Higher-Dimensional Stable Bernstein Theorems

We outline the strategy for the unconditional stable Bernstein theorem in dimensions \(n=3,4,5\). The argument has three main steps: a conformal change producing positive spectral bi-Ricci curvature, the construction of \(\mu\)-bubbles, and uniform area bounds for these bubbles via spectral Ricci estimates. The conformal and spectral curvature estimates are proved in §5.2.1; the \(\mu\)-bubble reduction is treated in §5.2.2, and the spectral Ricci estimates are recorded in §5.2.6.

Step 1: Conformal change and spectral bi-Ricci curvature. Let \(M^n\hookrightarrow \mathbb{R}^{n+1}\) be a complete, two-sided, stable minimal immersion. Fix \(0\in M\) and write \(r(x)=|x|\) for the Euclidean distance from the origin. Following Schoen–Simon–Yau and the recent stable Bernstein literature, we pass to the conformal metric

where \(g\) is the induced metric. Stability of \(M\) yields a spectral inequality for the conformal Laplacian with a curvature potential. To state it uniformly in dimension, we introduce the bi-Ricci curvature.

In dimension \(m=3\), the bi-Ricci curvature is independent of the chosen orthonormal pair:

This is why the three-dimensional part of the argument is often stated as a spectral scalar-curvature condition.

We use spectral curvature condition to mean a lower bound for the first eigenvalue of a Schrödinger operator whose potential is built from one of these curvature quantities. The coefficient depends on the dimension and on the normalization used in the \(\mu\)-bubble argument.

In the conformal metric \(\tilde g=r^{-2}g\), stability gives the concrete inputs needed below: for \(n=3\),

by the conformal scalar-curvature computation in §5.2.1, while for \(n=4\),

after the corresponding conformal bi-Ricci normalization.

Thus, after conformal reparametrization, the problem is reduced to studying a manifold with non-negative spectral bi-Ricci curvature (in dimension three this is precisely non-negative spectral scalar curvature). This replaces the extrinsic curvature input of the minimal hypersurface by an intrinsic spectral curvature hypothesis on \((M,\tilde{g})\).

Step 2: Construction of \(\mu\)-bubbles. The second step is to produce separating hypersurfaces inside a collar of \((M,\tilde{g})\) with controlled geometry. These are \(\mu\)-bubbles.

Classically, a soap bubble is a hypersurface of constant mean curvature (CMC) enclosing a prescribed volume—the archetypal prescribed mean curvature problem. More generally, one studies hypersurfaces whose mean curvature is a prescribed function of position and geometry; Gromov introduced and systematically used such constructions in his work on scalar curvature, naming them \(\mu\)-bubbles (the name reflects the measure \(\mu\) specifying the prescription). In the present setting, one considers a Riemannian manifold \((N,g)\) with boundary \(\partial N=\partial_+N\cup\partial_-N\) and seeks a domain \(\Omega\subset N\) whose boundary \(\Sigma=\partial\Omega\setminus\partial_-N\) (the \(\mu\)-bubble) solves a variational problem of prescribed mean curvature type: schematically, one minimizes a functional of the form

for a weight \(w\) (often the first eigenfunction of a suitable operator on \(N\)) and a carefully chosen function \(h\) determined by the curvature of \(N\). Under the spectral bi-Ricci condition from Step 1, such a minimizer exists and provides a \(\mu\)-bubble \(\Sigma\) separating \(\partial_-N\) from \(\partial_+N\) (Theorem 5.2.8).

In recent years, \(\mu\)-bubbles have become a central tool in positive scalar curvature geometry and have led to breakthroughs in the classification of manifolds with positive scalar curvature, the resolution of the Riemannian positive mass theorem in many settings, and the stable Bernstein program in dimensions three through five. The method is flexible: by choosing the prescription (the \(\mu\)-data and the function \(h\)), one encodes the desired curvature inequality into the Euler–Lagrange equation of the \(\mu\)-bubble.

