Proof. We first describe the possible contacts. Let \(\gamma(t)=\exp_x(tv)\), where \(v\in T_x\Sigma\) is a unit vector, and regard \(\gamma\) as a curve in \(\mathbb R^{m+2}\). The ambient differentiation formulas give \[\begin{align*} \langle N(x),\gamma(t)\rangle &=\frac{t^2}{2}A_x(v,v) +\frac{t^3}{6}(\nabla_vA)_x(v,v)+O(t^4), \tag{4}\\ 1-\langle x,\gamma(t)\rangle &=\frac{t^2}{2}+O(t^4). \tag{5}\end{align*}\] For example, \(\gamma''(0)=-x+A_x(v,v)N(x)\), while \(\langle x,\gamma'''(0)\rangle=0\). Compactness makes these expansions uniform on the unit tangent bundle for sufficiently small \(|t|\). Consequently, adjoining unit tangent directions at the diagonal extends the quotient continuously, with values \(|A_x(v,v)|\).

For completeness, this extension also gives continuity when the base point varies. If \(x_j\to x\) and nearly maximizing points \(y_j\) stay away from \(x\), a convergent subsequence gives an ordinary distinct-point contact. If \(y_j\to x\), write \(y_j=\exp_{x_j}(t_jv_j)\) and take a convergent subsequence of unit tangent directions. The uniform expansions show that its limiting quotient is at most \(\mu(x)\). This proves upper semicontinuity. Holding fixed any \(y\ne x\) whose quotient is within \(\varepsilon\) of \(\mu(x)\) proves lower semicontinuity, after letting \(\varepsilon\downarrow0\). The same compactness argument proves finiteness and shows that a maximum is realized either by distinct points or by a tangent direction. If \(\mu(x)=0\), the entire hypersurface lies in \(N(x)^\perp\), contrary to fullness from Lemma 5. Thus \(\mu>0\).

Fix an upper test \(\phi\) at \(x\). At a realizing contact, choose the sign of \(N\) to make the signed quotient positive, and change the sign of \(A\) at the same time. This choice is held fixed in a neighborhood. It requires no continuous choice of maximizing sign: for either fixed sign and for every nearby \(x'\) and every \(y'\ne x'\), \[ \frac{\langle N(x'),y'\rangle}{1-\langle x',y'\rangle} \le\mu(x')\le\phi(x'). \tag{6}\] Choose a principal orthonormal basis \(e_1,\ldots,e_m\) at \(x\), and write \(Ae_i=\lambda_i e_i\). The diagonal limits give \[ -\phi(x)\le\lambda_i\le\phi(x)\qquad(1\le i\le m). \tag{7}\] We shall prove the stronger inequality \[ \Delta\phi+(P-m)\phi \ge 2\sum_{\lambda_i<\phi} \frac{\phi_i^2}{\phi-\lambda_i} \ge\frac{|\nabla\phi|^2}{\phi} \quad\text{at }x, \tag{8}\] where \(\phi_i=d\phi(e_i)\) and every component with \(\lambda_i=\phi\) will be shown to vanish.

Distinct-point contact. Suppose \(y\ne x\) realizes the maximum with the chosen sign. On the product of a neighborhood of \(x\) with \(\Sigma\), set \[Z(x',y')=\phi(x')\bigl(1-\langle x',y'\rangle\bigr) -\langle N(x'),y'\rangle.\] By (6), \(Z\ge0\) and \(Z(x,y)=0\). Write \(c=\langle x,y\rangle\), \(s=1-c>0\), and \(d_i=\langle e_i,y\rangle\). The first derivatives at the minimum give \[ s\phi_i=(\phi-\lambda_i)d_i. \tag{9}\] Reflection across the hyperplane perpendicular to \(x-y\) sends \(x\) to \(y\), and sends \(N(x)\) to \[n_y=N(x)+\phi(x-y).\] Indeed \(\langle N(x),y\rangle=\phi s\), so this vector is perpendicular to \(y\) and has length one. The \(y\) first-derivative equation for \(Z\) shows that \(n_y\) is perpendicular to \(T_y\Sigma\). Therefore the reflected vectors \[\widetilde e_i=e_i+\frac{d_i}{s}(x-y)\] form an orthonormal basis of \(T_y\Sigma\), and \[\langle x,\widetilde e_i\rangle=d_i, \qquad \langle e_i,\widetilde e_i\rangle=1-\frac{d_i^2}{s}.\] The vector \(n_y\) may be either orientation of the normal at \(y\); only minimality at \(y\) enters the following trace computation.

Use the product connection to compute the Hessian of \(Z\). Denote its mixed entries in the indicated bases by \(Z_{x_i y_i}\). From (1), \[\begin{align*} \Delta_x Z &=s\Delta\phi-2\sum_i\phi_i d_i+m\phi c+P\phi s,\\ \Delta_y Z&=m\phi,\\ \sum_i Z_{x_i y_i} &=-\sum_i\phi_i d_i -\sum_i(\phi-\lambda_i) \left(1-\frac{d_i^2}{s}\right). \end{align*}\] In particular, the nonnegative sum of Hessians along \((e_i,\widetilde e_i)\) is \[\begin{align*} 0&\le\Delta_xZ+2\sum_iZ_{x_i y_i}+\Delta_yZ\\ &=s\bigl(\Delta\phi+(P-m)\phi\bigr) -4\sum_i\phi_i d_i +\frac{2}{s}\sum_i(\phi-\lambda_i)d_i^2\\ &=s\bigl(\Delta\phi+(P-m)\phi\bigr) -\frac{2}{s}\sum_i(\phi-\lambda_i)d_i^2, \end{align*}\] where we used \(\mathop{\mathrm{tr}}A=0\) and then (9). If \(\lambda_i=\phi\), (9) gives \(\phi_i=0\). Dividing by \(s\) and substituting that equation in all other directions gives the first inequality of (8).

Diagonal contact, including repeated eigenvalues. Suppose instead that a unit tangent direction realizes the maximum. With our choice of normal sign, the top eigenvalue is \(\phi(x)\). Let \(E\subset T_x\Sigma\) be its full eigenspace. For every unit \(v\in E\) and every local smooth unit extension \(V\) of \(v\), \[A(V,V)\le\mu\le\phi, \qquad A_x(v,v)=\phi(x).\] The first derivative at this contact, in any direction \(w\in T_x\Sigma\), is \[ d\phi(w)=(\nabla_wA)(v,v). \tag{10}\] The field-derivative term vanishes because \(A(v,\nabla_wV)=\phi\langle v,\nabla_wV\rangle=0\). For the geodesic with initial direction \(v\), formulas (4)–(5) give the signed quotient \[\frac{\langle N(x),\exp_x(tv)\rangle} {1-\langle x,\exp_x(tv)\rangle} =\phi(x)+\frac{t}{3}(\nabla_vA)(v,v)+O(t^2).\] It is at most the fixed value \(\mu(x)=\phi(x)\) for both signs of \(t\). Hence \((\nabla_vA)(v,v)=0\), and (10) implies \[ d\phi|_E=0. \tag{11}\] This holds for every vector in \(E\), not just for one principal vector. Moreover, polarizing (10) on \(E\) gives \((\nabla_wA)|_{E\times E}=d\phi(w)g|_{E\times E}\). These are pointwise statements; no smooth extension of the eigenspace is assumed.

