on min-max theory and the Willmore conjecture

Overview

Marques and Neves proved the Willmore conjecture: every embedded torus in ${\mathbb{R}}^3$ has Willmore energy at least

$$2{\pi }^2,$$

with equality only for stereographic images of the Clifford torus (Marques & Neves, 2014; Willmore, 1965). Their proof is not a direct Euler-Lagrange analysis of the Willmore functional. It replaces the surface by a canonical five-parameter sweepout of $S^3$, proves that the sweepout has min-max width strictly larger than the area of a round sphere, and then identifies the resulting minimal surface by a low-index rigidity theorem.

The paper is important because it joins three mechanisms that had mostly lived in separate parts of geometry: conformal invariance of the Willmore functional, Almgren-Pitts min-max theory for minimal hypersurfaces, and a topological degree carried by the genus of the original surface. The proof shows how an analytic inequality can be forced by topology after the right family of competitors is built.

Willmore Energy and the Spherical Formulation

Let $\Sigma \subset {\mathbb{R}}^3$ be a smooth closed immersed surface, and let

$$H=\frac{k_1+k_2}{2}$$

be the scalar mean curvature, with $k_1,k_2$ the principal curvatures. The Euclidean Willmore energy is

$${\mathcal{W}}_{{\mathbb{R}}^3}(\Sigma )={\int }_{\Sigma }{H}^{2}dA.$$

The round sphere is the elementary normalization. A sphere of radius $r$ has $H=1/r$ and area $4\pi r^2$, hence

$${\mathcal{W}}_{{\mathbb{R}}^3}(S_r^2)=4\pi .$$

Willmore’s conjecture asserted that if $\Sigma$ is an embedded torus, then

$${\mathcal{W}}_{{\mathbb{R}}^3}(\Sigma )\ge 2{\pi }^2.$$

The number $2{\pi }^2$ is forced by the model example. It is the area of the Clifford torus in the round three-sphere.

The proof is most naturally written in $S^3$. If

$$\Sigma \subset S^3$$

is a smooth closed surface with mean curvature $H$ computed inside $S^3$, Marques and Neves use the spherical Willmore energy

$$\mathcal{W}(\Sigma )={\int }_{\Sigma }(1+H^2)dA.$$

This convention is compatible with stereographic projection. If

$$\pi :S^3\setminus \{p\}\to {\mathbb{R}}^3$$

is stereographic projection and $\tilde{\Sigma }=\pi (\Sigma )$, then

$${\int }_{\tilde{\Sigma }}{\tilde{H}}^{2}d\tilde{A}={\int }_{\Sigma }(1+H^2)dA.$$

This identity is one expression of the conformal invariance of the Willmore functional (White, 1973; Li & Yau, 1982). Thus the Euclidean torus problem can be moved to $S^3$ without changing the relevant energy.

The Clifford torus is

$${\mathbb{T}}_{\mathrm{Cl}}=S^1\left(\frac{1}{\sqrt{2}}\right)\times S^1\left(\frac{1}{\sqrt{2}}\right)\subset S^3\subset {\mathbb{R}}^4.$$

It is minimal in $S^3$, so $H=0$. Its area is the product of the two circle lengths:

$$\mathrm{area}({\mathbb{T}}_{\mathrm{Cl}})={\left(\frac{2\pi }{\sqrt{2}}\right)}^2=2{\pi }^2.$$

Therefore

$$\mathcal{W}({\mathbb{T}}_{\mathrm{Cl}})=2{\pi }^2.$$

Stereographic projection gives the usual torus of revolution in ${\mathbb{R}}^3$ with major radius $\sqrt{2}$ and minor radius $1$.

Embeddedness and Li-Yau

The embeddedness hypothesis is essential. Li and Yau proved that if an immersion covers some point with multiplicity at least $k$, then its Willmore energy is at least $4\pi k$ (Li & Yau, 1982). A nonembedded immersion has a double point, hence energy at least $8\pi$, and

$$8\pi >2{\pi }^2.$$

Thus the essential remaining case is the embedded case below the $8\pi$ threshold.

Principal Theorems

Marques and Neves prove the following spherical form of the conjecture.

