,D_n)\leq 1/n^2.$

In other words, the Möbius energy of a trivial link can be arbitrarily small.

On the other hand, since a non-trivial link ${(\gamma_1,\gamma_2)}$ satisfies ${l(\gamma_1,\gamma_2)\in\mathbb{Z}-\{0\}}$, we deduce that

$\displaystyle E(\gamma_1,\gamma_2)\geq 4\pi|l(\gamma_1,\gamma_2)|\geq 4\pi$

for all non-triviallink ${(\gamma_1,\gamma_2)}$. For sake of comparison, note that, by direct computation, one can check that the Möbius energy of the Hopf link is ${E(h_1,h_2)=2\pi^2>4\pi}$.

Evidently, this motivates the search for the optimal (minimizing) configuration for the Möbius energy among all non-trivial links. In this direction, it was conjectured by M. Freedman, Z.-X. He and Z. Wang in 1994 that:

Freedman-He-Wang conjecture. The Hopf link ${(h_1,h_2)}$ is the optimal configuration for the Möbius energy among non-trivial links in the following sense:

• ${E(\gamma_1,\gamma_2)\geq E(h_1,h_2)=2\pi^2}$ for all non-trivial link ${(\gamma_1,\gamma_2)}$ and
• any non-trivial link ${(\gamma_1,\gamma_2)}$ with ${E(\gamma_1,\gamma_2)=E(h_1,h_2)=2\pi^2}$ is the Hopf link up to conformal transformations.

In 2002, Z.-X. He showed the existence of optimal configurations ${(m_1,m_2)}$ for the Möbius energy, i.e., the existence of non-trivial links ${(m_1,m_2)}$ such that

$\displaystyle E(m_1,m_2)=\inf\{E(\gamma_1,\gamma_2): (\gamma_1,\gamma_2) \textrm{ is a non-trivial link}\}$

and, furthermore, any optimal configuration ${(m_1,m_2)}$ is isotopic to the Hopf link. In particular, this reducesthe Freedman-He-Wang conjecture to the study of the Möbius energy of non-trivial links ${(\gamma_1,\gamma_2)}$ with linking number ${l(\gamma_1,\gamma_2)\in\{-1,+1\}}$.

In this direction, Ian Agol, Fernando Codá Marques and André Neves answered Freedman-He-Wang conjecture by establishing the following result (cf. Main Theorem 1.1 of their preprint):

Theorem 1 (I. Agol, F. Marques, A. Neves (2012)) Let ${(\gamma_1,\gamma_2)}$ be a non-trivial link in ${\mathbb{R}^3}$ with linking number ${l(\gamma_1,\gamma_2)\in\{-1,+1\}}$. Then, the Möbius energy of ${(\gamma_1,\gamma_2)}$ satisfies ${E(\gamma_1,\gamma_2)\geq 2\pi^2}$. Moreover, if ${E(\gamma_1,\gamma_2)=2\pi^2}$, then ${(\gamma_1,\gamma_2)}$ is the standard Hopf link up to conformal transformations.

The attentive reader of this blog may notice that the “numerology” in Freedman-He-Wang’s conjecture superficially resembles the one of Willmore conjecture: indeed, one has in both conjectures an “energy” (Möbius/Willmore) whose lower bound on “non-trivial objects” (non-trivial links/non-zero genera compact surfaces) is ${4\pi}$ from the “general theory”, but one wishes to show that the actual “optimal configuration” is an specific object (Hopf link/Clifford torus) with energy ${2\pi^2}$. In particular, it is not hard to imagine that the arguments of Agol, André and Fernando have something to do with the work of André and Fernando on Willmore’s conjecture.

In fact, the basic strategy of Agol, André and Fernando is the same used for the Willmore conjecture (as described, e.g., in this previous post here): one wishes to associated to a link ${(\gamma_1,\gamma_2)}$ with ${|l(\gamma_1,\gamma_2)|=1}$ a “continuous” ${5}$-parameter family of “surfaces” (integral ${2}$-currents with zero boundary) in ${S^3}$, that is, a “continuous” map ${\Phi}$ from ${[0,1]^5=[0,1]^4\times [0,1]}$ to the space ${\mathcal{Z}_2(S^3)}$ of “surfaces” in ${S^3}$ such that