Fix \(e_1\in E\) and choose a smooth unit field \(V\) with \(V(x)=e_1\). Put \[T_{ji}=(\nabla_{e_j}A)(e_1,e_i),\qquad \beta_{ji}=\langle\nabla_{e_j}V,e_i\rangle.\] The first derivatives \(\nabla_{e_j}V\) may be prescribed arbitrarily in \(e_1^\perp\): prescribe the corresponding jets of any local field and normalize it. Unit length determines precisely the second-derivative contraction that occurs in the calculation, \[\langle e_1,\Delta V\rangle =-\sum_j|\nabla_{e_j}V|^2.\] Thus, in a geodesic orthonormal frame at \(x\), \[ \Delta\bigl(A(V,V)\bigr) =(\Delta A)_{11} +4\sum_{j,i}\beta_{ji}T_{ji} +2\sum_{j,i}(\lambda_i-\phi)\beta_{ji}^2. \tag{12}\] No other second derivative of \(V\) contributes, since \(Ae_1=\phi e_1\). Choose the free first jets by \[\beta_{ji}=\frac{T_{ji}}{\phi-\lambda_i} \quad\text{if }\lambda_i<\phi, \qquad \beta_{ji}=0\quad\text{if }\lambda_i=\phi.\] The minimum of \(\phi-A(V,V)\) then yields \[\begin{align*} \Delta\phi &\ge(\Delta A)_{11} +2\sum_{\lambda_i<\phi} \frac{\sum_jT_{ji}^2}{\phi-\lambda_i}\\ &\ge(m-P)\phi +2\sum_{\lambda_i<\phi} \frac{\phi_i^2}{\phi-\lambda_i}. \end{align*}\] In the last step we used Simons’ identity and Codazzi: \(T_{1i}=(\nabla_{e_i}A)(e_1,e_1)=\phi_i\). Equation (11) accounts for every omitted top-eigenspace gradient component. This proves the first inequality of (8) also at a repeated diagonal contact.

In both cases, (7) gives \(0<\phi-\lambda_i\le2\phi\) in each retained direction. The vanished components in the other directions therefore give the second inequality of (8). Dividing by \(\phi\) and subtracting \(|\nabla\phi|^2/\phi^2\) proves (3). ◻

Proof. If the inequality failed, then \[c=\frac{1}{V_\Sigma}\int_\Sigma(m-P)\,dV>0.\] Since \(m-P-c\) has mean zero, the Poisson equation on the closed connected manifold has a smooth solution. Indeed, the kernel of \(\Delta\) consists of constants by integration by parts; the compact elliptic Fredholm alternative and regularity (Taylor, n.d.-a, sec. 10, equations (10.4)–(10.6)) give \[ \Delta h=m-P-c. \tag{14}\] Let \(K=\max_\Sigma(\log\mu-h)\) and let \(x\) attain this maximum. The smooth positive function \(\phi=\exp(h+K)\) is an upper test for \(\mu\) at \(x\). Lemma 6 and (14) imply \[m-P(x)\le\Delta\log\phi(x)=m-P(x)-c,\] a contradiction. This argument uses only continuity and positivity of \(\mu\); it does not differentiate \(\mu\) or assume any weak regularity for its gradient. ◻

Proof. Suppose \(a\ne0\) and \(\langle a,N\rangle=0\) everywhere. Set \(f=\langle a,x\rangle\). The ambient Gauss formula gives \[\nabla^2f=-fg+\langle a,N\rangle A=-fg.\] This function is nonconstant: a constant solution would be zero, whereas \(f\equiv0\) contradicts fullness. Obata’s sphere-rigidity theorem (Obata 1962, Theorem A) applies to the complete connected manifold \((\Sigma,g)\) and shows that it is isometric to the unit round \(S^m\). The scalar Gauss equation for a minimal hypersurface, \(\operatorname{Scal}_\Sigma=m(m-1)-P\), then gives \(P=0\). This contradicts the hypothesis that \(\Sigma\) is not totally geodesic. ◻

Proof. Parametrize the transformations by \(z=(\tanh\tau)a\) in the open unit ball of \(\mathbb R^{m+2}\), with the identity at \(z=0\). Consider \[F(z)=\frac{\int_\Sigma u\,T_{\tau,a}(x)\,dV} {\int_\Sigma u\,dV}.\] As \(z\to a_*\in S^{m+1}\), the integrand tends to \(-a_*\) at every point with \(x\ne a_*\). This remains true for a moving direction \(a\to a_*\). The measure \(u\,dV\) assigns zero mass to a point, and the integrands have length one. Dominated convergence extends \(F\) continuously to the closed ball, with \(F(a_*)=-a_*\). If \(F\) had no zero, its normalization \(F/|F|\) would extend the antipodal map from the boundary sphere to the ball. The antipodal map has nonzero degree, so such an extension is impossible. A zero lies in the open ball and proves (17).

To prove (18), fix \(a\) and abbreviate \(f=\langle a,x\rangle\), \(\eta=\langle a,N\rangle\), and \(\beta=\beta_{\tau,a}(x)\). Orthogonal decomposition of \(a\) into position, tangent, and normal components gives \[|\nabla f|^2=1-f^2-\eta^2,\qquad \Delta f=-mf,\qquad \nabla\beta=S\beta^2\nabla f.\] Integrating a divergence on the closed hypersurface yields \[\begin{align*} 0&=\frac1m\int_\Sigma\operatorname{div}(\beta^m\nabla f)\,dV\\ &=\int_\Sigma\beta^{m+1} \bigl(S(1-f^2-\eta^2)-f(C-Sf)\bigr)\,dV\\ &=\int_\Sigma\beta^{m+1}(S-Cf-S\eta^2)\,dV. \end{align*}\] Differentiation in \(\tau\) therefore gives the exact formula \[ \frac{d}{d\tau}\int_\Sigma\beta^m\,dV =m\int_\Sigma\beta^{m+1}(Cf-S)\,dV =-mS\int_\Sigma\beta^{m+1}\eta^2\,dV. \tag{20}\] By Lemma 8, \(\eta\) is not identically zero. The derivative is strictly negative for every \(\tau>0\). Starting from \(\beta_{0,a}=1\) proves (18) and its strictness. Finally, Hölder’s inequality, including the identity case \(m=2\), gives \[\int_\Sigma\beta^2\,dV \le V_\Sigma^{\,1-2/m} \left(\int_\Sigma\beta^m\,dV\right)^{2/m} \le V_\Sigma.\] The second inequality is strict for \(\tau>0\), proving (19). ◻

Proof. By (1), the \(m+2\) normal coordinate functions satisfy \(LN=mN\); they are linearly independent by Lemma 8. The largest eigenvalue \(\lambda_1\) of \(L\) is simple and has a strictly positive eigenfunction \(u\). Apply the first-eigenfunction theorem to the shifted Schrödinger operator \(K-L\), with \(K>1+\max_\Sigma(m+P)\); its positivity uses connectedness and the strong maximum principle (Taylor, n.d.-b, Proposition 2.9). Testing its Rayleigh quotient with the constant function gives \[\lambda_1\ge m+\frac{1}{V_\Sigma}\int_\Sigma P\,dV>m.\] These eigenfunctions prove \(\mathop{\mathrm{ind}}(\Sigma)\ge m+3\).