Willmore bound in $S^3$

Let $\Sigma \subset S^3$ be an embedded closed surface of genus $g\ge 1$. Then

$$\mathcal{W}(\Sigma )\ge 2{\pi }^2.$$

Equality holds if and only if $\Sigma$ is the image of the Clifford torus under a conformal transformation of $S^3$ (Marques & Neves, 2014).

A second theorem, proved first, is a sharp area lower bound for minimal surfaces.

Minimal surface area bound

Let $\Sigma \subset S^3$ be an embedded closed minimal surface of genus $g\ge 1$. Then

$$\mathrm{area}(\Sigma )\ge 2{\pi }^2.$$

Equality holds if and only if $\Sigma$ is the Clifford torus up to an isometry of $S^3$ (Marques & Neves, 2014).

The bridge between the two statements is a min-max theorem attached to any embedded positive-genus surface. Given such a surface $\Sigma$, Marques and Neves construct a homotopy class $\Pi$ of five-parameter sweepouts whose width satisfies

$$4\pi <\mathbf{L}(\Pi )\le \mathcal{W}(\Sigma ).$$

Almgren-Pitts theory then produces an embedded minimal surface whose area is the width, possibly with multiplicity. Under the energy threshold relevant to the proof, this surface has multiplicity one and positive genus. The minimal area theorem then gives the Willmore inequality.

The entire proof is organized around making the displayed inequality rigorous.

Almgren-Pitts Width

A sweepout should be thought of as a family of surfaces that begins with the empty surface, fills the ambient manifold, and ends again with the empty boundary. In geometric measure theory the correct objects are integral currents and varifolds, because min-max limits can develop singular behavior before regularity theory is applied.

Let

$${\mathcal{Z}}_2(S^3)$$

denote the space of integral $2$-cycles in $S^3$, equipped first with the flat topology. A parameterized family is a map

$$\Phi :I^m\to {\mathcal{Z}}_2(S^3).$$

The mass

$$\mathbf{M}(\Phi (x))$$

is the area of the current, counted with multiplicity. If $\Pi$ is the homotopy class of $\Phi$ relative to its boundary data, the width is

$$\mathbf{L}(\Pi )=\underset{\Psi \in \Pi }{\inf }\underset{x\in I^m}{\sup }\mathbf{M}(\Psi (x)).$$

This is the geometric analogue of a mountain-pass value: every representative of the class must cross a surface of area at least $\mathbf{L}(\Pi )$.

The Almgren-Pitts theorem says, in the form used here, that if

$$\mathbf{L}(\Pi )>\underset{x\in \partial I^m}{\sup }\mathbf{M}(\Phi (x)),$$

then the width is realized by a smooth embedded minimal surface in $S^3$, possibly with integer multiplicity (Almgren, 1962; Pitts, 1981; Simon, 1983). The strict inequality over the boundary is essential: otherwise the min-max value could be achieved by a trivial boundary slice rather than by an interior critical point.

Several technical points are suppressed in this summary. The natural families constructed from a smooth surface are continuous in the flat topology, but flat convergence can lose area by cancellation. Almgren-Pitts discretization and interpolation replace the continuous family by maps on cubical complexes that are controlled in mass. The proof also needs a no-concentration condition: area cannot disappear into arbitrarily small balls during interpolation. These issues are not cosmetic. Without them the topological degree carried by the family would not be stable under the limiting procedure.

The Canonical Family

Let

$$B^4=\{v\in {\mathbb{R}}^4:|v|<1\}$$

be the unit ball whose boundary is $S^3$. For $v\in B^4$, define the conformal transformation

$$F_v:S^3\to S^3,F_v(x)=\frac{1-|v{|}^2}{|x-v{|}^2}(x-v)-v.$$

When $v\ne 0$, this is a conformal dilation of $S^3$ fixing the two antipodal points $v/|v|$ and $-v/|v|$. For an oriented embedded surface

$$\Sigma \subset S^3,$$

write

$${\Sigma }_v=F_v(\Sigma ).$$

Let $d_v$ be the signed distance function to ${\Sigma }_v$ in $S^3$. The associated parallel surface is

$${\Sigma }_{(v,t)}=\partial \{x\in S^3:d_v(x)<t\},(v,t)\in B^4\times [-\pi ,\pi ].$$

At $t=0$ this is the conformal image ${\Sigma }_v$. At $t=\pm \pi$ the boundary is empty as a cycle. Thus the parameter $v$ gives four conformal directions, while $t$ gives one normal-distance direction. This is the five-dimensional family.

