The Solomon–Yau least-volume problem: Clifford products and min–max

Source and verification

This article explains family 349 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Minimal hypersurfaces and the Clifford threshold

A smooth immersion $F:M^m\to S^{m+1}$ is minimal if its mean-curvature vector vanishes. Equivalently, its induced $m$-dimensional volume is stationary under smooth variations. We call this volume its area, even when $m>2$. A closed manifold is compact without boundary. All ambient spheres below have radius one.

An equator is a totally geodesic copy of $S^m$ in $S^{m+1}$: its second fundamental form vanishes. It is the least-area closed minimal hypersurface. The Solomon–Yau problem asks for the next area threshold when the image is not totally geodesic.

The comparison examples are the minimal Clifford products. For $k+\ell =m$, with $k,\ell \ge 1$, put

$${\mathcal{C}}_{k,\ell }=S^k\left(\sqrt{\frac{k}{m}}\right)\times S^{\ell }\left(\sqrt{\frac{\ell }{m}}\right).$$

The squared radii sum to one, so this product lies in the unit sphere. With one choice of normal, its principal curvatures are $\sqrt{\ell /k}$ with multiplicity $k$ and $-\sqrt{k/\ell }$ with multiplicity $\ell$. Their sum is zero, proving minimality. Their squared sum is $m$.

Let ${\sigma }_j={\mathrm{Vol}}_j(S^j)$. The product areas and the least Clifford area are

$$\begin{array}{rl}|{\mathcal{C}}_{k,\ell }|& ={\sigma }_k{\sigma }_{\ell }{\left(\frac{k}{m}\right)}^{k/2}{\left(\frac{\ell }{m}\right)}^{\ell /2},\\ a_m& =\underset{1\le k<m}{\min }|{\mathcal{C}}_{k,m-k}|.\end{array}$$

Manuscript result

For every $m\ge 2$, if $M^m$ is closed and connected and $F:M\to S^{m+1}$ is a smooth minimal immersion whose image is not totally geodesic, then

$$\mathrm{Vol}(M,F^{\ast }{g}_{\mathrm{round}})\ge a_m.$$

A Clifford product minimizing $a_m$ attains the threshold. The volume is measured on the domain, including covering multiplicities. (OpenAI, 2026)

The statement is a lower bound for all such immersions. We do not add a classification of every equality case to the displayed theorem. The manuscript separately proves an index-rigidity result for embedded hypersurfaces, which is one of its proof ingredients.

Prerequisites. The exposition uses the first and second variation of area, elementary elliptic equations, weak convergence of measures, and basic covering-space topology. The needed notions of varifold, density, finite perimeter, and cohomological detection are introduced where they enter.

Proof architecture

A hypothetical counterexample first becomes embedded by a density argument. A family of distance levels provides a uniform gap above the equator and a topological obstruction to contracting away all its area. Lower-dimensional volume bounds then give enough regularity to attain a least counterexample. Its extra unstable direction lowers the entire family below that minimum, while min–max produces a smaller smooth counterexample. The contradiction closes the dimension induction. (OpenAI, 2026)

🏷️ Normalized areas and the immersion reduction

Set ${\vartheta }_m=a_m/{\sigma }_m$. The manuscript proves that these ratios strictly decrease for $m\ge 2$, with

$${\vartheta }_2=\frac{\pi }{2}<2,{\vartheta }_4\le \frac{3}{2}.$$

For orientation, the small-dimensional comparisons are:

Hypersurface dimension	Clifford example	Area divided by equator area
$2$	$S^1(1/\sqrt{2})\times S^1(1/\sqrt{2})$	$\pi /2$
$3$	$S^1(1/\sqrt{3})\times S^2(\sqrt{2/3})$	$8/(3\sqrt{3})$
$4$	$S^2(1/\sqrt{2})\times S^2(1/\sqrt{2})$	$3/2$

These are direct area calculations, not a numerical verification of the volume theorem. In particular $a_2=2{\pi }^2$.