Suppose that equality holds. The positive spectrum consists of the simple eigenvalue \(\lambda_1>m\) and the eigenvalue \(m\) with multiplicity \(m+2\). The spectral theorem therefore gives \[ \int_\Sigma\phi L\phi\,dV\le m\int_\Sigma\phi^2\,dV \qquad\text{whenever }\int_\Sigma u\phi\,dV=0. \tag{22}\] Choose the conformal map supplied by Lemma 9 for the weight \(u\), and write \(v=T_{\tau,a}(x)\). Each coordinate \(v_\alpha\) satisfies the orthogonality in (22). Since \(|v|=1\) and \(\sum_\alpha|\nabla v_\alpha|^2=m\beta^2\), summing those inequalities and integrating by parts gives \[ \int_\Sigma P\,dV \le m\int_\Sigma\beta^2\,dV\le mV_\Sigma. \tag{23}\] Theorem 7 forces equality throughout. The strict part of (19) implies \(\tau=0\), so \(v=x\).

We now use equality in the individual spectral inequalities. For each \(\alpha\), the defect \[d_\alpha=m\int_\Sigma x_\alpha^2\,dV -\int_\Sigma x_\alpha Lx_\alpha\,dV\] is nonnegative by (22), and (23) shows \(\sum_\alpha d_\alpha=0\). Thus every defect vanishes. In the spectral decomposition on \(u^\perp\), all eigenvalues are at most \(m\); equality in its quadratic form implies \(Lx_\alpha=mx_\alpha\) for every \(\alpha\). On the other hand, (1) gives \(Lx=Px\). Taking the inner product of \((P-m)x=0\) with \(x\) yields \[ P=m\quad\text{everywhere on }\Sigma. \tag{24}\] The pointwise conclusion uses spectral equality as well as the integrated curvature equality.

The last identity in (1) now gives \(\nabla A=0\). We finish by deriving the product classification. Parallelness makes the eigenvalues constant and their eigendistributions parallel. If \(U\) and \(W\) are unit vectors in distinct eigendistributions, the curvature tensor preserves each of those distributions, so \(\langle R(U,W)W,U\rangle=0\). The Gauss equation gives \[1+\lambda\kappa=0\] for any two distinct eigenvalues \(\lambda,\kappa\). There are at most two distinct eigenvalues: a third would have to equal \(-1/\lambda\) as well. Since \(A\) is nonzero and trace-free, there are exactly two, of opposite signs. After changing \(N\) if necessary, write them as \[\lambda=\sqrt{\frac{\ell}{k}},\qquad \kappa=-\frac1\lambda=-\sqrt{\frac{k}{\ell}}, \qquad k+\ell=m,\quad k,\ell>0,\] where \(k\) and \(\ell\) are their multiplicities.

Let \(E_\lambda\) and \(E_\kappa\) denote the parallel tangent eigendistributions. Consider the ambient subspaces, initially depending on \(x\), \[\mathcal E(x)=E_\lambda(x)\oplus\operatorname{span}\{\lambda N(x)-x\}, \qquad \mathcal F(x)=E_\kappa(x)\oplus\operatorname{span}\{\kappa N(x)-x\}.\] They are orthogonal and their dimensions sum to \(m+2\), because \(\langle\lambda N-x,\kappa N-x\rangle=1+\lambda\kappa=0\). For any tangent vector \(w\) and any local section \(V\) of \(E_\lambda\), ambient differentiation gives \[D_wV=\nabla_wV+\langle w,V\rangle(\lambda N-x), \qquad D_w(\lambda N-x)=-(\lambda A+I)w.\] Both expressions lie in \(\mathcal E(x)\): the second vanishes on \(E_\kappa\) and maps \(E_\lambda\) to itself. Hence \(\mathcal E\) is a constant subspace of \(\mathbb R^{m+2}\) along the connected hypersurface; the same holds for its orthogonal complement \(\mathcal F\). The projections of \(x\) to these fixed spaces have lengths \[|\pi_{\mathcal E}x|=\frac1{\sqrt{1+\lambda^2}} =\sqrt{\frac{k}{m}}, \qquad |\pi_{\mathcal F}x|=\frac1{\sqrt{1+\kappa^2}} =\sqrt{\frac{\ell}{m}}.\] Thus the image of \(\Sigma\) lies in the displayed Clifford product. The inclusion is a local diffeomorphism onto this product, since both manifolds have dimension \(m\). Its image is open, and compactness makes it closed. The product is connected because \(k,\ell\ge1\), so the image is the entire product. The final assertion follows from the definition of \(a_m\). ◻

Proof. We use the small-filling theorem in the following precise form: there are constants \(\nu>0\) and \(c\ge1\), depending on the sphere, such that if \(\mathcal F(T-T')<\nu\), then there is an \((m+1)\)-chain \(R\) in the sphere with \[ \partial R=T-T',\qquad \mathbf M(R)\le c\mathcal F(T-T'). \tag{26}\] This is the isoperimetric choice described in (Marques and Neves 2017, sec. 3.1, p. 11); all locators for this source refer to arXiv:1311.6501v2. Applied to a sufficiently fine partition of a flat-continuous path from zero to \(T\), these fillings sum to a top-dimensional chain with boundary \(T\). The preceding representation therefore writes \(T=\partial[E]\) with \(E\) of finite perimeter. Conversely, intersecting any such \(E\) with the increasing spherical caps \(\{y_{m+2}<t\}\), \(-1\le t\le1\), gives sets of finite perimeter continuous in volume, from the empty set to \(E\). Thus every \(\partial[E]\) belongs to this path component.

If \(\partial[E]=\partial[F]\), then \(\partial[E\triangle F]=0\). By (25), the characteristic function of \(E\triangle F\) has zero distributional gradient. Connectedness of the sphere makes it constant almost everywhere. Hence \(F=E\) or \(F=E^c\).

For the covering topology, fix \(T=\partial[E]\) and choose \(\delta>0\) so that \(2\delta<\nu\) and \(4c\delta<\sigma_{m+1}\). For \(T'\) in the flat \(\delta\)-ball about \(T\), choose the small filling \(R(T,T')\) from (26). This choice is unique: two such choices differ by zero or the whole sphere, and their total mass is smaller than \(\sigma_{m+1}\). If \(T'\) and \(T''\) are in the ball, the same argument applied to \[R(T,T')+R(T,T'')+R(T',T'')\] shows that this cycle is zero: its mass is at most \(4c\delta\). Consequently the filling branch determined by \[[E_{T'}]=[E]+R(T,T')\] satisfies \(d(E_{T'},E_{T''})\le c\mathcal F(T'-T'')\). It and its complementary branch are continuous and exhaust the preimage of the ball. They are disjoint open branches in that preimage, distinguished by volume distance from \(E\) less than, or greater than, \(\sigma_{m+1}/2\). Together with (25), this proves the assertion. ◻

Proof. Radial projection identifies the centrally symmetric convex body \(Q\) with the closed \(p\)-ball, respecting antipodes. Its boundary-antipodal quotient is therefore \(\mathbb{RP}^{p}\). Complementary fillings have the same mod-two boundary, so (25) and the quotient property give a flat-continuous descended map \(\Phi\). The fillings themselves specify local branches of its pulled-back cover, not a single filling on the quotient. At a boundary seam one uses \(E_q\) in one collar and the complementary choice in the collar of \(-q\); these choices agree along the identified boundary and are continuous in volume.