Step 3: Volume and diameter estimates for \(\mu\)-bubbles. The third step controls the size of each \(\mu\)-bubble \(\Sigma\). This is analogous in spirit to the dimension reduction in the Schoen–Yau proof of the positive mass theorem: one does not work directly on the full ambient manifold, but on a hypersurface of one dimension lower whose geometry is more rigid.

After Step 2, each \(\mu\)-bubble \(\Sigma\) is a closed hypersurface in \(N\) (of dimension \(m-1\)) lying in a thin collar between \(\partial_-N\) and \(\partial_+N\). Using the construction and the spectral bi-Ricci hypothesis on \(N\), one shows that \(\Sigma\) itself carries a spectral non-negative Ricci condition: the first eigenvalue of an operator of the form \(-\Delta^\Sigma+\alpha\,\mathrm{Ric}\) is non-negative for some \(\alpha\in(0,2)\) depending on \(n\). One then studies manifolds with spectral non-negative Ricci curvature by methods similar to Bray’s proof of the Bishop–Gromov comparison: spectral bounds on \(-\Delta+\mathrm{Ric}\) imply diameter and volume upper bounds (Theorem 5.2.13 and the discussion in §5.2.6). In particular, each component of \(\partial\Omega\) has area and intrinsic diameter bounded by constants depending only on the spectral bound, not on the size of the collar.

Conclusion of the proof. Combining Steps 1–3, one obtains separating \(\mu\)-bubbles in long conformal annuli with uniform \(\tilde g\)-area and diameter bounds. As explained in the proof of the Euclidean volume-growth estimate in §5.2.6, these conformal estimates imply

for every \(\rho\gt{}0\). Thus the \(\mu\)-bubble argument supplies the area-growth hypothesis needed for the stable Bernstein theorems with area growth, and the cited classification results force \(M\) to be an affine hyperplane. This completes the proof sketch; the conformal and spectral bi-Ricci estimates used in Steps 1–3 are proved in §5.2.1.

Conformal change of the metric on minimal hypersurfaces

Let \(M^n\hookrightarrow \mathbb{R}^{n+1}\) be a complete, two-sided, stable minimal immersion with induced metric \(g=\varphi^*\delta\). Fix \(0\in M\) and write \(r(x)=|x|\). Throughout we use the conformal metric

We use the bi-Ricci convention fixed above: \(\mathrm{BiRic}\) denotes the minimum over orthonormal two-frames, and in dimension three \(\mathrm{BiRic}=\frac12 R\).

Under a general conformal change \(\tilde{g}=f^{-2}g\), we first relate the intrinsic Hessian on \(M\) to the ambient Euclidean Hessian.

Proof. Write the ambient gradient \(Df=\nabla^M f+\langle Df,\nu\rangle\,\nu\), so \(\nabla^M f\) is the tangential part. By direct computation,

◻

For the conformal metric \(\tilde{g}=e^{2\varphi}g\), the change of Riemannian curvatures is given by

where

For \(\varphi=-\log r\), we have

So

For the conformal metric \(\tilde{g}=r^{-2}g\) with \(r(x)=|x|\), \(\varphi=-\log r\), the conformal curvatures satisfy

We compute \(\mathrm{BiRic}\) on \(M\) using the Gauss equation.

Proof. Using the Gauss equation, we compute

where the last line follows since \(\mathrm{tr}\,A=0\) for a minimal hypersurface. ◻

Now, we choose \(\tilde{e}_1\) and \(\tilde{e}_2\) such that \(\widetilde{\mathrm{BiRic}}(\tilde{e}_1,\tilde{e}_2)\) takes the minimum value. Then, we have

Proof. Recall that

Since \(\langle \vec x,\nu\rangle = r\,dr(\nu)\), we use Young’s inequality to obtain