• ${\Phi(x,0)=\Phi(x,1)=0}$ is the “trivial surface” for any ${x\in [0,1]^4}$,
• ${\Phi(x,t)}$ is an oriented round sphere in ${S^3}$ for any ${x\in\partial[0,1]^4}$, ${t\in [0,1]}$,
• for each ${x\in\partial[0,1]^4}$, the ${1}$-parameter family ${\{\Phi(x,t)\}_{t\in[0,1]}}$ of surfaces in ${S^3}$ is a standard sweepout of ${S^3}$, i.e., ${\Phi(x,t)=\partial B_{r(t)}(h(x))}$ where ${h(x)\in S^3}$, ${B_r(p)\subset S^3}$ is the standard ball of radius ${r}$ centered at ${p\in S^3}$ and ${r(t)}$ is a decreasing function of ${t\in[0,1]}$ with ${r(0)=\pi}$, ${r(1/2)=\pi/2}$ and ${r(1)=0}$,
• ${\sup\{\textrm{area}(\Phi(x,t)): (x,t)\in[0,1]^5\}\leq E(\gamma_1,\gamma_2)}$.

Once one disposes of such a ${\Phi}$, the min-max theory (i.e., Theorem 4 in this post hereand its variants) of André and Fernando applies to show the existence of a closed minimal surface ${\Sigma\subset S^3}$ of genus ${g\geq 1}$ with

$\displaystyle \textrm{area}(\Sigma)\leq\sup\{\textrm{area}(\Phi(x,t)): (x,t)\in[0,1]^5\}$

and this completes the proof of the first statement of Theorem 1 because André and Fernando showed (cf., e.g., Theorem 3 in this post here) that any closed minimal surface ${\Sigma\subset S^3}$ of genus ${g\geq 1}$ has area ${2\pi^2\leq \textrm{area}(\Sigma)}$, so that

$\displaystyle 2\pi^2\leq \textrm{area}(\Sigma)\leq\sup\{\textrm{area}(\Phi(x,t)): (x,t)\in[0,1]^5\}\leq E(\gamma_1,\gamma_2)$

However, the implementation of the strategy (or more precisely, the construction of the family ${\Phi}$) by Agol, André and Fernando is technically different from the case of the Willmore conjecture.

Thus, the plan for the rest of this post is to highlight some points in the construction of Agol-Marques-Neves family ${\Phi}$ and its application to the proof of (the first statement of) Theorem 1.

2. Agol-Marques-Neves canonical family of a link

For sake of convenience, we will think of our links ${(\gamma_1,\gamma_2)}$ inside ${\mathbb{R}^4}$: indeed, by starting with a link in ${\mathbb{R}^3}$, we can use the stereographic projection to see it inside ${S^3\subset\mathbb{R}^4}$. In principle, this procedure seems artificial, but it will facilitate our task of constructing the family ${\Phi}$ of “surfaces” in ${S^3}$ (adapted to the min-max theory of André and Fernando).

Given a link ${(\gamma_1,\gamma_2)}$ in ${\mathbb{R}^4}$, its Gauss map ${g=G(\gamma_1,\gamma_2):S^1\times S^1\rightarrow S^3}$ is

$\displaystyle g(s,t):=\frac{\gamma_1(s)-\gamma_2(t)}{|\gamma_1(s)-\gamma_2(t)|}$

In the concrete case of the Hopf link ${(h_1,h_2)}$, some of the vectors ${g(s,t)}$ can be “visualized” by looking at unit vectors inside the segments forming the ruled surface (Möbius band) shown in the picture below (also from the blog Sketches of Topology):

In general, the surface ${C=g(S^1\times S^1)\subset S^3}$ is naturally related to the Möbius energy of the link ${(\gamma_1,\gamma_2)}$ via:

$\displaystyle \textrm{area}(C)\leq E(\gamma_1,\gamma_2)$

In fact, as it is shown in Lemma 2.1 of the article of Agol, André and Fernando, a direct computation reveals that the Jacobianof ${g}$ satisfies the estimate:

$\displaystyle |\textrm{Jac}\, g(s,t)|\leq \frac{|\gamma_1'(s)|\cdot|\gamma_2'(t)|}{|\gamma_1(s)-\gamma_2(t)|^2}$

Therefore,

$\displaystyle \begin{array}{rcl} \textrm{area}(C)&=&\int_{(s,t)\in S^1\times S^1}\textrm{Jac}\, g(s,t) \,ds\,dt\leq \int_{(s,t)\in S^1\times S^1}|\textrm{Jac}\, g(s,t)| \,ds\,dt \\ &\leq& \int_{(s,t)\in S^1\times S^1}\frac{|\gamma_1'(s)|\cdot|\gamma_2'(t)|}{|\gamma_1(s)-\gamma_2(t)|^2} \,ds\,dt:= E(\gamma_1,\gamma_2) \end{array}$