The monotonicity follows from a convenient continuous interpolation. Define

$$d_j={\sigma }_j{\left(\frac{j}{2\pi e}\right)}^{j/2},j>0,$$

using the gamma-function formula for ${\sigma }_j$. Then the Clifford ratio is $d_k{d}_{\ell }/d_{k+\ell }$. The logarithm of $d_j$ is strictly convex: its second derivative is positive by the midpoint comparison between a reciprocal-square series and its integral. Consequently, for fixed $k>0$, the ratio decreases strictly as $\ell$ increases. Increasing one factor dimension in a minimizing pair gives ${\vartheta }_{m+1}<{\vartheta }_m$. The two consequences needed later are $a_m<2{\sigma }_m$ in all dimensions and ${\vartheta }_m\le 3/2$ for $m\ge 4$. (OpenAI, 2026)

To treat an immersion, form its parametrized Euclidean cone

$$(r,z)⟼rF(z),r>0.$$

A varifold records tangent planes and an area measure; an integral varifold has integer sheet multiplicities almost everywhere. Stationarity means its first area variation vanishes. The cone is stationary away from its vertex by minimality of $F$, and the inner-boundary contribution from truncating at radius $\epsilon$ is $O({\epsilon }^m)$, so it is stationary at the vertex as well.

For a $d$-dimensional varifold $V$, its density is

$$\Theta (V,z)=\underset{r\downarrow 0}{\lim }\frac{\Vert V\Vert (B_r(z))}{{\omega }_d{r}^d},$$

when the limit exists; ${\omega }_d$ is the Euclidean unit-ball volume and $\Vert V\Vert$ is the area measure. A smooth sheet has density one. The cone’s exact mass formula gives

$$\Theta ({\mathcal{C}}_F,0)=\frac{\mathrm{Vol}(M,F^{\ast }{g}_{\mathrm{round}})}{{\sigma }_m}.$$

If one image point has $q$ preimages, its nonzero cone point has density $q$. This includes tangent intersections and coincident sheets. The stationary monotonicity formula says that centered mass ratios increase with radius. Their large-radius limit, even when centered away from the vertex, equals the vertex density by comparison of translated balls. Therefore

$$\mathrm{Vol}(M,F^{\ast }{g}_{\mathrm{round}})\ge q{\sigma }_m.$$

A noninjective immersion has $q\ge 2$, already exceeding $a_m$. Any counterexample is thus injective, and compactness makes it an embedding. This reduction preserves the domain-volume convention rather than replacing it by the measure of the image. (OpenAI, 2026)

🏷️ Average curvature from a two-point comparison

Now let ${\Sigma }^m\subset S^{m+1}$ be embedded, connected, minimal, and not an equator. It separates the sphere and has a global unit normal $N$. Write $x$ for its position, $A=-dN$ for its shape operator, and $P=|A{|}^2$. Minimality gives $\mathrm{tr}A=0$. With $\Delta =\mathrm{div}\nabla$, the basic equations include

$$\begin{array}{rl}\Delta x& =-mx,\\ \Delta N& =-PN,\\ \Delta A& =(m-P)A,\\ \frac{1}{2}\Delta P& =|\nabla A{|}^2+(m-P)P.\end{array}$$

The last two are Simons’ identities. The manuscript first proves the integrated estimate

$${\int }_{\Sigma }P\ge m|\Sigma |.$$

It uses the two-sided tangent-ball quantity

$$\mu (x)=\underset{y\in \Sigma ,y\ne x}{\sup }\frac{|⟨N(x),y⟩|}{1-⟨x,y⟩}.$$

The absolute value is essential: both signs of principal curvature must be controlled. Taylor expansion along a geodesic from $x$ gives diagonal limiting values $|A_x(v,v)|$. Adjoining these tangent directions makes the maximum finite and continuous. It is strictly positive because a non-equatorial minimal hypersurface is full: its position vectors span the ambient Euclidean space.