Combined with (5.2.1) and the fact that

we have

Now we compute, using the fact that \(\operatorname{Tr} A=0\),

where we took

in the last line. Hence, for \(n\geq 3\), we have

Therefore,

and the proposition follows. ◻

Proof. Recall that in the conformal metric, we have

Then the stability inequality for \(M\) implies

for any \(f \in C_c^{0,1}(N,\tilde g)\). We take

for \(\psi \in C_c^{0,1}(N,\tilde g)\). Then

so

We have

Since

we have

Finally, we use integration by parts and the displayed formula for \(\tilde\Delta\log r\) below to compute

Recall that we have the following

Altogether, we obtain

as desired. ◻

Proof. Combining Proposition 5.2.6 with Proposition 5.2.5, for every \(\psi\in C_c^{0,1}(N,\tilde g)\) we get

where

Since \(0\leq |dr|^2\leq1\) and \(C_1\leq0\) for \(3\leq n\leq5\), the right-hand side is bounded from below by

This proves the claimed spectral lower bound. ◻

Construction of the \(\mu\)-Bubble

The constant \(1\) is only a normalization and can be replaced by any positive constant after rescaling the metric.

Second variation of weighted area functional

Given a domain \(\Omega\) in \(N\), we do a variation \(\Omega_t\) under vector field \(V\) and denote \(\Sigma_t=\partial\Omega_t\backslash \partial_-N\) and \(\Sigma=\Sigma_0\). We choose \(V\) such that \(\nabla_{V}V=0\) and \(V=\phi \nu\) along \(\Sigma\). (Note that in general, we do not have \(V\bot \Sigma_t\) for every \(t\).)

The first variation is straightforward.

Hence, the stationary gives us the formula

For the derivative of \(\int_{\Sigma} w\,\mathrm{div}^\Sigma(V^T)\,d\mathcal{H}^{n-1}\), we rewrite it using integration by parts as

After differentiating,

where we have used \(\nabla_{V}V=0\). Note that we have

Then, the derivative of \(\int_{ \Sigma} \left< \nabla^\Sigma w, \phi \nabla_{V}\nu \right>\) is

Now, we can compute the second variation. Note that we can use \(V\left< V, \nu \right> =0\).

For the Hessian, we have

using integration by parts, we have

This is the weighted-area second variation formula used below.

Properties of the minimizer

Proof. Suppose \(\Omega\) is the minimizer to \(\mathcal{A}\). In the spectral bi-Ricci application the weight in the functional is

and we consider

Then

Hence

The first variation gives

Put

Then \(H=h-av\), and hence

We also need to compare the ambient and intrinsic Laplacians of the weight \(W=w^a\). Along \(\Sigma\),

Thus

Since \(\Sigma\) is stable for the weighted functional, the second variation formula with weight \(W\) gives, after moving the potential terms to the right-hand side,

The final term is obtained by combining the Laplacian comparison contribution \(-a(a-1)v^2\) with the \(ahv\) term. Since \(0\lt{}a\leq2\), it is nonnegative and can be discarded.

It remains to estimate the left-hand side of (5.2.2). After substituting \(\phi=w^{-a/2}\psi\), it becomes

and Young’s inequality with \(\varepsilon=(4-a)^{-1}\) gives the estimate. Therefore,

Combining this estimate with (5.2.2) and the spectral inequality \(-a\Delta^Nw+\mathrm{BiRic}_Nw\geq w\) gives

Now, we need to use \(|A|^{2}+\mathrm{Ric}_N(\nu,\nu)\) to bound \(\mathrm{BiRic}_N\).

This is the extension of the usual Schoen–Yau trick. Recall that we have (for \(n=3\))

This is exactly the same as the inequality in the lemma.