The decisive estimate is

$$\mathrm{area}({\Sigma }_{(v,t)})\le \mathcal{W}({\Sigma }_v)=\mathcal{W}(\Sigma )$$

for all $(v,t)$ (Ros, 1999; Heintze & Karcher, 1978; Marques & Neves, 2014). The equality on the right is conformal invariance. The inequality on the left is a Heintze-Karcher type comparison, in the form used by Ros: the areas of all parallel boundaries of a conformal image are controlled by the Willmore energy of the original surface.

This estimate is the analytic reason the construction proves an inequality. If the canonical family has nontrivial width, then its maximal area cannot be pushed below a certain min-max value; but the same maximal area is bounded above by $\mathcal{W}(\Sigma )$.

Boundary Blow-Up and Continuity

The family

$$(v,t)\mapsto {\Sigma }_{(v,t)}$$

is not continuous if one simply lets $v$ approach $\partial B^4=S^3$. The singular behavior occurs when $v$ approaches a point of the original surface $\Sigma$. The limit then depends not only on the boundary point but also on the angle at which $v$ approaches it.

More concretely, if $p\in \Sigma$ and $N(p)$ is the chosen unit normal, a sequence of parameters may approach $p$ in the form

$$v_n=|v_n|(\cos (s_n)p+\sin (s_n)N(p)),|v_n|\to 1,s_n\to 0.$$

The limiting surface depends on the ratio between $s_n$ and $1-|v_n|$. Different approach angles produce different geodesic spheres. Thus the raw compactification $B^4\times [-\pi ,\pi ]$ forgets data needed for continuity.

Marques and Neves resolve this by blowing up ${\overline{B}}^4$ along $\Sigma$. The blow-up remembers the missing angular variable. After this modification they construct a continuous map

$$C:{\overline{B}}^4\times [-\pi ,\pi ]\to {\mathcal{Z}}_2(S^3)$$

with the following properties:

• $C(v,t)={\Sigma }_{(v,t)}$ away from the blown-up boundary region,
• $C(v,\pi )=C(v,-\pi )=0$ as a cycle,
• for boundary parameters $v\in S^3$, the cycle $C(v,t)$ is a geodesic sphere,
• the mass estimate remains valid:

$$\mathbf{M}(C(v,t))\le \mathcal{W}(\Sigma ).$$

The construction also produces a continuous center map

$$\overline{Q}:S^3\to S^3.$$

For each boundary point $v\in S^3$, there is a special value $s(v)$ for which

$$C(v,s(v))$$

is a great sphere centered at $\overline{Q}(v)$. The unoriented space of great spheres in $S^3$ is

$$\mathcal{T}\simeq {\mathbb{RP}}^3,$$

because a great sphere is determined by its unit normal vector up to sign.

This blow-up step is one of the main conceptual contributions of the paper. It turns a discontinuous conformal degeneration into a continuous family whose boundary is geometrically simple: only round spheres appear there.

The Genus Degree

The center map is not arbitrary. Marques and Neves prove

$$\mathrm{deg}(\overline{Q})=g,$$

where $g$ is the genus of $\Sigma$ (Marques & Neves, 2014). This is the point at which the topology of the original surface enters the min-max argument.

The computation is explicit. Near $\Sigma$, use coordinates

$$G(p,t)=\cos tp+\sin tN(p),p\in \Sigma .$$

On the blown-up collar the center map has a model form involving both $p$ and $N(p)$. If $k_1,k_2$ are the principal curvatures, the Jacobian contribution contains

$$\left(-\frac{t}{\epsilon }+\frac{\sqrt{{\epsilon }^2-t^2}}{\epsilon }{k}_1\right)\left(-\frac{t}{\epsilon }+\frac{\sqrt{{\epsilon }^2-t^2}}{\epsilon }{k}_2\right)\frac{-1}{\sqrt{{\epsilon }^2-t^2}}.$$

After integration in $t$, the cross term vanishes and the Gauss equation

$$K=1+k_1{k}_2$$

converts the remaining curvature integral into Euler characteristic. The collar contributes

$$-{\pi }^2\chi (\Sigma )={\pi }^2(2g-2).$$

The two complementary regions of $S^3$ contribute total volume

$$\mathrm{vol}(S^3)=2{\pi }^2.$$

Thus

$${\int }_{S^3}{\overline{Q}}^{\ast }(dV)=2{\pi }^2+{\pi }^2(2g-2)=2{\pi }^{2}g=g{\int }_{S^3}dV.$$

This proves $\mathrm{deg}(\overline{Q})=g$.