Instead of differentiating this maximum, the paper proves an inequality for every smooth positive upper test $\varphi$ touching $\mu$ from above at $x$:

$$\Delta \log \varphi (x)\ge m-P(x).$$

At a distinct-point contact, reflect the tangent frame at $x$ across the hyperplane perpendicular to $x-y$. The reflected frame is tangent at the contact point $y$. The Hessian of the nonnegative two-point contact function, traced in these paired directions, gives

$$\begin{array}{rl}\Delta \varphi +(P-m)\varphi & \ge 2\sum _{{\lambda }_i<\varphi }\frac{{\varphi }_i^2}{\varphi -{\lambda }_i}\\ & \ge \frac{|\nabla \varphi {|}^2}{\varphi }.\end{array}$$

Here ${\lambda }_i$ are principal curvatures. The omitted components vanish when ${\lambda }_i=\varphi$, and $-\varphi \le {\lambda }_i\le \varphi$ bounds the other denominators. A diagonal contact needs a separate calculation: vary a unit vector in the entire top eigenspace, including repeated eigenvalues, and use the first-derivative contact identities and Codazzi. No smooth eigenframe through a repeated eigenvalue is assumed. Dividing the resulting inequality by $\varphi$ gives the logarithmic upper-test statement. (OpenAI, 2026)

The integrated conclusion now follows without assigning weak derivatives to $\mu$. If the mean $c$ of $m-P$ were positive, solve

$$\Delta h=m-P-c.$$

At a maximum of $\log \mu -h$, a suitable constant multiple of $e^h$ is a smooth upper test for $\mu$. Its logarithmic Laplacian is $m-P-c$, contradicting the upper-test inequality. Hence $c\le 0$ and $\int P\ge m|\Sigma |$.

🏷️ Index rigidity and the extra unstable direction

The Jacobi operator and area second variation are

$$\begin{array}{rl}L& =\Delta +m+P,\\ Q(w)& =-{\int }_{\Sigma }wLw.\end{array}$$

The Morse index is the number of positive eigenvalues of $L$, counted with multiplicity: it counts independent directions in which area decreases to second order. This sign convention matters.

The $m+2$ functions $⟨a,N⟩$, as $a$ ranges over an ambient basis, are independent and satisfy $L⟨a,N⟩=m⟨a,N⟩$. Their independence follows from the hypersurface equations and sphere rigidity: a missing normal direction would make the hypersurface totally geodesic. The largest eigenvalue is simple, has a positive eigenfunction $u$, and exceeds $m$. Thus

$$\mathrm{ind}(\Sigma )\ge m+3.$$

Suppose equality holds. On the orthogonal complement of $u$, the quadratic form of $L$ is at most $m$ times the squared $L^2$ norm. Conformally move the ambient sphere so that every transformed coordinate function is orthogonal to $u$. If $\beta$ is the conformal length factor, summing these spectral inequalities gives

$${\int }_{\Sigma }P\le m{\int }_{\Sigma }{\beta }^2\le m|\Sigma |.$$

The last inequality is strict for any nonidentity conformal transformation. It follows from an exact variation formula for its area factor; normal-coordinate independence gives the strictness. The average-curvature lower bound therefore forces the transformation to be the identity and equality in every individual spectral test.

That final point is stronger than equality of integrated curvature. It gives $L{x}_{\alpha }=m{x}_{\alpha }$ for each coordinate, whereas $Lx=Px$. Since $|x|=1$, it follows that $P=m$ pointwise. Simons’ identity gives $\nabla A=0$. Its parallel eigendistributions and the Gauss equation force two opposite principal curvatures, with product minus one and multiplicities $k,m-k$; the resulting image is the Clifford product ${\mathcal{C}}_{k,m-k}$. Consequently an embedded hypersurface of area below $a_m$ must have

$$\mathrm{ind}(\Sigma )>m+3.$$

This extra direction will remove the maximum of a family with $m+3$ parameters. (OpenAI, 2026)

🏷️ Distance levels transformed by Lorentz boosts

The family is built from the signed Euclidean distance $\phi$ to the cone over $\Sigma$. It is homogeneous of degree one and $1$-Lipschitz. Fullness and ${\int }_{\Sigma }x=0$ imply a stronger bound $|\phi (z)|\le K|z|$ for some $K<1$.