The segment \(b\mapsto(0,b)\) becomes a closed loop in this quotient. Its displayed lift starts at the empty set and ends at the whole sphere. Hence \(\Phi^*\lambda\) is the nonzero degree-one class \(\alpha\) of \(\mathbb{RP}^{p}\). The projective cohomology ring is \[H^*(\mathbb{RP}^{p};\mathbb Z_2) =\mathbb Z_2[\alpha]/(\alpha^{p+1}),\qquad |\alpha|=1\] (Hatcher 2002, Theorem 3.19, p. 220). In particular \((\Phi^*\lambda)^p=\alpha^p\ne0\). ◻

Proof. First consider the constrained class of finite-perimeter sets \[\mathcal A_m=\left\{E\subset S^{m+1}: \mathop{\mathrm{Vol}}(E)=\frac{\sigma_{m+1}}2,\quad \int_E y\,d\mathop{\mathrm{Vol}}(y)=0\right\}.\] It is nonempty: the union of two antipodal caps, with their common radius chosen to give half the spherical volume, belongs to it. Define \[ \gamma_m=\inf_{E\in\mathcal A_m}\operatorname{Per}(E). \tag{28}\] This number is finite. A minimizing sequence has bounded perimeter; compactness for sets of finite perimeter on the compact sphere yields a subsequence converging in \(L^1\) to a set \(E\) of finite perimeter. The volume and all centroid coordinates pass to the limit, while perimeter is lower semicontinuous. Thus \(E\in\mathcal A_m\) realizes (28). These are the usual BV compactness and lower-semicontinuity statements in local charts (Simon 2018, chap. 2, Theorem 2.6, pp. 54–55); a finite chart cover gives the compact-sphere assertion used here.

Spherical isoperimetry says that a half-volume set has perimeter at least \(\sigma_m\), with equality only for a hemisphere modulo null sets (Bögelein et al. 2017, sec. 2.3, equation (2.2), p. 5). The centroid of the hemisphere \(\{y\cdot a>0\}\), \(|a|=1\), is \[\int_{y\cdot a>0} y\,d\mathop{\mathrm{Vol}}(y)=\frac{\sigma_m}{m+1}a\ne0.\] The minimizer in \(\mathcal A_m\) therefore cannot attain \(\sigma_m\), proving \(\gamma_m>\sigma_m\).

Now pull the filling cover back by \(\Phi\). On this cover the function \[ (x,E)\longmapsto \left(\mathop{\mathrm{Vol}}(E)-\frac{\sigma_{m+1}}2, \int_E y\,d\mathop{\mathrm{Vol}}(y)\right)\in\mathbb R^{m+3} \tag{29}\] is continuous and changes sign under \(E\mapsto E^c\). If it never vanished, normalization would give an equivariant map to \(S^{p-1}\) and hence a quotient map \(f:X\to\mathbb{RP}^{p-1}\). Fiberwise this identifies the filling cover with the pullback of \(S^{p-1}\to\mathbb{RP}^{p-1}\): each of its two points is sent to one of the two antipodal unit vectors. It follows that \(\Phi^*\lambda=f^*\alpha\), so \((\Phi^*\lambda)^p=f^*(\alpha^p)=0\), contrary to detection. At a zero of (29) the filling belongs to \(\mathcal A_m\), and its cycle has mass \(\operatorname{Per}(E)\ge\gamma_m\) by (25). ◻

Proof. Embed \(\mathbb{RP}^{p}\) in the Euclidean space of real symmetric \((p+1)\times(p+1)\) matrices by \([u]\mapsto uu^t\), \(|u|=1\). Each such matrix has a simple top eigenvalue \(1\), separated from the remaining eigenvalues \(0\). On a neighborhood \(U\) of this compact image the top eigenvalue remains simple, and its eigenline gives a continuous retraction to \(\mathbb{RP}^{p}\). Choose a sufficiently fine finite cubical grid in a larger cube, and let \(K\) be the union of the closed cubes meeting the image, together with their faces. Then \(K\subset U\). After an affine rescaling it is a finite cubical subcomplex of a subdivided unit cube.

Let \(i:\mathbb{RP}^{p}\to K\) be the inclusion and \(r:K\to\mathbb{RP}^{p}\) the restricted retraction. Since \(r\circ i=\mathrm{id}\), the map \(r^*\) is injective on cohomology. Thus \(\Phi\circ r\) detects whenever \(\Phi\) does. Surjectivity of \(r\) makes the two cycle images identical, proving both remaining assertions. The dimension of \(K\) may exceed \(p\). Only its finiteness and its detecting cohomology class are needed here. ◻

Proof. We specify both the imported results and the width comparison. For a discrete map \(\phi\) on the vertices of a cubical subdivision, its fineness is \[\mathbf f(\phi)=\max\{ \mathbf M(\phi(v)-\phi(w)):v,w\text{ are adjacent vertices}\}.\] An \((X,\mathbf M)\)-homotopy sequence consists of uniformly mass-bounded discrete maps whose successive maps are joined, after subdivision, by discrete homotopies of fineness tending to zero. Two sequences represent the same class if their corresponding maps are joined by such homotopies with fineness tending to zero. These are the conventions of (Marques and Neves 2017, secs. 2.2–2.6, pp. 6–7).

The discretization theorem (Marques and Neves 2017, Theorem 3.9, p. 13), applied using (27), provides such a sequence \(\mathcal S=\{\phi_i\}\) with \(\mathbf f(\phi_i)\to0\) and \[\limsup_i\max_v\mathbf M(\phi_i(v))\le D_0.\] For any sufficiently fine discrete map, the Almgren extension \(\widehat\phi:X\to\mathcal Z_m^0(S^{m+1};\mathbb Z_2)\) is continuous in the mass norm and satisfies \[ \sup_X\mathbf M(\widehat\phi) \le\max_v\mathbf M(\phi(v))+C_X\mathbf f(\phi), \tag{31}\] where \(C_X\) is fixed (Marques and Neves 2017, Theorem 3.10, p. 14). Moreover, the extensions of \(\phi_i\) are flat homotopic to \(\Phi\) for all sufficiently large \(i\) (Marques and Neves 2017, Corollary 3.12(ii), p. 15).

Let \(\Pi\) be the class of \(\mathcal S\), and write \[\mathbf L(\Pi)=\inf_{\mathcal S'\in\Pi}\mathbf L(\mathcal S'), \qquad \mathbf L(\mathcal S')= \limsup_i\max_v\mathbf M(\phi_i'(v)), \quad \mathcal S'=\{\phi_i'\}.\] Certainly \(\mathbf L(\Pi)\le D_0\). Fix one competitor \(\mathcal S'\in\Pi\). Its homotopies to \(\mathcal S\) have fineness tending to zero. Therefore their Almgren extensions are flat homotopic for all sufficiently large indices, by (Marques and Neves 2017, Corollary 3.12(i), p. 15). Each \(\widehat\phi_i'\) is then detecting. Lemma 14 and (31) give \[\gamma_m\le\sup_X\mathbf M(\widehat\phi_i') \le\max_v\mathbf M(\phi_i'(v))+C_X\mathbf f(\phi_i').\] Taking the upper limit proves \(\gamma_m\le\mathbf L(\mathcal S')\). Since the competitor was arbitrary, \[0<\gamma_m\le\mathbf L(\Pi)\le D_0.\] The index beyond which the extensions detect may depend on \(\mathcal S'\); no threshold uniform over all competitors is used. This is the continuous/discrete detection compatibility also recorded in (Marques and Neves 2017, Lemma 4.6, p. 16).