Proof. Suppose \(e_1\in T_p\Sigma\) is a unit direction where \(Ric_\Sigma\) attains its minimum. Using the Gauss equation, we have

Note that we have

By the choice of \(e_1\), \(\mathrm{Ric}_\Sigma=\mathrm{Ric}_\Sigma(e_1,e_1)\). Hence, we have

Equivalently,

which is the desired inequality. ◻

Now, we go back to our inequality. When \(3\leq n\leq 4\), we have \(\frac{6-n}{4}\geq \frac{1}{2}\) and therefore

Together with the condition for \(h\), \(1+h^{2}-2|\nabla^N h|\geq0\), we have

Hence

This is the desired inequality. ◻

Completion of the \(\mu\)-Bubble Construction

To prove Theorem 5.2.8, choose \(h\) by

where \(\tilde{d}(x,\partial_-N)\) is a smoothing of \(d_N(x,\partial_-N)\) with \(\mathrm{Lip}\,\tilde{d}\leq 2\). We denote \(\varphi(x)=\frac{\tilde{d}(x,\partial_-N)}{4}-\frac{\pi}{2}\). Then

To show the existence of the minimizer, fix a domain \(\Omega_0\) containing \(\partial_-N\) but not \(\partial_+N\) such that \(\partial \Omega_0\) lies in \(\tilde{d}(x,\partial_-N)\lt{} 4\pi\). We consider minimizing the following (relative) energy functional

The direct method gives a minimizer in the corresponding relative homology class. Since \(3\leq n\leq4\), the free boundary part \(\Sigma=\partial\Omega\setminus\partial_-N\) is smooth after the usual regularity theory for prescribed-mean-curvature hypersurfaces. The choice \(d_N(\partial_+N,\partial_-N)\geq5\pi\) gives enough room for the smoothing \(\tilde d\) and hence for the above choice of \(h\). Applying the stability inequality from the preceding proposition proves Theorem 5.2.8.

Diameter estimate and volume estimate under spectral Ricci curvature bound

The following result of Antonelli–Xu [AX24] gives the radius/diameter and volume estimates under a spectral Ricci lower bound. In this subsection \(\mathrm{Ric}_N(v,v)\) denotes the Ricci tensor on a unit vector; when \(\mathrm{Ric}_N\) appears without arguments in a spectral inequality, it means the smallest eigenvalue of the Ricci tensor. The bi-Ricci notation is the one fixed above. In the low-dimensional range considered here, the minimizers appearing in the \(\mu\)-bubble arguments are smooth. The parameter \(\alpha\) below denotes the coefficient of the spectral Laplacian term.

Proof. This is precisely the complete-case corollary in [AX24], based on Xu’s subcritical spectral Bonnet–Myers theorem. The latter gives compactness and the uniform bound \(\mathrm{diam}(N)\leq C(n,\alpha)\lambda^{-1/2}\) in the stated range. Once compactness is known, Theorem 5.2.13 applies and gives (5.2.4) and (5.2.5). Since \(N\) is the quotient of \(\widetilde N\) by a finite group of deck transformations, the same volume upper bound also holds for \(N\). ◻

Even in the larger sharp range \(0\leq\alpha\leq (n-1)/(n-2)\), if \(N\) is complete and the positive function \(u\) in (5.2.3) satisfies \(0\lt{}\inf_N u\leq\sup_Nu\lt{}\infty\), then Corollary 5.2.17 below proves compactness directly. Thus noncompact examples in the super-subcritical range must have \(u\) degenerating at infinity.

Weighted geodesic proof of the diameter estimates.

We also record the weighted-geodesic calculation, since it gives another way to see the Bonnet–Myers part.

For a positive function \(u\) on \((M^m,g)\), we define the weighted geodesic distance from \(p\) to \(q\) by

where the infimum is taken over piecewise smooth curves from \(p\) to \(q\). As above, \(\mathrm{Ric}_M\) in a spectral inequality denotes the smallest eigenvalue of the Ricci tensor, while \(\mathrm{Ric}_M(T,T)\) denotes the tensor on the vector \(T\).