Every closest-point contact for a nonzero $z$ has the form

$$z=rx+hN(x),h=\phi (z),$$

where

$$\begin{array}{c}r=\sqrt{|z{|}^2-h^2}>0,\\ rI-hA(x)⪰0.\end{array}$$

The matrix inequality follows from the second derivative of the height function $⟨z,x⟩$ at a maximizing point on $\Sigma$. It holds at every closest point, even if that point is not unique.

A Lorentz boost is a linear transformation in one time coordinate and $m+2$ spatial coordinates preserving $|z{|}^2-h^2$. For a direction $a\in S^{m+1}$ and parameter $\tau \ge 0$, set

$$\begin{array}{c}C=\cosh \tau ,S=\sinh \tau ,\\ B=I+(C-1)a{a}^T.\end{array}$$

The transformation used here is

$$\begin{array}{rl}z& =By+bSa,\\ h& =Cb+S⟨a,y⟩.\end{array}$$

Transform the graph $h=\phi (z)$ and restrict the new spatial variable $y$ to the unit sphere. The graph remains single-valued, say $b={\psi }_v(y)$ with $v=(\tanh \tau )a$. Strict monotonicity in the new time variable follows from the Lipschitz bound and $C-S>0$. The bound $K<1$ permits compactification to $|v|=1$, where

$${\psi }_a(y)=-⟨a,y⟩.$$

Thus infinite boosts produce ordinary spherical caps. Define the fillings $E_{v,b}=\{y:{\psi }_v(y)<b\}$ for

$$(v,b)\in {\overline{B}}^{m+2}\times [-1,1].$$

The endpoint fillings are empty and full. (OpenAI, 2026)

The decisive area estimate takes place on the fixed source $\Sigma$. Put $r=\sqrt{1-b^2}$. The inverse boost parametrizes each possible level contact by a map $Y_{v,b}$ with

$$d{Y}_{v,b}=O(rI-h_{v,b}A),$$

where $O$ is an isometry on the relevant tangent space and $h_{v,b}$ is the contact height on the source. Restrict to the source set where $rI-h_{v,b}A$ is positive semidefinite. Its eigenvalues are nonnegative and sum to $mr$, by minimality. Therefore

$$0\le \det (rI-h_{v,b}A)\le r^m.$$

Every level point has at least one such source contact. The area formula, allowing multiple preimages, gives a perimeter bound at every level:

$$\mathrm{Per}(E_{v,b})\le r^m|\Sigma |(|v|<1).$$

On $|v|=1$ the bound is $r^m{\sigma }_m$. No unique closest-point assumption and no regular-value exception are imposed. The finite area of each level also implies that its ambient volume is zero, so uniform convergence of the graph functions makes the fillings continuous in volume. Their boundaries are consequently continuous in flat distance, the distance that allows a small filling between nearby cycles. (OpenAI, 2026)

Min–max additionally requires no concentration of mass: small ambient balls must contain uniformly small boundary mass over the whole parameter space. The restricted determinant is bounded by one on the fixed compact source. For finite boost limits, the area formula makes its limiting Jacobian zero almost everywhere on any point fiber. For infinite boosts, the source maps converge smoothly away from $\{N=a\}$. That exceptional normal fiber lies in the zero set of the nontrivial analytic coordinate function $⟨a,x⟩$ and is source-null. Dominated convergence then gives uniform nonconcentration, even with moving boost direction. Only restricted Jacobians are bounded; unrestricted source maps can degenerate near the exceptional set.

🏷️ Projective detection and a gap above the equator

Use mod-two boundaries, so choosing a region or its complement gives the same cycle. A set of finite perimeter is a measurable region whose indicator has finite distributional variation; the mass of its boundary cycle is its perimeter. On the sphere every cycle in the flat path component of zero has precisely two such fillings, complementary modulo null sets. These choices form a two-sheeted covering of the cycle space.

Let $\lambda$ denote the degree-one cohomology class recording this cover. Along a loop of cycles, it is nonzero precisely when a continuous filling returns as the complement of its initial filling. A family $\Phi$ is detecting for $p=m+3$ if

$$({\Phi }^{\ast }\lambda {)}^p\ne 0.$$

Here the power is the cup product; it provides a topological obstruction preserved by continuous deformation of the family.