The Almgren–Pitts existence theorem in (Marques and Neves 2017, Theorem 2.14, p. 10) now gives a stationary integral varifold at mass \(\mathbf L(\Pi)\), almost minimizing in annuli. That theorem has no dimension restriction. Its low-dimensional regularity consequence is (Marques and Neves 2017, Theorem 2.11, pp. 8–9), which gives the stated smooth embedded support for \(2\le m\le6\). These results are the mod-two form of the Almgren–Pitts theory; the integrality input is (Pitts 1981, Theorem 3.13), and the regularity extension is (Schoen and Simon 1981, Theorem 4). This proves (30). For projective parameters, first apply Lemma 15. ◻

Proof. Define \(H(z)=\max_{x\in\Sigma}\langle z,x\rangle\). Lemma 5 gives \(\int_\Sigma x\,d\mu=0\) and fullness. Thus \(H(z)>0\) for \(z\ne0\): a nonpositive coordinate function of mean zero would vanish identically. Compactness gives \(c=\min_{|z|=1}H(z)>0\). Minimizing first along each ray of the cone gives \[\mathop{\mathrm{dist}}(z,\mathcal C)^2=|z|^2-H(z)^2 \le(1-c^2)|z|^2.\] This proves (36) with \(K=\sqrt{1-c^2}\) and excludes the vertex as a closest point to nonzero \(z\).

A closest ray is determined by a maximum \(x\) of \(\langle z,\cdot\rangle\), and its radial coordinate is \(H(z)\). At this maximum the tangential component of \(z\) vanishes, so \(z=H(z)x+\widetilde hN(x)\), where \(|\widetilde h|=|h|\). The segment from \(z\) to its closest point meets no earlier cone point. Its side near contact therefore agrees with the side of \(z\), proving \(\widetilde h=h\) with our signed-distance convention. Finally, at this height maximum the spherical immersion formula gives \[\nabla^2_\Sigma\langle z,x\rangle=-H(z)I+hA\le0.\] These arguments apply to each maximum separately and therefore also apply when there are several closest points. ◻

Proof. For fixed \(v,y\), consider \[F_{v,y}(b)=Cb+S\langle a,y\rangle-\varphi(By+bSa).\] The Lipschitz property of \(\varphi\) gives, when \(b_2>b_1\), \[F_{v,y}(b_2)-F_{v,y}(b_1)\ge(C-S)(b_2-b_1)>0.\] The same estimate implies limits \(-\infty\) and \(+\infty\) at the two ends of the real line. Hence there is exactly one zero, and the local positive lower bound for \(C-S\) proves continuous dependence of that zero on finite parameters and \(y\).

At a graph point with \(|y|=1\), (36) and Lorentz invariance give \[ 1-b^2=|z|^2-h^2\ge(1-K^2)|z|^2, \qquad |h|\le\frac{K}{\sqrt{1-K^2}}\sqrt{1-b^2}. \tag{40}\] Equality \(|b|=1\) would force \(z=h=0\), contrary to \(|y|=1\) and invertibility of the boost. Thus \(|b|<1\). Since \(b+\langle v,y\rangle=h/C\), we also obtain \[ \sup_{|y|=1}|\psi_v(y)+\langle v,y\rangle| \le\frac{K}{C\sqrt{1-K^2}}. \tag{41}\] This tends to zero as \(|v|\to1\), uniformly in \(y\) and in the boost direction, proving the claimed extension. ◻

Proof. Take \(y\in\Lambda_{v,b}\) and transform \((y,b)\) to \((z,h)\) by (38). Lemma 19 supplies a closest point \(r_zx\). Lorentz invariance gives \(r_z^2=|z|^2-h^2=1-b^2\), so \(r_z=r\). The inverse formulas then show that \(h=h_{v,b}(x)\), \(y=Y_{v,b}(x)\), and \(x\in T_{v,b}\). Therefore \[ \Lambda_{v,b}\subset Y_{v,b}(T_{v,b}). \tag{47}\] The nonnegative eigenvalues of \(rI-h_{v,b}A\) have sum \(mr\), because \(\mathop{\mathrm{tr}}A=0\). Their arithmetic–geometric mean inequality proves (46). The area formula (Federer 1969, sec. 3.2.3) (Federer 1978, sec. 3, pp. 299–300), with multiplicities counted in its source integral, now yields the second inequality in (45). The condition defining \(T_{v,b}\) is only necessary for a closest-point contact. Additional points in this source set, and multiple preimages of a level point, increase the upper bound and cause no difficulty.

In particular \(\mathcal H^m(\Lambda_{v,b})<\infty\). Since \(E_{v,b}\) is open, its essential boundary is contained in its topological boundary, which is contained in \(\Lambda_{v,b}\). The finite-essential-boundary criterion (Federer 1978, sec. 4, p. 302), applied in coordinate charts, gives finite perimeter. The intrinsic reduced-boundary representation (Bögelein et al. 2017, sec. 2.2) then gives \[\operatorname{Per}(E_{v,b};U) =\mathcal H^m(\partial^*E_{v,b}\cap U) \le\mathcal H^m(\Lambda_{v,b}\cap U).\] This proves the first inequality with constant one.

For \(|v|<1\) the endpoint sublevels are empty and full, respectively. For \(|v|=1\), formula (39) makes them round caps, whose boundary masses are \(\sigma_m(1-b^2)^{m/2}\), with zero mass at \(b=\pm1\). These cases also have finite perimeter. ◻

Proof. We prove the sequential form of the assertion. Consider \(u_j=(v_j,b_j)\to(v_*,b_*)\), \(y_j\to y_*\), and \(\rho_j\downarrow0\). The case \(|b_*|=1\) follows directly from (32). Parameters on \(|v|=1\) are harmless as well: an \(m\)-sphere of Euclidean radius \(r\) in any affine \((m+1)\)-plane has at most \(c_m\rho^m\) area in a ball of radius \(\rho\), uniformly in \(r\) and the plane. For \(r\le2\rho\) use its total area; for \(r>2\rho\) the intersection lies in a spherical cap with angular radius at most \(\arcsin(\rho/r)\), and the same bound follows by the cap area formula. Spherical balls in the ambient unit sphere lie in Euclidean balls of the same radius. We may therefore pass to a subsequence with \(|v_j|<1\) and \(|b_*|<1\).

Put \(Y_j=Y_{v_j,b_j}\) and \(T_j=T_{v_j,b_j}\). The local perimeter bound reduces the desired assertion to \[ \int_\Sigma I_j\,d\mu\longrightarrow0, \qquad I_j=\mathbf1_{T_j}\, \mathbf1_{Y_j^{-1}(B_{\rho_j}(y_j))}\,J_mY_j. \tag{50}\] The restricted bound gives \(0\le I_j\le1\) on the fixed compact source.