Also, for later use, let us remark that (as it is also shown in Lemma 2.1 of the article of Agol, André and Fernando) ${C=\partial B_{\pi/2}(p)\subset S^3}$ whenever the link ${(\gamma_1,\gamma_2)}$ is contained in an affine hyperplane of ${\mathbb{R}^4}$ with normal vector ${p\in S^3}$. Actually, a little bit more is true: ${C=l(\gamma_1,\gamma_2)\cdot \partial B_{\pi/2}(p)}$ in the sense that ${C}$ must be thought (as an integral ${2}$-current) as the set ${\partial B_{\pi/2}(p)}$ with multiplicity ${l(\gamma_1,\gamma_2)}$.

At this point, we dispose of a single surface ${C=G(\gamma_1,\gamma_2)(S^1\times S^1)\subset S^3}$ naturally related to the link ${(\gamma_1,\gamma_2)}$. Evidently, this is still far from the end of the argument because we need a ${5}$-parameter family ${\Phi}$.

Evidently, we can gain ${4}$ parameters by applying conformal transformations to ${C}$ by hoping that this is a harmless due to the conformal invariance of Möbius energy. Formally, one proceeds as follows: given ${v\in\mathbb{R}^4}$, consider the conformal transformation ${F_v:\mathbb{R}^4-\{v\}\rightarrow\mathbb{R}^4}$,

$\displaystyle F_v(x)=(x-v)/|x-v|^2$

For later use, observe that if ${v\in B^4:=\{w\in\mathbb{R}^4: |w|<1\}}$, then ${F_v(S^3_1(0))=S^3_{1/(1-|v|^2)}(c(v))}$ where ${c(v)=v/(1-|v|^2)}$ and ${S^3_r(x):=\{y\in\mathbb{R}^4: |y-x|=r\}}$. Then, one consider the surfaces ${G(F_v\circ \gamma_1,F_v\circ\gamma_2)(S^1\times S^1)}$ obtained from the image of the Gauss map of the links ${(F_v\circ \gamma_1,F_v\circ\gamma_2)}$ derived from ${(\gamma_1,\gamma_2)}$ via conformal transformations ${F_v}$ with ${v\notin\gamma_1(S^1)\cup\gamma_2(S^1)}$.

Now, we can add an extra parameter without affecting the area bound by the Möbius energy by conveniently dilating the curve ${F_v\circ \gamma_2}$. Concretely, given ${w\in\mathbb{R}^4}$ and ${\lambda\in\mathbb{R}}$, let ${D_{w,\lambda}(x):=\lambda(x-w)+w}$ the dilation by ${\lambda}$ centered at ${w}$. Recall that our initial link ${(\gamma_1,\gamma_2)}$ in ${\mathbb{R}^3}$ can be seen inside ${S^3\subset \mathbb{R}^4}$ (via stereographic projection). In this situation, for each ${v\in B^4}$ and ${z\in (0,1)}$, we define

$\displaystyle C(v,z):=g_{(v,z)}(S^1\times S^1)$

where ${g_{(v,z)}=G(F_v\circ\gamma_1, D_{(c(v), a(v,z))}\circ F_v\circ\gamma_2)}$ is the Gauss map of the link

$\displaystyle (F_v\circ\gamma_1, D_{(c(v), a(v,z))}\circ F_v\circ\gamma_2)$

with ${a(v,z)=1+(1-|v|^2) b(v,z)}$ and ${b(v,z)=\frac{2z-1}{(1-|v|^2+z)(1-z)}}$.

Remark 1 Actually, ${C(v,z)}$ must be thought as an integral ${2}$-current, i.e., it is not simply the surface ${g_{(v,z)}(S^1\times S^1)}$, but, as “usual”, let’s ignore this technical issue by thinking of integral ${2}$-currents as surfaces.

Concerning the definition of ${C(v,z)}$, the following comments may help why it was defined in this way. Firstly, for each ${v\in B^4}$, ${(0,1)\ni z\mapsto a(v,z)\in (0,\infty)}$ is a non-decreasing reparametrization of ${(0,\infty)}$ such that ${a(v,1/2)=1}$. In particular, we have that ${g_{(v,1/2)}=G(F_v\circ\gamma_1, F_v\circ\gamma_2)}$ and, a fortiori, ${C(v,1/2)}$ is the “surface” associated to the link ${(F_v\circ\gamma_1, F_v\circ\gamma_2)}$. Furthermore, it is shown in Lemma 2.5 of Agol, André and Fernando article that the area bound by the Möbius energy is not affected, i.e.,

$\displaystyle \textrm{area}(C(v,z))\leq \int_{(s,t)\in S^1\times S^1}|\textrm{Jac}\, g_{(v,z)}|\,ds\,dt\leq E(\gamma_1,\gamma_2)$

Finally, ${C(v,z)}$ is a continuous family of “surfaces”.