Proof. Let \(T=\alpha'\) and put

Then \(L_u^\alpha\) is exactly the distance function of the conformal metric \(\tilde g\). Write

Along \(\alpha\) we have, by the definition of \(\tilde g\) and of the distance function \(\rho\),

We first compute \(\tilde\Delta\rho\) in the conformal metric. Choose a \(\tilde g\)-parallel orthonormal frame along \(\alpha\),

and write \(\tilde E_i=u^{-\alpha}e_i\). Thus \(e_1,\ldots,e_{m-1},T\) is \(g\)-orthonormal and \(e_i\perp T\). Let \(\phi\) be a test function with \(\phi(0)=0\) and \(\phi(l)=1\), and set

These fields vanish at \(p\) and equal \(\tilde E_i(q)\) at \(q\). Hence the ordinary second variation formula for the \(\tilde g\)-distance gives, for \(1\leq i\leq m-1\),

The missing \(m\)-th direction is radial and contributes nothing, since \(\tilde\nabla^2\rho(\tilde T,\tilde T)=0\) away from the cut locus. Summing the preceding inequality over the transverse directions gives

Here we used \(d\tilde s=u^\alpha ds\), \(\tilde T\phi=u^{-\alpha}\phi_s\), and \(\widetilde{\mathrm{Ric}}(\tilde T,\tilde T)=u^{-2\alpha}\widetilde{\mathrm{Ric}}(T,T)\).

Now return to the original metric. The conformal Laplacian formula for \(\tilde g=e^{2\log u^\alpha}g\) gives, at \(q\),

Combining (5.2.10) and (5.2.11), we get

Now rewrite the curvature term in (5.2.12) in the original metric. The conformal Ricci formula is

The conformal connection formula is

Applying it to the \(\tilde g\)-geodesic equation \(\tilde\nabla_{\tilde T}\tilde T=0\) gives

and therefore

Using this in the Ricci formula gives the pointwise identity along \(\alpha\)

The endpoint term in (5.2.12) is written in the same \(u(q)^{2\alpha}\int u^{-\alpha}(\cdots)\,ds\) scale as

because \(\phi(0)=0\) and \(\phi(l)=1\). Substituting (5.2.13) into (5.2.12) and expanding this total derivative gives the original-metric formula

Using (5.2.6) and \(\mathrm{Ric}_M(T,T)\geq \mathrm{Ric}_M\), along the curve

Together with

this gives

Only at this point do we change the test function. The subcritical estimate uses the substitution

Thus

Inserting this into (5.2.15) cancels the weight \(u^\alpha\) and gives

Since \(|\nabla u|^2\geq u_s^2\), this implies

When \(\alpha\lt{}4/(m-1)\), Cauchy’s inequality gives

This proves (5.2.7); the sine choice gives (5.2.8).

For the bounded-\(u\) diameter estimate one uses a different exponent. Put

Then

Substituting this into (5.2.15), the cross term cancels exactly and we obtain

If \(\alpha\leq (m-1)/(m-2)\), the two terms involving derivatives of \(u\) are nonpositive. Dropping them gives (5.2.9). ◻

Proof. Assume first that two points \(p,q\) are joined by an \(L_u^\alpha\)-minimizing curve \(\alpha:[0,l]\to M\) parametrized by \(g\)-arclength. Let \(x=\alpha(l/2)\). The weighted excess

has a local minimum at \(x\). Using the two subsegments of \(\alpha\) as barriers and applying (5.2.8) to each half, we get, in the viscosity sense,

Therefore \(l\leq \pi\sqrt{C_{m,\alpha}/((m-1)\lambda)}\). Since \(d_g(p,q)\leq l\), this gives (5.2.17).