The passage to unoriented great spheres doubles this degree. If

$$\pi :S^3\to \mathcal{T}\simeq {\mathbb{RP}}^3$$

sends a unit vector to the corresponding unoriented great sphere, then the relevant boundary slice of the min-max family represents

$$2g\in H_3({\mathbb{RP}}^3,\mathbb{Z}).$$

For $g\ge 1$ this class is nonzero. The sweepout therefore carries a homological obstruction that cannot be removed by a homotopy preserving the boundary conditions.

The Five-Parameter Sweepout

After choosing a homeomorphism from the cube $I^4$ to the blown-up four-ball, the continuous family is written as

$$\Phi :I^5\to {\mathcal{Z}}_2(S^3).$$

Its properties are the operative input to the width argument:

• $\Phi$ is continuous in the flat topology,
• $\Phi (I^4\times \{0\})=\Phi (I^4\times \{1\})=0$,
• ${\sup }_{I^5}\mathbf{M}(\Phi )\le \mathcal{W}(\Sigma )$,
• ${\max }_{\partial I^5}\mathbf{M}(\Phi )=4\pi$,
• if a boundary slice has mass $4\pi$, then it is a great sphere,
• on the middle boundary slice $\partial I^4\times \{1/2\}$, the induced map to $\mathcal{T}\simeq {\mathbb{RP}}^3$ has homology class $2g$.

The number $4\pi$ appears because every boundary surface is a round sphere or the zero cycle, and the largest round spheres in $S^3$ are great spheres of area $4\pi$. Positive genus is detected because the middle boundary slice winds nontrivially through the space of great spheres.

Thus the family has a simple boundary and a nontrivial interior. This is exactly the configuration where min-max theory is effective.

The Width Gap

The central topological estimate is

$$\mathbf{L}(\Pi )>4\pi$$

for the homotopy class $\Pi$ associated with a positive-genus surface. The boundary already has maximum mass $4\pi$, so the statement says that any representative of the class must pass strictly above the largest boundary sphere.

The proof is by contradiction. Suppose there were representatives

$${\Phi }_i\in \Pi$$

with

$$\underset{I^5}{\sup }\mathbf{M}({\Phi }_i)\le 4\pi +o(1).$$

Near mass $4\pi$, the only possible limiting boundary objects relevant to the argument are great spheres. This uses compactness, the structure of the boundary family, and the fact that round spheres are isolated at the maximal boundary area.

One then considers the region of the parameter cube lying below the great-sphere transition. Its boundary contains a hypersurface $R(i)$ separating the bottom face $I^4\times \{0\}$ from the top face $I^4\times \{1\}$. On $R(i)$ the family can be projected, up to small error, to the space

$$\mathcal{T}\simeq {\mathbb{RP}}^3$$

of great spheres. In the rigorous proof, these regions are replaced by cubical singular chains to make the homological conclusion compatible with the discrete Almgren-Pitts setup.

The boundary of $R(i)$ includes the nontrivial middle boundary slice

$$\partial I^4\times \{1/2\}.$$

If the near-minimizing representatives existed, the projected chain in $\mathcal{T}$ would fill that slice. Therefore the class

$$2g\in H_3({\mathbb{RP}}^3,\mathbb{Z})$$

would vanish. This contradicts the degree computation. Hence the width is strictly larger than $4\pi$.

This argument is a useful template: the lower bound for a variational quantity comes not from estimating every surface in the family, but from proving that the family cannot be homotoped into the low-energy region without killing a homology class.