By analogy with the work of André and Fernando, the family ${C(v,z)}$ is called the canonical family of the link ${(\gamma_1,\gamma_2)}$.

At this stage, we are happy because ${C(v,z)}$, ${(v,z)\in B^4\times (0,1)}$, is a continuous ${5}$-parameter family of ”surfaces” whose areas are controlled by the Möbius energy of the link ${(\gamma_1,\gamma_2)}$. However, this is not sufficient to apply the min-max theory of André and Fernando because of the non-compactness of the parameter space ${B^4\times (0,1)}$ (compare with Subsection 2.1 of this post).

Recall that, in the context of the Willmore conjecture, one could not extend the canonical family (denoted ${\Sigma_{(v,t)}}$) to the closure ${\overline{B}^4\times [0,1]}$ of ${B^4\times (0,1)}$ because of an issue of “angle of convergence” (cf. Subsection 2.2 of this post), and one was obliged to perform a blowup procedure (cf. Subsection 2.3 of this post) to get around this difficulty. Fortunately, in the context of Freedman-He-Wang conjecture, there is no angle of convergence issue and, as it is shown in Proposition 3.1 of the article of Agol, André and Fernando, the canonical family ${C(v,z)}$ can be continuously extended to ${\overline{B}^4\times [0,1]}$ and this extension satisfies:

• ${C(v,0)=C(v,1)=0}$ is the trivial surface for all ${v\in\overline{B}^4}$,
• if ${v\in S^3-(\gamma_1(S^1)\cup \gamma_2(S^1))}$ and ${0, then ${C(v,z)=g_{(v,z)}(S^1\times S^1)}$
• for every ${v\in S^3}$, ${C(v,1/2)=\partial B_{\pi/2}(v)}$ (or, more precisely, as an integral ${2}$-current, ${C(v,1/2)}$ is ${\partial B_{\pi/2}(v)}$ with multiplicity ${l(\gamma_1,\gamma_2)}$).

Here, the optimistic reader may think that we are really lucky that ${C(v,z)}$ extends without any blowup procedure and hence it remains only to apply the min-max theory. However, a careful inspection of the article of André and Fernando (or alternatively the second item of the list properties of the canonical family after blowup in Subsection 2.3 of this post) reveals that it is important to know that the map ${S^3\times [0,1]\ni (v,z)\mapsto C(v,z)}$ is a (non-trivial) map in the set of (oriented) geodesic spheres of ${S^3}$, and, unfortunately, this is not the case for ${C(v,z)}$.

On the other hand, this technical problem is not hard to overcome. In Proposition 4.1 of their article, Agol, André and Fernando show that ${C(v,z)}$ is confined in specific hemispheres of ${S^3}$: more concretely, they show that ${C(v,z)\subset \overline{B}_{\pi/2}(v)-B_{r(z)}(v)}$ if ${z\in[1/2,1]}$ and ${C(v,z)\subset \overline{B}_{\pi/2}(-v) - B_{\pi-r(z)}(-v)}$ if ${z\in[0,1/2]}$, where ${r(z):=\arccos(b(z)/\sqrt{|b(z)|^2+c^2})}$ and ${c>0}$ is an adequate constant. In particular, if we give ourselves a little room by extending the definition of the canonical family to ${\overline{B}_2^4\times [0,1]}$, we can use the extra place ${\overline{B}_2^4-B_1^4=\{x\in\mathbb{R}^4:1\leq|x|\leq 2\}}$ between ${B_1^4}$ and ${\overline{B}_2^4}$ to apply a retraction to ${C(v,z)}$ in order to convert it into a geodesic sphere. Formally, one proceeds as follows. For ${v\in S^3}$, ${z\in[0,1]}$ and ${\lambda\in[0,\pi/2]}$, consider the retraction