If the weighted minimizer is not realized, take a minimizing sequence. The usual Calabi limiting argument gives a broken minimizing object made of finite minimizing segments, weighted minimizing rays, and weighted minimizing lines. At the point where the corresponding excess vanishes, replace any ray by the point at parameter \(t\) on that ray and use the finite-segment barrier above. The right hand side contains the term \(-(m-1)\lambda t/2\) for that ray and is negative for \(t\) sufficiently large, contradicting the local minimum of the excess. Thus the non-realized case cannot occur, and the same diameter bound holds for all pairs of points. Hopf–Rinow then implies compactness. ◻

Proof. Because \(u\) is bounded above and below, the weighted metric \(u^{2\alpha}g\) is complete, so \(L_u^\alpha\)-minimizers exist. Let \(\alpha:[0,l]\to M\) be an \(L_u^\alpha\)-minimizer from \(p\) to \(q\), parametrized by \(g\)-arclength, and let \(x=\alpha(l/2)\). As above, the weighted excess has a local minimum at \(x\).

Use (5.2.9) on each half of \(\alpha\) with

Since \(\alpha(m-3)/(m-1)\geq0\), we estimate the positive term by \(u_+^{\frac{m-3}{m-1}\alpha}\) and the negative term by \(u_-^{\frac{m-3}{m-1}\alpha}\). Thus

Hence

Since \(d_g(p,q)\leq l\) and \(p,q\) were arbitrary, this proves (5.2.18). Compactness follows from Hopf–Rinow. ◻

Volume estimate.

Proof. By the diameter part, \(\widetilde N\) is compact; hence the minimizer \(E\) exists for every \(v_0\in(0,\int_{\widetilde N}\tilde u^\theta)\). We suppress tildes. Put

Let \(\Sigma=\partial E\) and take a smooth normal variation with variational field \(\varphi\nu\). We extend \(\varphi\) to a neighborhood of \(\Sigma\) and use the same symbol for the extension. With the convention used here,

Also

Thus

and differentiating this last expression gives the following formula. In the following displays, the same color marks the terms with the same origin: magenta terms cancel by integration by parts or by the normal derivative of \(\varphi=u^{-\alpha}\); green terms are the spectral-Ricci pair; blue terms come from \(|A_\Sigma|^2\); red terms come from \(u_\nu^2\); orange terms come from \(Hu_\nu\); and purple terms come from the surviving part of the last line.

The last line is just the derivative of the factor \(u^\alpha\varphi\,d\mu_t\) in \(A'(t)\), rewritten with a factor \(u^\theta\):

The volume constraint implies that

where \(A_v\) is the area of this variation written as a function of \(V\). Choose \(\varphi=u^{-\alpha}\) and set

Then \(H=X-\alpha Y\). For \(\varphi=u^{-\alpha}\), the tangential \(\Delta^\Sigma u\) term cancels after integration by parts with the tangential gradient term

Hence the first two lines of (5.2.21), together with (5.2.3) and \(|A_\Sigma|^2\geq H^2/(n-1)\), are bounded above by

The last line of (5.2.21) is

because \(\class{note-ax-cancel}{\varphi_\nu=-\alpha u^{-\alpha}Y}\) and \(\class{note-ax-last}{H+\alpha Y=X}\). Therefore

Expanding \(H=X-\alpha Y\) gives

The inverse-function chain rule gives

Substituting the formulas for \(V',V''\) and using \(\theta=2\alpha/(n-1)\) yields

Here we used \(\alpha\leq(n-1)/(n-2)\) and \(u\geq1\). Holder’s inequality gives

Thus

Since \(A_v\) is an upper barrier for \(I\) at \(v_0\), this is exactly the viscosity inequality (5.2.20). ◻

Proof. Set

Applying the chain rule to positive upper test functions for \(I\), the preceding viscosity inequality is equivalent to

in the viscosity sense. The small-volume assumption gives

For \(\zeta\gt{}0\) define

and define the model profile \(I_\zeta\) by

Let \(\psi_\zeta=I_\zeta^{n/(n-1)}\). Since

direct differentiation gives

The existence interval of \(I_\zeta\) has length

For \(\zeta=|S^{n-1}|\), this is exactly \(\lambda^{-n/2}|S^n|\).