If \(|v_*|<1\), the maps \(Y_j\) converge smoothly on \(\Sigma\) to \(Y_*=Y_{v_*,b_*}\). Away from the fiber \(Y_*^{-1}(y_*)\), the ball indicator in (50) eventually vanishes. The area formula applied to the fiber gives \[\int_{Y_*^{-1}(y_*)}J_mY_*\,d\mu=0,\] because a singleton has zero \(\mathcal H^m\) measure. Thus \(J_mY_*=0\) almost everywhere on that fiber, and smooth convergence gives \(I_j\to0\) there almost everywhere as well. Dominated convergence completes this case.

It remains to consider \(v_*=a\) with \(|a|=1\). Write \(a_j=v_j/|v_j|\), \(t_j=S_j/C_j=|v_j|\), and \(f_j=\langle a_j,x\rangle\), \(g_j=\langle a_j,N\rangle\). The cancellation identity \[ Bx+SfN'=x+f\,\frac{SN-(C-1)a}{D} \tag{51}\] gives \[Y_j=r_j\left[x+ f_j\frac{t_jN-(1-C_j^{-1})a_j}{1-t_jg_j}\right]+b_jN'_j.\] On compact subsets of \(\Sigma\setminus Z_a\), where \(Z_a=\{x:N(x)=a\}\), the denominators stay uniformly positive. Also \[N'_j= \frac{C_j^{-1}N+(1-C_j^{-1})g_ja_j-t_ja_j}{1-t_jg_j} \longrightarrow-a.\] The maps and all their source derivatives therefore converge locally to \[ Y_\infty(x)=r_*\left[x+ \frac{\langle a,x\rangle}{1-\langle a,N\rangle}(N-a) \right]-b_*a, \qquad r_*=\sqrt{1-b_*^2}. \tag{52}\] This computation allows the directions \(a_j\) to vary.

The exceptional normal fiber \(Z_a\) is source-null. Indeed \(Z_a\subset\{x:\langle a,x\rangle=0\}\), whereas \(\langle a,x\rangle\) is a nontrivial analytic function by Lemma 5. Its zero set has zero \(m\)-dimensional measure. No rank assumption on the Gauss map is involved.

The point fiber \(F=Y_\infty^{-1}(y_*)\), defined off \(Z_a\), may instead have positive source measure. On a countable cover by relatively compact coordinate patches in \(\Sigma\setminus Z_a\), the area formula gives \[\int_{F\cap K}J_mY_\infty\,d\mu=0\] for each such patch \(K\). Hence \(J_mY_\infty=0\) almost everywhere on \(F\). Off \(F\cup Z_a\) the shrinking-ball indicator is eventually zero; on \(F\), local smooth convergence gives \(J_mY_j\to0\) almost everywhere. We conclude that \(I_j\to0\) almost everywhere on \(\Sigma\), and \(0\le I_j\le1\) again permits dominated convergence. In particular, convergence of the indicators \(\mathbf1_{T_j}\) is unnecessary. Only the restricted Jacobians are bounded; no bound for the unrestricted \(J_mY_j\) near \(Z_a\) has been used.

Finally, if (49) failed, compactness of \(Q\) and \(S^{m+1}\) would supply sequences of the above form whose ball masses stay bounded below by a positive constant. The sequential result excludes them. ◻

Proof. Suppose otherwise. There are \(u_j=(v_j,b_j)\in Q\setminus U\) with \(\mathbf M(\widehat\Phi(u_j))\to|\Sigma|\). The mass bound forces \(b_j\to0\). Parameters on the boost boundary have mass at most \(\sigma_m\), which is strictly smaller by (53); discard those terms. After taking a subsequence, \(v_j\to v_*\ne0\). Set \(t_*=|v_*|\in(0,1]\) and \(a=v_*/|v_*|\). Discard finitely many terms so that \(v_j\ne0\). For each \(j\), set \(t_j=|v_j|\), \(a_j=v_j/|v_j|\), \(C_j=(1-t_j^2)^{-1/2}\), \(r_j=\sqrt{1-b_j^2}\), \(f_j=\langle a_j,x\rangle\), and \(g_j=\langle a_j,N\rangle\).

The open set \(\{A\ne0\}\) is nonempty. The analytic function \(f=\langle a,x\rangle\) cannot vanish on that entire open set, by fullness and connectedness. Choose a compact source patch \(K_0\) of positive measure on which \(|A|\) and \(|f|\) are bounded below by positive constants. It is also disjoint from \(\{N=a\}\), because \(N=a\) implies \(f=0\). On this fixed patch, the formula for \(h_j=h_{v_j,b_j}\) becomes \[h_j=\frac{b_j/C_j+t_jr_jf_j}{1-t_jg_j} \longrightarrow h_*:=\frac{t_*f}{1-t_*\langle a,N\rangle},\] uniformly. Its limit is finite and nonzero on \(K_0\), including when \(t_*=1\).

For a real symmetric matrix \(M\), define its truncated determinant by \[\mathcal D(M)= \begin{cases} \det M,&M\ge0,\\ 0,&M\not\ge0. \end{cases}\] This is continuous: every boundary point of the positive-semidefinite cone is singular, so its determinant vanishes. At every \(x\in K_0\), the matrix \(I-h_*(x)A(x)\) has trace \(m\) and is not the identity. If it is positive semidefinite, strictness in the arithmetic–geometric mean inequality gives determinant less than one; if it is not, \(\mathcal D\) is zero. Compactness of \(K_0\) and continuity therefore give \(\delta>0\) with \[\mathcal D(I-h_*A)\le1-2\delta\quad\hbox{on }K_0.\] For all large \(j\) this yields \(\mathcal D(r_jI-h_jA)\le r_j^m-\delta\) on \(K_0\). On all of \(\Sigma\), the restricted determinant is at most \(r_j^m\). Thus the area formula estimate gives \[\mathbf M(\widehat\Phi(u_j)) \le\int_\Sigma\mathcal D(r_jI-h_jA)\,d\mu \le r_j^m|\Sigma|-\delta\mu(K_0),\] contradicting convergence to \(|\Sigma|\). The proof supplies a uniform loss on a fixed portion of the source; it makes no upper-semicontinuity assumption about area in the flat topology. ◻

Proof. Use a tubular neighborhood of the smooth cone on a compact annulus containing \(\{sx:x\in\Sigma,\ 1/2\le s\le3/2\}\), constructed inside a slightly larger annulus. Compact embeddedness supplies a uniform \(\epsilon>0\) so that, for these central radii and \(|h|<\epsilon\), the point \(sx+hN(x)\) has unique closest cone point \(sx\), and signed distance \(h\). This is a tube for the whole cone: after shrinking \(\epsilon\), points at radii outside the larger annulus cannot be closest, and uniqueness within that annulus follows from its tubular neighborhood.