$\displaystyle R_{(v,\lambda,t)}: \overline{B}_{\pi/2}(v) - B_{\lambda}(v)\rightarrow \overline{B}_{\pi/2}(v) - B_{\lambda}(v)$

given by

$\displaystyle R_{(v,\lambda,t)}(x):=\exp_v\left(\left(1-t)+\frac{\lambda}{d(x,v)}t\right)\cdot \exp_v^{-1}(x)\right)$

where ${\exp}$ is the usual exponential mapand ${d(.,.)}$ is the standard (round) metric of ${S^3}$. Some of the main features of ${R_{(v,\lambda,t)}(x)}$ are listed below:

• ${R_{(v,\lambda,0)}}$ is the identity map ${R_{(v,\lambda,0)}(x)=x}$ for all ${x\in\overline{B}_{\pi/2}(v) - B_{\lambda}(v)}$;
• the restriction of ${R_{(v,\pi/2,t)}}$ to ${\partial B_{\pi/2}(v)}$ is the identity map for all ${v\in S^3}$ and ${z\in [0,1]}$,
• ${R_{(v,\lambda,1)}(\overline{B}_{\pi/2}(v) - B_{\lambda}(v))\subset \partial B_{\lambda}(v)}$.

Intuitively, these properties mean that, by fixing ${v}$ and ${\lambda}$, the map ${R_{(v,\lambda,t)}}$ start to concentrate the wholeregion ${\overline{B}_{\pi/2}(v) - B_{\lambda}(v)}$ near ${\partial B_{\lambda}(v)}$ as the parameter ${t\in[0,1]}$ approaches ${t=1}$. Geometrically, this is expressed by the following picture

Thus, if we can “insert” the retractions ${R_{(v,r(z),z)}}$ in the extension of the canonical family ${C(v,z)}$ to ${\overline{B}_2^4\times [0,1]}$, then we are in good shape because ${R_{(v,r(z),1)}}$ certainly will “convert” ${C(v,z)\subset \overline{B}_{\pi/2}(v)-B_{r(z)}(v)}$ into the geodesic sphere ${\partial B_{r(z)}(v)}$.

Concretely, one defines the extension of ${C(v,z)}$ from ${\overline{B}_1^4\times [0,1]}$ to ${\overline{B}_2^4\times [0,1]}$ as follows: for ${(v,z)}$ with ${1\leq |v|\leq 2}$ and ${z\in[0,1]}$, we let

$\displaystyle C(v,z) = \left\{\begin{array}{cc} R_{(v/|v|,r(z),|v|-1)}(C(v/|v|,z)) & \textrm{if } z\in[1/2,1] \\ R_{(-v/|v|,\pi-r(z),|v|-1)}(C(v,z)) & \textrm{if } z\in [0,1/2]\end{array}\right.$

The (new) canonical family ${C(v,z)}$ is continuous: indeed, since ${R_{(v,\lambda,z)}}$ depends continuously on ${v,\lambda,z}$, ${C(v,z)}$ is continuous on ${\{(v,z): 1<|v|\leq 2, z\in[0,1]\}}$, and, more importantly, the definitions of ${C(v,z)}$ on ${\overline{B}_1^4\times[0,1]}$ and ${(\overline{B}_2^4-B_1^4)\times[0,1]}$ “glue” together in a continuous way at ${S^3\times [0,1]}$ because ${R_{(v/|v|,\lambda,|v|-1)}=R_{(v,\lambda,0)}}$ is the identity for all ${v\in S^3}$ and ${\lambda\in[0,\pi/2]}$.

Also, the restriction of the (new) canonical family ${C(v,z)}$ to ${\partial B_2^4\times[0,1]}$ is a (non-trivial) map to the set of (oriented) geodesic spheres of ${S^3}$ because, for ${v\in\partial B_2^4}$ (i.e., ${|v|=2}$),

$\displaystyle C(v,z)=R_{(v/|v|,r(z),|v|-1)}(C(v/|v|,z))=R_{(v/|v|,r(z),1)}C(v/|v|,z)=\partial B_{r(z)}(v)$

if ${z\in[1/2,1]}$, and

$\displaystyle C(v,z)=R_{(v/|v|,r(z),|v|-1)}(C(v/|v|,z))=R_{(-v/|v|,\pi-r(z),1)}C(v/|v|,z)= \partial B_{\pi-r(z)}(-v)$

if ${z\in[0,1/2]}$. Actually, in order to be completely honest, we must say that, for ${|v|=2}$, the integral ${2}$-current ${C(v,z)}$ is the set ${\partial B_{r(z)}(v)}$ or ${\partial B_{\pi-r(z)}(v)}$ with multiplicity ${l(\gamma_1,\gamma_2)}$.