Assume by contradiction that \(V\gt{}\lambda^{-n/2}|S^n|\). Choose \(\zeta\gt{}|S^{n-1}|\) so close to \(|S^{n-1}|\) that \(V_\zeta\lt{}V\). Then \((\psi_\zeta)'_+(0)\gt{}\psi'_+(0)\), so \(\psi\lt{}\psi_\zeta\) on \((0,\delta)\) for some \(\delta\gt{}0\). If the two functions first meet at \(a\in(0,V_\zeta)\), then on \((0,a)\) we have \(\psi\lt{}\psi_\zeta\). Since the function \(s\mapsto -n\lambda s^{(2-n)/n}\) is increasing on \((0,\infty)\), the difference \(w=\psi-\psi_\zeta\) satisfies \(w''\lt{}0\) in the viscosity sense on \((0,a)\). Thus \(w\) is strictly concave there. But \(w(0)=w(a)=0\) and \(w\lt{}0\) on \((0,a)\), which is impossible for a concave function. Hence \(\psi\lt{}\psi_\zeta\) on \((0,V_\zeta)\). Since \(\psi_\zeta(V_\zeta)=0\) while \(V_\zeta\lt{}V\) and \(I\gt{}0\) on \((0,V)\), continuity gives a contradiction at \(V_\zeta\). Therefore \(V\leq \lambda^{-n/2}|S^n|\). ◻

Proof. Normalize \(\min_{\widetilde N}\tilde u=1\) and set

The profile \(I\) satisfies (5.2.20). Since \(\tilde u\) attains its minimum at some point \(\tilde p\), small geodesic balls centered at \(\tilde p\) satisfy

Since these balls are admissible competitors for \(I\), it follows that

Lemma 5.2.19 applied with \(V=V_0\) therefore gives

Since \(\tilde u\geq1\), we have

If equality holds, then \(\int_{\widetilde N}\tilde u^\theta=|\widetilde N|\) and \(\tilde u\geq1\), hence \(\tilde u\equiv1\). The spectral inequality becomes the pointwise Ricci bound \(\mathrm{Ric}_{\widetilde N}\geq(n-1)\lambda\), and the equality case in the classical Bishop–Gromov theorem gives that \(\widetilde N\) is the round sphere of radius \(\lambda^{-1/2}\). ◻

We need a comparison of distance.

Proof. Let \(\alpha\) be a curve joining \(p\) and \(q\) with length \(D+\varepsilon\). Then, we have

◻

Proof. Let \(M^n\hookrightarrow\mathbb{R}^{n+1}\) be as in the theorem, where \(n=3\) or \(n=4\). We fix any point \(x_0\in M\) and suppose \(x_0=0\) after a translation. Let \(\tilde{g}=r^{-2}g\), where \(r(x)=|x|\).

The preceding spectral estimates and \(\mu\)-bubble constructions give the following dimension-dependent constants. There exist \(L_n,A_n,D_n\lt{}\infty\) such that, whenever \(N\) is a compact collar in \((M,\tilde{g})\) with \(d_{\tilde{g}}(\partial_-N,\partial_+N)\geq L_n\), one can find a smooth \(\mu\)-bubble component \(\Sigma\) which separates \(\partial_-N\) from \(\partial_+N\), lies in \(\tilde B_{L_n}(\partial_-N)\), and satisfies

For \(n=3\) and \(n=4\) this follows from the \(\mu\)-bubble reduction in Theorem 5.2.8 together with the spectral Ricci estimates in §5.2.6.