On the boundary of the compact parameter cylinder, the distance-family fillings at $(v,b)$ and $(-v,-b)$ are complementary. Identifying these boundary antipodes gives ${\mathbb{RP}}^{m+3}$. The path from the empty filling to the whole sphere closes to its projective generator, so the pulled-back covering is the nontrivial projective double cover. Its degree-one class has nonzero $(m+3)$-rd power. This explains the parameter count without imposing a topological hypothesis on $\Sigma$. (OpenAI, 2026)

Detection also prevents the family from collapsing to equators. Consider fillings satisfying two constraints:

$$\begin{array}{c}\mathrm{Vol}(E)=\frac{1}{2}{\sigma }_{m+1},\\ {\int }_{E}yd\mu (y)=0.\end{array}$$

Let ${\gamma }_m$ be the infimum of their perimeters. Such regions exist, for example suitable pairs of antipodal caps. Compactness for sets of finite perimeter attains the infimum. Spherical isoperimetry gives perimeter at least ${\sigma }_m$, with equality only for a hemisphere. A hemisphere has nonzero centroid, so

$${\gamma }_m>{\sigma }_m.$$

Every detecting family contains a filling with the two constraints. Otherwise the vector consisting of its volume error and its $m+2$ centroid coordinates is a nowhere-zero odd map on the filling double cover. Normalizing would produce a map to $S^{p-1}$ and factor the covering class through ${\mathbb{RP}}^{p-1}$, where its $p$-th power vanishes. That contradicts detection.

After replacing projective parameters by a finite cubical complex, the Almgren–Pitts interface therefore supplies a stationary integral varifold with mass between ${\gamma }_m$ and the maximum family mass. The lower bound applies to every competitor in the min–max homotopy class, since sufficiently fine extensions preserve detection. In dimensions $2\le m\le 6$, the imported theorem also gives smooth embedded support with integer multiplicities. In higher dimensions, smoothness is supplied separately below. No min–max index estimate or multiplicity-one theorem is assumed. (OpenAI, 2026)

🏷️ Removing the unique maximum

The center $(v,b)=(0,0)$ represents $\Sigma$. The level factor $r^m$ loses area when $b\ne 0$. For a nonzero boost with $b$ tending to zero, a fixed positive-area source patch has both $A\ne 0$ and $⟨a,x⟩\ne 0$. On that patch, the contact matrix has fixed trace but is not scalar, so arithmetic–geometric mean gives a uniform strict determinant loss. This handles finite and infinite boost limits and proves a uniform area gap outside every neighborhood of the center. It does not infer upper semicontinuity of mass from flat continuity.

Near the center, a tubular neighborhood of the cone makes the entire level a smooth embedded boundary with unique source contacts. Its first normal variations are

$$\begin{array}{c}{w}_{d,t}(x)=t+⟨d,x⟩,\\ (d,t)\in {\mathbb{R}}^{m+2}\times \mathbb{R}.\end{array}$$

They span a $p=m+3$ dimensional space, and

$$\begin{array}{rl}{\int }_{\Sigma }{w}_{d,t}L{w}_{d,t}& =m|\Sigma |t^2\\ & +{\int }_{\Sigma }P(t+⟨d,x⟩{)}^2\\ & >0\end{array}$$

for a nonzero parameter. Thus area has a strictly negative Hessian in these directions.

If the index exceeds $p$, choose one additional unstable function orthogonal to this space for the bilinear form $\int \varphi L\psi$. Flow the nearby fillings by an ambient extension of that normal variation. With $u$ denoting the old parameters and $s$ the new one, Taylor’s theorem gives

$$\mathcal{A}(u,s)\le |\Sigma |-c(|u{|}^2+s^2)$$

near zero. Replace $s=0$ locally by $s=\epsilon \beta (u)$, where the cutoff equals one at the center and vanishes outside a small neighborhood. This graph misses the origin; the uniform gap outside that neighborhood controls the rest of the family. Its total maximum is strictly below $|\Sigma |$.