Low-Index Rigidity

The minimal surface area theorem is proved by applying the canonical family to a least-area positive-genus minimal surface. Existence of such a minimizer is proved by geometric measure theory compactness and regularity arguments; Simon’s lectures provide background for the compactness framework, and his work on Willmore minimization belongs to the surrounding existence theory for related variational problems (Simon, 1983; Simon, 1993; Marques & Neves, 2014).

Let $\Sigma \subset S^3$ be an embedded minimal surface of least area among all embedded minimal surfaces of genus at least one. The Clifford torus is an admissible competitor, so

$$\mathrm{area}(\Sigma )\le 2{\pi }^2.$$

The claim is that the Morse index of $\Sigma$ is at most five. The number five is not accidental: it is the dimension of the canonical family.

If the index were greater than five, then there would be enough independent negative directions for the area functional to perturb the five-parameter family near its maximal slice and lower the maximum area. The perturbed family has the same boundary data and the same positive-genus topological obstruction, so the width remains larger than $4\pi$. Almgren-Pitts theory would then produce an embedded minimal surface $\hat{\Sigma }$ with

$$4\pi <\mathrm{area}(\hat{\Sigma })<\mathrm{area}(\Sigma ).$$

Multiplicity must be one: every embedded minimal surface in $S^3$ has area at least $4\pi$, so a multiplicity at least two would have area at least $8\pi$, while $\mathrm{area}(\Sigma )\le 2{\pi }^2<8\pi$. The resulting surface also has positive genus, because the only embedded minimal two-spheres in $S^3$ are great spheres of area $4\pi$ (Almgren, 1966). This contradicts the least-area choice of $\Sigma$.

Therefore

$$\mathrm{index}(\Sigma )\le 5.$$

Urbano’s theorem classifies closed minimal surfaces in $S^3$ with low index: the totally geodesic sphere has index one, and the Clifford torus is the only non-spherical example with index at most five (Urbano, 1990). Since $\Sigma$ has positive genus, it must be the Clifford torus. This proves

$$\mathrm{area}(\Sigma )\ge 2{\pi }^2$$

for every embedded minimal surface of positive genus, with equality only for the Clifford torus.

Completion of the Willmore Inequality

Now let $\Sigma \subset S^3$ be any embedded closed surface of genus at least one. If

$$\mathcal{W}(\Sigma )\ge 8\pi ,$$

then the desired inequality is immediate because

$$8\pi >2{\pi }^2.$$

Thus assume

$$\mathcal{W}(\Sigma )<8\pi .$$

Construct the canonical five-parameter family and let $\Pi$ be its homotopy class. The previous sections give

$$4\pi <\mathbf{L}(\Pi )\le \underset{I^5}{\sup }\mathbf{M}(\Phi )\le \mathcal{W}(\Sigma ).$$

By Almgren-Pitts theory, $\mathbf{L}(\Pi )$ is realized by an embedded minimal surface $\hat{\Sigma }$, possibly with multiplicity:

$$\mathrm{area}(\hat{\Sigma })=\mathbf{L}(\Pi ).$$

The same threshold argument as above gives multiplicity one. Since the area is strictly larger than $4\pi$, the surface is not a great sphere and therefore has positive genus (Almgren, 1966). The minimal surface area theorem applies:

$$2{\pi }^2\le \mathrm{area}(\hat{\Sigma })=\mathbf{L}(\Pi )\le \mathcal{W}(\Sigma ).$$

This proves the Willmore conjecture for embedded positive-genus surfaces in $S^3$, hence for embedded tori in ${\mathbb{R}}^3$ by stereographic projection.

The rigidity statement follows from the equality case. If $\mathcal{W}(\Sigma )=2{\pi }^2$, the min-max surface has area $2{\pi }^2$ and is therefore the Clifford torus by the minimal surface theorem. The argument then forces the original surface to be a conformal image of that torus. This recovers precisely the Euclidean equality family: stereographic projections of conformal images of ${\mathbb{T}}_{\mathrm{Cl}}$.

Technical Framework

The proof depends on several layers of regularity and topology that are easy to miss if one only reads the outline.

First, the canonical family is naturally continuous in the flat topology, not in the mass norm. Flat convergence permits cancellation: two nearby sheets with opposite orientations can converge to zero while their areas remain large. The Almgren-Pitts discretization theorem replaces flat-continuous families by discrete maps with small fineness and mass control. Interpolation then returns continuous representatives without losing the width information (Almgren, 1962; Pitts, 1981).