For any \(\rho\gt{}0\), choose \(R\) large such that

Set \(N=B_R^M\setminus B_\rho^M\), with \(\partial_-N=\partial B_\rho^M\) and \(\partial_+N=\partial B_R^M\), and apply the preceding paragraph. If \(x\in\Sigma\), then there is \(y\in\partial B_\rho^M\) with \(d_{\tilde{g}}(x,y)\leq L_n\). By the distance comparison lemma,

Therefore \(\Sigma\subset B_{C_n\rho}\) in the Euclidean metric. Since \(g=r^2\tilde{g}\), the induced measures on the \((n-1)\)-dimensional hypersurface \(\Sigma\) satisfy \(d\mu_g=r^{n-1}d\mu_{\tilde{g}}\), and hence

Since \(M\) is simply connected and has one end [CSZ97], the side of \(\Sigma\) containing \(B_\rho^M(x_0)\) is a compact region. Let \(\Omega_\rho\) denote this region; then \(B_\rho^M(x_0)\subset\Omega_\rho\) and \(\partial\Omega_\rho=\Sigma\). The Michael–Simon isoperimetric inequality for minimal submanifolds [MS73] gives

Consequently,

◻

Mazet’s proof in the ambient space \(\mathbb{R}^6\)

Here \(\mathbb{R}^6\) means the hypersurface dimension is \(5\). Mazet [Maz24] proves that every complete, connected, two-sided stable minimal immersion

is flat. The proof follows the Chodosh–Li–Minter–Stryker strategy [CLMS25], but with one extra parameter in the curvature quantity. This extra parameter is the weighted bi-Ricci curvature.

Let \((N^m,g)\) be a Riemannian manifold, and let \(\{e_1,\ldots,e_m\}\) be an orthonormal basis. For \(\alpha\in\mathbb{R}\), Mazet defines the \(\alpha\)-weighted bi-Ricci curvature by

Its pointwise minimum is

When \(\alpha=1\), this is the usual bi-Ricci curvature:

Thus \(\alpha\) lets one change the relative weight of the curvature directions which are tangent to the eventual \(\mu\)-bubble.

The first step is the same conformal change as before. Remove the points where the immersion hits the origin and set

The Gulliver–Lawson observation is that \((N,\tilde g)\) is complete. Mazet’s main spectral estimate says that, in dimension \(5\), one can choose

so that the stability inequality implies the following weighted bi-Ricci spectral lower bound:

Equivalently,

So \((N,\tilde g)\) does not have a pointwise lower bound \(\widetilde{\Lambda}_\alpha\geq\delta\), but it has the corresponding spectral lower bound.

The second step is to build a weighted \(\mu\)-bubble in a long annulus of \((N,\tilde g)\). The bubble is a hypersurface

which separates the inner and outer boundary of the annulus. The second variation of the weighted \(\mu\)-bubble turns the spectral \(\operatorname{BRic}_\alpha\) bound on \(N\) into a spectral Ricci bound on \(\Sigma\). More precisely, if

then the induced metric on \(\Sigma\) satisfies an inequality of the form

For Mazet’s parameters,

This is the numerical point that allows the Antonelli–Xu spectral Bishop–Gromov theorem [AX24] to be applied to the \(4\)-dimensional bubble. It gives a uniform \(\tilde g\)-volume bound for \(\Sigma\).

Finally one returns to the original metric \(g\). If \(\Sigma\) is chosen around \(B_\rho^M(p_0)\), the conformal collar bound gives

so the \(\tilde g\)-volume bound for \(\Sigma\) becomes a \(g\)-area bound

Since \(M\) has one end, one can choose the relevant component of \(\Sigma\) to enclose \(B_\rho^M(p_0)\). The Michael–Simon–Brendle isoperimetric inequality [MS73, Bre21] on minimal hypersurfaces then gives

This is the Euclidean volume growth needed by the Schoen–Simon–Yau stable Bernstein theorem. Hence \(M^5\) is flat. In short, the new feature in \(\mathbb{R}^6\) is that the weighted curvature \(\operatorname{BRic}_{40/43}\) creates just enough spectral Ricci positivity on the \(4\)-dimensional \(\mu\)-bubble for Antonelli–Xu’s volume estimate to close.