For any actual level point \(y\in\Lambda_{v,b}\), the boost gives \[|h|=|Cb+S\langle a,y\rangle|\le C|b|+S\longrightarrow0 \quad\hbox{as }(v,b)\to(0,0).\] Every closest cone point has radius \(r=\sqrt{1-b^2}\to1\). Thus all actual contacts lie in this one fixed tube, so each is unique. Conversely, for every source point \(x\), the inverse formula gives \[|h_{v,b}(x)|\le\frac{|b|+Sr}{C-S}\longrightarrow0\] uniformly on \(\Sigma\). Consequently \(z=rx+h_{v,b}(x)N(x)\) lies in the same tube and really has signed distance \(h_{v,b}(x)\). Its transformed point \(Y_{v,b}(x)\) is therefore an actual level point. This proves equality with the entire level and uniqueness of its source representation, not merely existence of one nearby component.

For these parameters the matrix \(rI-h_{v,b}A\) is uniformly positive definite, so (44) gives an immersion. Uniqueness of the contact and invertibility of the boost give injectivity; compactness then gives an embedding. All the formulas are smooth in \((v,b,x)\) near the center.

In the tube \(d\varphi=N\cdot dz\). Differentiating the graph relation at one of the level points yields \[0=dh-N\cdot dz =D\bigl(db-N'\cdot dy\bigr).\] The derivative with respect to \(b\) is \(D>0\), so the graph is smooth near that point and its Euclidean gradient is \(N'\). Projecting onto the tangent space of the unit sphere and using (43) proves (54). Each level point therefore has a neighborhood in which \(E_{v,b}\) is one side of a regular smooth hypersurface. This proves the claimed topological and reduced boundary identities, with multiplicity one. ◻

Proof. At a spherical point \(x\) with tangent plane \(P\subset x^\perp\), assign to \(rx\) the plane \(\mathbb Rx\oplus P\) and the measure \(r^d\,dr\,dV(x,P)\). This defines the cone and gives (58). The polar map is a diffeomorphism away from the origin; rectifiability and integer multiplicity there follow from the area formula. The origin has zero mass.

For stationarity, first take a smooth ambient field \(X\) supported away from the origin and write \(X(rx)=f(r,x)x+Y_r(x)\), where \(Y_r(x)\perp x\). The trace of its derivative on the cone plane is \[\operatorname{div}_{\mathbb Rx\oplus P}X(rx) =\partial_r f(r,x)+\frac{d}{r}f(r,x) +\frac1r\operatorname{div}^{S}_{P}Y_r(x).\] After multiplication by \(r^d\) and integration, the first two terms integrate to the endpoint values of \(r^df\), and the last term vanishes by spherical stationarity for each \(r\). Thus the cone is stationary off the origin. For a general compactly supported \(X\), insert a radial cutoff that is zero on \(B_\varepsilon\) and one outside \(B_{2\varepsilon}\), with derivative bounded by \(c/\varepsilon\). Formula (58) bounds the omitted first-variation terms by \(c_X(\varepsilon^{d+1}+\varepsilon^d)\), which tends to zero.

For the last assertion, \([\mathcal C(V)]\) has zero boundary on every annulus: an odd component multiplicity gives a smooth mod-two product cylinder, and an even multiplicity gives zero. Truncate this chain inside radius \(\varepsilon\). Its difference from the full chain has mass at most \(\mathbf M(V)\varepsilon^{d+1}/(d+1)\), whereas its inner boundary has mass at most \(\mathbf M(V)\varepsilon^d\). The truncated chains therefore converge locally flat to \([\mathcal C(V)]\) and their boundaries converge locally flat to zero. Continuity of the boundary operator proves \(\partial[\mathcal C(V)]=0\) also at the vertex. ◻

Proof. Integrality gives density at least one at \(\|V\|\)-almost every point. Every neighborhood of \(z\) has positive mass, so choose such points \(x_i\to z\). If \(\overline{B_r(z)}\subset U\) and \(d_i=|x_i-z|<r\), monotonicity at \(x_i\) and ball inclusion give \[\|V\|(B_r(z))\geq\|V\|(B_{r-d_i}(x_i)) \geq\omega_k(r-d_i)^k.\] Let \(i\to\infty\), then \(r\downarrow0\). This proves the asserted lower bound also at support points that are exceptional for rectifiability.

For a cone, its homogeneous mass measure gives \(\|\mathcal C\|(B_R(0))=\omega_k\Theta(\mathcal C,0)R^k\). For \(R>|z|\), the inclusions \[B_{R-|z|}(0)\subset B_R(z)\subset B_{R+|z|}(0)\] identify the limit of the mass ratios centered at \(z\) as \(R\to\infty\) with \(\Theta(\mathcal C,0)\). Monotonicity proves the upper bound in (60). Tangent-cone existence and the monotonicity formula give the stated tangent density, which is positive by the first part. ◻

Proof. Write \(T_i=(\eta_{z,\rho_i})_\#\mathcal C\to T\), where \(\eta_{z,\rho_i}(y)=(y-z)/\rho_i\) and \(\rho_i\downarrow0\). For fixed \(b\in\mathbb R\), dilation invariance of \(\mathcal C\) by the positive factors \(1+b\rho_i\) implies invariance of \(T_i\) under \[y\longmapsto(1+b\rho_i)y+bz.\] Passing to the limit shows that \(T\) is invariant under translation by \(bz\). Its weight consequently disintegrates as \(\|T\|=\mathcal L^1\otimes\nu\) on \(\mathbb Re\oplus e^\perp\).

We check that this is an integral varifold product. At almost every rectifiable point of \(T\), a measure blow-up is an integer multiple of the measure on its approximate tangent plane. These blow-ups retain invariance under translations in the \(e\)-direction, so that plane contains \(e\). The tangential Jacobian of the coordinate \(\pi(te+w)=t\) is therefore one almost everywhere. Rectifiable coarea gives, for almost every \(t\), a \((k-1)\)-rectifiable slice with the original integer multiplicities; its tangent plane is the intersection of the tangent plane of \(T\) with \(e^\perp\). Product disintegration identifies the weight of this slice, after translation to \(e^\perp\), with \(\nu\). Thus \(\nu\) defines an integral varifold \(W\) and the tangent-plane identification gives \(T=\mathbf v(\mathbb Re)\times W\) as varifolds. Here we use rectifiable coarea (Federer 1978, sec. 3, p. 300).

For a compactly supported field \(Y\) on \(e^\perp\) and a compactly supported smooth function \(\chi\) on \(\mathbb R\) with \(\int\chi\ne0\), test stationarity of \(T\) with \(X(te+w)=\chi(t)Y(w)\). Since \(Y\) is perpendicular to \(e\), \[0=\delta T(X)=\left(\int_{\mathbb R}\chi(t)\,dt\right)\delta W(Y).\] Hence \(W\) is stationary. Dilation invariance of the tangent cone \(T\) and the product formula imply dilation invariance of \(W\). Writing \(\theta=\Theta(W,0)\), integration over a ball gives \[\|T\|(B_1) =\theta\omega_{k-1} \int_{-1}^1(1-t^2)^{(k-1)/2}\,dt =\theta\omega_k.\] This proves (61); Lemma 26 makes \(W\) nonzero.