Second, the proof sometimes forgets orientations and works with the associated varifolds $|\Phi (x)|$. This is necessary because compactness for minimal surfaces is varifold compactness, while the topological obstruction is initially expressed using currents. The boundary map to

$$\mathcal{T}\simeq {\mathbb{RP}}^3$$

uses unoriented great spheres, which is why the degree $g$ becomes the homology class $2g$.

Third, the no-concentration estimate for the canonical family prevents mass from accumulating in very small balls. This condition is required for the interpolation machinery to preserve the supremal mass and for the discretized homotopies to represent the same min-max class.

Finally, the regularity conclusion is special to the dimension. In a three-dimensional ambient manifold, Almgren-Pitts min-max surfaces are smooth embedded minimal surfaces after the regularity theory is applied. The clean final statement would be much more delicate in higher dimensions, where singular sets may appear.

Scope and Limitations

The theorem is sharp for embedded positive-genus surfaces. It does not classify all critical points of the Willmore functional. There are many Willmore surfaces in $S^3$, including constructions not conformal to minimal surfaces; the proof identifies the global minimizer in each positive-genus embedded class below the decisive threshold.

The argument also does not give a direct deformation of an arbitrary torus to the Clifford torus. The min-max surface is produced indirectly from the canonical family. Its role is to certify a lower bound for the original surface, not to provide a flow or an algorithm.

The construction is very specific to codimension one surfaces in $S^3$. The conformal group, the tube-area comparison, the topology of great spheres, and Urbano’s low-index classification all enter in essential ways. The method is transferable, but the exact theorem is not a formal consequence of general min-max theory alone.

Transferable Mechanisms

The paper contains several reusable mechanisms.

The first is an energy-controlled sweepout. Instead of varying a surface by an arbitrary homotopy, Marques and Neves use conformal images and parallel surfaces so that every slice satisfies an area bound by the Willmore energy. This is the bridge from an analytic functional to a min-max width, and it depends on the Ros-Heintze-Karcher comparison (Ros, 1999; Heintze & Karcher, 1978; Marques & Neves, 2014).

The second is boundary resolution by blow-up. A degenerating conformal family can fail to be continuous because the limit remembers approach directions. Blowing up the parameter space records those directions and turns the boundary into a tractable geometric model. This pattern is useful whenever compactifying a family loses the data needed for continuous limiting objects (Marques & Neves, 2014).

The third is a homological lower bound for width. The class $2g\in H_3({\mathbb{RP}}^3,\mathbb{Z})$ prevents the sweepout from being pushed into the region of area at most $4\pi$. This is an instance of the general Almgren-Pitts philosophy that nontrivial topology in a cycle space forces minimal hypersurfaces (Almgren, 1962; Pitts, 1981).

The fourth is the combination of existence with rigidity. Min-max theory produces a critical object, but the sharp constant comes from Urbano’s classification of low-index minimal surfaces in $S^3$ (Urbano, 1990). Similar strategies appear whenever a variational argument can produce an object with bounded index and an independent theorem classifies all such objects.

The fifth is the use of an energy threshold to eliminate multiplicity and degeneration. The inequalities $4\pi <\mathbf{L}(\Pi )<8\pi$ force the min-max limit to be a single positive-genus surface rather than a multiple cover of a sphere. This threshold logic is backed by the Li-Yau multiplicity estimate and by the area lower bound for minimal surfaces in $S^3$ (Li & Yau, 1982; Almgren, 1966).

See Also

• on degree of mapping and Lipschitz constant — the Willmore proof uses degree as a quantitative obstruction: the center map has degree $g$, and the induced great-sphere map carries the class $2g$.
• on convergence of graph Laplacian to manifold’s Laplacian — the min-max principle there is spectral rather than geometric, but both arguments turn a constrained variational class into an unavoidable critical value.
• on Weyl’s asymptotic law — Weyl-type variational methods and Almgren-Pitts widths share the theme that parameter-space topology controls critical values, although the analytic settings are very different.
• on orbit closures in moduli space — both proofs rely on compactness plus rigidity: possible limiting objects are constrained until a topological or geometric invariant forces the desired classification.

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