The cutoff gives a flat homotopy fixed near the parameter boundary, preserving detection. In the modified region the cycles form a compact smooth family, preserving nonconcentration. Both properties are necessary for the final min–max application. (OpenAI, 2026)

🏷️ Low-density cones and the dimension induction

Fix a hypersurface dimension $m$, assuming the theorem only in dimensions $2,\dots ,m-1$. We need regularity below density ${\vartheta }_m$.

For a stationary integral cone, density at any support point is at least one and at most the vertex density. At a nonzero point, every tangent cone splits off the radial line:

$$\begin{array}{c}T=\mathbb{R}\times W,\\ \Theta (W,0)=\Theta (T,0).\end{array}$$

The splitting follows because dilations about the original vertex become translations along the radial direction after blowing up at the nonzero point. Rectifiable slicing shows that $W$ remains stationary and integral.

There is one low-dimensional qualification. A cone is cyclic modulo two when reducing its multiplicities modulo two gives a chain with zero boundary. This property passes to the relevant limits and sliced factors. In dimensions $m=2,3$ it is imposed in the cone argument to exclude a three-ray junction. Indeed, a one-dimensional stationary cone is a union of weighted rays with balanced directions and density half its total integer ray weight. For $m\ge 4$, density below ${\vartheta }_m\le 3/2$ forces total weight below three, hence two opposite rays. For $m=2,3$, density below two allows weight three, but mod-two cyclicity forces even weight and again leaves two opposite rays. (OpenAI, 2026)

Now induct on the Euclidean cone dimension $k$, keeping $m$ fixed. A tangent at a nonzero point splits into a line and a lower-dimensional cone. If that factor is a multiplicity-one plane, Allard regularity makes the original cone smooth and multiplicity one near the point, without a dimension or stability restriction. Its spherical link is therefore smooth. Each link component contributes normalized area at least one, so total density below two forces connectedness.

For $k\le m$, a non-equatorial link of dimension $k-1\ge 2$ would violate the already known volume bound:

$$\Theta (\mathcal{C},0)\ge {\vartheta }_{k-1}\ge {\vartheta }_m.$$

Thus all such cones are planes; the one-dimensional link case is a great circle. At $k=m+1$, the lower-dimensional tangent factors are now planes, giving a smooth connected multiplicity-one link of dimension $m$. The unknown dimension-$m$ volume theorem is not applied to that final link. This is the distinction between the outer volume induction and the inner cone induction. (OpenAI, 2026)

🏷️ Attainment and the final contradiction

Suppose embedded non-equatorial counterexamples exist, and let $H$ be their infimal area. Their detecting distance families and area bounds imply

$${\sigma }_m<{\gamma }_m\le H<a_m.$$

Take cones over a minimizing sequence. Their exact centered ball masses are

$$\begin{array}{c}\Vert {\mathcal{C}}_i\Vert (B_R(0))=\frac{H_i{R}^{m+1}}{m+1},\\ H_i\to H.\end{array}$$

Integral-varifold compactness gives a stationary conical limit. Homogeneity makes every positive-radius sphere have zero limit mass: otherwise infinitely many dilated spheres would have positive mass in one compact annulus. Weak convergence can therefore be evaluated on every centered ball, preserving the exact value $H$, rather than only a lower-semicontinuity inequality. The low-density cone result makes its link a smooth connected embedded minimal hypersurface of multiplicity one and area exactly $H$.

The least counterexample has index greater than $m+3$, since equality would make it a Clifford product of area at least $a_m$. Remove the center of its detecting family as above. Min–max then produces a stationary integral varifold with

$${\gamma }_m\le \mathbf{M}(V)<H<a_m.$$

For $m=2,3$, choose the smooth low-dimensional min–max output; its closed components make its cone cyclic modulo two. For $m\ge 4$, stationarity and integrality suffice. In both cases the cone induction makes $V$ a smooth connected multiplicity-one minimal hypersurface. Its area is above the equator but below the least counterexample, a contradiction.