If \(\partial[\mathcal C]=0\), every \(T_i\) is cyclic, and (59) implies \(\partial[T]=0\). The mod-two slicing identity for the boundary (White 1999, sec. 3, equation (1) and Proposition 3.1) gives \[\partial\langle[T],\pi,t\rangle =\langle\partial[T],\pi,t\rangle=0 \quad\text{for almost every }t,\] where signs disappear modulo two. For the locally flat chain, apply the identity on a countable exhaustion by relatively compact regions; restriction compatibility gives one full-measure set of levels where it holds locally everywhere. Choose one such level where the preceding rectifiable slice description also holds. The slice there is the translate of \([W]\) to \(te+e^\perp\), because \(T\) is the product just established. Translation back proves \(\partial[W]=0\). No assertion about whether zero is a good slicing level is needed. ◻

Proof. Apply Allard’s interior regularity theorem (Allard 1972, sec. 8), in the formulation of (Simon 2018, chap. 5, hypotheses (5.1) and Theorem 5.2), on a sufficiently small rescaled ball from the sequence giving that tangent. Its hypotheses hold as follows. Integrality supplies the almost-everywhere density lower bound one. Stationarity supplies zero generalized mean curvature and zero first variation, hence the required mean-curvature bound for any exponent greater than \(k\). Convergence to the multiplicity-one plane supplies normalized mass arbitrarily close to \(\omega_k\) and arbitrarily small height and tangent-plane excess on a fixed smaller ball. The center remains in the support. Allard’s theorem therefore gives a single \(C^{1,\alpha}\) graph near the center. On that connected graph the stationary integral multiplicity is constant; its tangent multiplicity is one, so the constant is one. The higher-regularity conclusion of (Simon 2018, chap. 5, Theorem 6.1), applied with the smoothly prescribed mean curvature identically zero, makes the graph smooth; that statement permits arbitrary codimension. This theorem has no stability hypothesis and no upper bound on \(k\). ◻

Proof. Hold \(m\) fixed throughout. We prove (i) by induction in \(k\) and then obtain (ii) from the final induction step.

An integral one-dimensional cone is a finite union of rays with positive integer weights: radial slicing of its conical rectifiable measure gives a zero-dimensional integral measure on the unit circle, and local finiteness makes its total weight finite. Write the distinct unit ray directions as \(u_j\), their weights as \(q_j\), and \(N=\sum_jq_j\). Stationarity at the origin says \[\sum_jq_ju_j=0,\qquad \Theta(\mathcal C,0)=N/2.\] If \(m\geq4\), Lemma 2 gives \(\vartheta_m\leq3/2\), so the strict bound (62) gives \(N<3\). Nonzero stationarity excludes \(N=1\), leaving two opposite rays of weight one. If \(m=2\) or \(3\), the same lemma gives \(\vartheta_m<2\), hence \(N<4\); cyclicity says that the boundary coefficient \(N\) at the origin is even. Again \(N=2\) and balance gives a multiplicity-one line. This proves the base case in both branches.

Suppose next that \(2\leq k\leq m+1\) and that (i) has been proved in dimension \(k-1\). At any nonzero support point \(z\), Lemma 27 expresses a tangent as \(\mathbf v(\mathbb R)\times W\). Its factor is nonzero, stationary and integral, is cyclic when required, and has density at most \(\Theta(\mathcal C,0)<\vartheta_m\) by Lemma 26. The induction hypothesis makes \(W\) a multiplicity-one plane. Thus this tangent of \(\mathcal C\) is a multiplicity-one plane, and Lemma 28 proves smooth embedded multiplicity-one structure near \(z\).

It follows that \(\Lambda=\mathop{\mathrm{spt}}\|\mathcal C\|\cap S^k\) is a smooth closed embedded \((k-1)\)-manifold. The radial direction lies in the cone tangent plane and is normal to the sphere, so this intersection is transverse. Polar coarea gives \(\mathcal C=\mathcal C(\mathbf v(\Lambda))\) with multiplicity one. Reversing the angular first-variation calculation in Lemma 25 shows that \(\Lambda\) is minimal in \(S^k\).

Each connected component \(\Lambda_j\) is compact and minimal, and its cone is stationary. Lemma 26 applied to that cone, together with (58), yields \[ \frac{\mathop{\mathrm{Vol}}(\Lambda_j)}{\sigma_{k-1}}\geq1. \tag{63}\] Since the total cone density is less than \(\vartheta_m<2\), there is only one component.

If \(k=2\), a closed connected embedded minimal curve in \(S^2\) is a great circle by the geodesic equation, so its cone is a plane. If \(3\leq k\leq m\) and \(\Lambda\) were not totally geodesic, the outer induction in dimension \(k-1\) would give \[\Theta(\mathcal C,0) =\frac{\mathop{\mathrm{Vol}}(\Lambda)}{\sigma_{k-1}} \geq\vartheta_{k-1}\geq\vartheta_m,\] contrary to (62). Thus \(\Lambda\) is a great sphere and \(\mathcal C\) a multiplicity-one plane, completing (i). For \(k=m+1\), the already proved plane assertion in dimension \(m\) gives exactly the smoothness and connectedness asserted in (ii). We do not apply the dimension-\(m\) volume statement to this last link. ◻

Proof. Lemma 25 and the mass bound put \(\mathcal C(V)\) under part (ii) of Proposition 29. Its smooth multiplicity-one link identifies \(V\) by the polar cone formula. An equator has area \(\sigma_m\). ◻

Proof. For every member of \(\mathcal E_m\), the family of Proposition 18 is detecting and has maximum mass at most its area. Lemma 14 therefore gives \(\mathop{\mathrm{Vol}}(\Sigma)\geq\gamma_m>\sigma_m\). Nonemptiness gives \(H<a_m\), proving (64).

Choose \(\Sigma_i\in\mathcal E_m\) with areas \(H_i\to H\) and set \(\mathcal C_i=\mathcal C(\mathbf v(\Sigma_i))\). These cones are stationary, integral, and cyclic by Lemma 25. Their exact masses satisfy \[ \|\mathcal C_i\|(B_R(0))=\frac{H_iR^{m+1}}{m+1} \qquad(R>0). \tag{65}\] Consequently their masses are uniformly bounded on every compact set. Allard’s integral-varifold compactness theorem and a diagonal subsequence give a stationary integral limit \(\mathcal C\) in all of \(\mathbb R^{m+2}\). Every fixed dilation leaves each \(\mathcal C_i\) invariant, so by continuity of pushforward it leaves \(\mathcal C\) invariant as well. White’s theorem in the form (59) gives \(\partial[\mathcal C]=0\).

We verify that the limiting mass is exactly \(H\) at the spherical level. The weight of the conical limit is homogeneous of degree \(m+1\). It has zero mass on every sphere of positive radius: otherwise dilation would produce infinitely many disjoint spheres with uniformly positive mass in a fixed compact annulus. Weak convergence can thus be evaluated on every centered ball. Passing to the limit in (65) yields \[\|\mathcal C\|(B_R(0))=\frac{HR^{m+1}}{m+1}, \qquad \Theta(\mathcal C,0)=\frac H{\sigma_m}<\vartheta_m.\] In particular \(\mathcal C\) is nonzero. This exact identity retains the infimum value; a lower-semicontinuity bound would not give the same conclusion by itself.

Proposition 29(ii) now makes the link \(\Sigma_*=\mathop{\mathrm{spt}}\|\mathcal C\|\cap S^{m+1}\) a smooth closed connected embedded minimal hypersurface, with multiplicity one. Its area is exactly \(H\). The bounds \(\sigma_m<H<a_m\) exclude an equator and put \(\Sigma_*\) in \(\mathcal E_m\), as required. ◻