Finally, during the process of extending ${C(v,z)}$, we have not disrupted the area bound by the Möbius energy: indeed, since the derivatives of the retractions ${R_{(v,\lambda,z)}}$ are linear maps of norm ${\leq1}$, one has

$\displaystyle \begin{array}{rcl} \textrm{area}(C(v,z))&:=&\textrm{area}(R_{(v/|v|,r(z) \textrm{ or }\pi-r(z),z)}(C(v/|v|,z)))\leq \textrm{area}(C(v/|v|,z)) \\ &\leq& E(\gamma_1,\gamma_2) \end{array}$

for ${1\leq |v|\leq 2}$ and ${z\in [0,1]}$.

Now, we are ready to use the min-max theory of André and Fernando to complete the proof of the first part of Theorem 1.

3. Sketch of proof of the first part of Theorem 1

Let ${(\gamma_1,\gamma_2)}$ a link in ${\mathbb{R}^3}$ with linking number ${l(\gamma_1,\gamma_2)=1}$. By using the stereographic projection, we may think of ${(\gamma_1,\gamma_2)}$ inside ${S^3\subset\mathbb{R}^4}$ and we can associate to ${(\gamma_1,\gamma_2)}$ the canonical family ${C(v,z)}$, ${(v,z)\in\overline{B}_2^4\times[0,1]}$.

In order to fit the notations of the min-max theory of André and Fernando, let us choose ${f:[0,1]^4\rightarrow\overline{B}_2^4}$ a orientation-preserving homeomorphism and let’s define ${\Phi:[0,1]^5\rightarrow\mathcal{Z}_2(S^3)}$ as ${\Phi(x,t)=C(f(x),t)}$ for ${(x,t)\in[0,1]^4\times[0,1]=[0,1]^5}$.

Since ${2\pi^2<8\pi}$, we can also assume (without loss of generality) that ${E(\gamma_1,\gamma_2)<8\pi}$. In this setting, we have that the continuous ${5}$-parameter family ${\Phi}$ satisfies the following properties:

• ${\Phi(x,0)=0=\Phi(x,1)}$ is the trivial surface for every ${x\in I^4}$,
• ${\sup\{\textrm{area}(\Phi(x,t)): (x,t)\in [0,1]^5\}\leq E(\gamma_1,\gamma_2)<8\pi}$,
• ${\Phi(x,t)=\partial B_{r(t)}(f(x)/|f(x)|)}$ for all ${x\in\partial I^4}$ and ${t\in[0,1]}$,
• by identifying the space ${\mathcal{T}}$ of (unoriented) totally geodesic spheres in ${S^3}$ with ${\mathbb{RP}^3}$ (via ${\partial B_{\pi/2}(v)\mapsto \pm v}$), the map ${\Psi}$ given by

$\displaystyle \partial [0,1]^4\times \{1/2\}\ni (x,1/2)\mapsto \Phi(x,1/2)=\partial B_{\pi/2}(f(x)/|f(x)|)\in \mathcal{T}$

satisfies ${0\neq \Psi_*(\partial [0,1]^4\times \{1/2\})=2\cdot l(\gamma_1,\gamma_2)=2\in H^3(\mathbb{RP}^3,\mathbb{Z})}$

Essentially from these conditions, it is possible to use the min-max theory of André and Fernando, or more precisely Corollary 9.2 of their article (see also Theorem 2 and the arguments in Section 4 of this post here) to deduce the existence of a embedded closed minimal surface ${\Sigma\subset S^3}$ of genus ${g\geq 1}$ such that

$\displaystyle \textrm{area}(\Sigma)\leq\sup\{\textrm{area}(\Phi(x,t)):(x,t)\in[0,1]^5\}$

Now, one completes the argument by noticing that André and Fernando showed (cf. Theorem 3 in this link here) that any such ${\Sigma\subset S^3}$ has ${\textrm{area}(\Sigma)\geq 2\pi^2}$, so that

$\displaystyle 2\pi^2\leq \textrm{area}(\Sigma)\leq \sup\{\textrm{area}(\Phi(x,t)):(x,t)\in[0,1]^5\}\leq E(\gamma_1,\gamma_2)$

Of course, this completes our (vague and intuitive) discussion of (part of) Theorem 1. For a more serious approach to this result, please see the original article of Agol, André and Fernando: indeed, it is relatively short (19 pages), well-written and the reader will find in it as much details as it is possible to give without entering too much in the min-max theory of André and Fernando.