The outer induction starts at $m=2$, where no lower-dimensional volume hypothesis is used. The immersion reduction then recovers the full stated scope in every dimension. The argument depends on the reported all-level distance-family construction and cone regularity mechanism as well as the imported min–max and compactness theorems; the displayed contradiction alone is not a verification of those inputs. (OpenAI, 2026)

🏷️ Related regularity and variation arguments

• Stationary varifold singularities introduces stationarity, density, and tangent limits. Here a small-density induction is used to obtain complete smoothness for the particular limits, rather than only a dimension bound on their singular sets.
• Closed geodesic infinitude provides another variational argument in which topology must survive deformation past critical objects; the parameter spaces and compactness mechanisms are different.
• The collection guide records the graduate prerequisites and reading order for the differential geometry families.

🏷️ Key steps and proof outlines

An upper-test inequality avoids differentiating a maximum

The two-sided tangent-ball quantity $\mu$ is defined by a supremum and need not be differentiable. The manuscript proves its logarithmic Laplacian inequality by testing every smooth function touching it from above. Distinct-point contacts use reflected tangent frames; diagonal contacts use variations in the full top eigenspace, so repeated principal curvatures do not require a smooth eigenframe.

The integrated curvature consequence is a useful general device. If $c=\mathbf{E}(m-|A{|}^2)>0$, solve $\Delta h=m-|A{|}^2-c$. At a maximum of $\log \mu -h$, the function $C{e}^h$ touches $\mu$ from above. Its logarithmic Laplacian is smaller than the required value by $c$, a contradiction. Thus $\int |A{|}^2\ge m|\Sigma |$ follows without first proving weak differentiability of $\mu$. (OpenAI, 2026)

Orthogonality produces a genuinely additional negative direction

The central family has normal variations $w=t+⟨d,x⟩$, spanning a space $W$ of dimension $p=m+3$. Its area Hessian is negative definite because $\int wLw>0$ for nonzero $w$. If the Morse index exceeds $p$, take a finite-dimensional positive spectral subspace of $L$ with dimension greater than $p$. The $p$ linear equations

$${\int }_{\Sigma }\eta Lw=0(w\in W)$$

have a nonzero solution $\eta$ there. Hence $\int \eta L\eta >0$, and the Hessian is negative definite on $W\oplus \mathbb{R}\eta$. This supplies one extra variation independent of all original parameters, rather than a direction already represented by the Lorentz-distance family.

Extend $\eta N$ to an ambient vector field and flow the nearby domains. Taylor’s theorem gives area at most $|\Sigma |-c(|u{|}^2+s^2)$. Replacing $s=0$ by $s=\epsilon \beta (u)$ produces a compact parameter graph missing the origin, so its maximum is strictly lower. Outside the modified neighborhood, the separately proved uniform strict loss applies. The deformation is fixed near the parameter boundary and preserves the detecting class. (OpenAI, 2026)

Smooth central levels are needed for the deformation argument

The area Hessian above concerns the entire actual boundary, not just one smooth branch of a distance level. The central tubular-neighborhood theorem guarantees unique contacts and smooth embedded boundaries for all nearby parameters. On that compact smooth family, finitely many uniformly controlled graph charts bound mass in a radius-$r$ ball by $C{r}^m$, proving nonconcentration. Away from the modified region, the original family supplies the same property. These facts make the deformed family admissible for min–max.

Finally, the cone argument uses two inductions with different purposes. For fixed target hypersurface dimension, lower-dimensional cone factors are forced to be planes using already established lower-dimensional volume bounds. This regularizes the final-dimensional link without applying the theorem being proved to that link. Attainment then provides a least candidate. The index theorem either identifies its Clifford structure or gives the extra unstable direction, whose deformed detecting family produces a smaller candidate and contradicts minimality. Keeping regularity, attainment, and index reduction in this order avoids a circular use of smoothness.

🏷️ Manuscripts and verification scope

• The Solomon–Yau least-volume theorem — pinned manuscript and companion source.

No family scope page was present at lean/docs/349.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The Solomon–Yau least-volume theorem.