Regularity and Compactness Results for Stable Minimal Hypersurfaces

The first result is the generalized Bernstein theorem from Schoen–Simon–Yau.

Theorem 4.1.1

Suppose \(M \subset \mathbb{R}^{n+1}\) is a complete, stable, minimal hypersurface without boundary and with at most (intrinsic) Euclidean volume growth. Then, if \(n \leq 5\), \(M\) must be an affine hyperplane.

Note that the intrinsic Euclidean volume growth condition is weaker than the extrinsic Euclidean volume growth condition. This is also a key condition for the Stable Bernstein Theorem without area growth condition.

Theorem 4.1.2

Let \(\{M_k\}\) be a sequence of embedded, stable, orientable minimal hypersurfaces in \(B_2^{n+1}(0)\) with the following properties:

\(0\in \bar{M}_k\) for each \(k\).
\(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\) for each \(k\).
\(\mathcal{H}^n(M_k \cap B_2^n(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(k\).

Then, up to a subsequence, \(M_k\) converges in the varifold sense to a stable minimal hypersurface \(M\) in \(B_2^{n+1}(0)\), which is smooth except for a closed singular set of Hausdorff dimension at most \(n-7\).

In addition, for \(n=7\), the singular set is discrete. \(M_k\) converges to a stable minimal hypersurface \(M\) in the varifold sense means that the varifold \(|M_k|\) converges to the varifold \(|M|\) in the sense of measures of weak limit.

Recall that a closed set \(S\) is of Hausdorff dimension at most \(k\) if for any \(\varepsilon\gt{}0\), we have \(\mathcal{H}^{k+\varepsilon}(S)=0\).

Corollary 4.1.3

Suppose \(M \subset \mathbb{R}^{n+1}\) is a complete, stable, embedded minimal hypersurface without boundary and with at most extrinsic Euclidean volume growth. Then, if \(n \leq 6\), \(M\) must be an affine hyperplane.

The dimension \(n\leq 6\) is sharp, as we already proved that Simons’ cone is stable in \(\mathbb{R}^8\).

Bellettini’s work completes the corresponding area-growth statement in the borderline dimension \(n=6\) for stable immersions. To state the result in a form that is independent of whether one measures volume intrinsically or extrinsically, we also record the comparison theorem of Florit–Simon.

Theorem 4.1.4

Let \(\Sigma^d\hookrightarrow \mathbb{R}^N\) be a complete, connected, smooth minimal immersion, and let \(p\in \Sigma\). Define the intrinsic and extrinsic area densities by

\[ \mathbf{M}_R^{\mathrm{int}}(\Sigma,p):=\frac{|B_R^\Sigma(p)|}{R^d}, \qquad \mathbf{M}_R^{\mathrm{ext}}(\Sigma,p):=\frac{\mathrm{Area}(\Sigma\cap B_R^{\mathbb{R}^N}(p))}{R^d}, \]

where \(B_R^\Sigma(p)=\{x\in \Sigma: d_\Sigma(x,p)\lt{}R\}\), and the extrinsic area is counted with multiplicity. Then \(\mathbf{M}_R^{\mathrm{int}}(\Sigma,p)\) and \(\mathbf{M}_R^{\mathrm{ext}}(\Sigma,p)\) are monotone nondecreasing in \(R\), and

\[ \lim_{R\to\infty}\mathbf{M}_R^{\mathrm{int}}(\Sigma,p) =\lim_{R\to\infty}\mathbf{M}_R^{\mathrm{ext}}(\Sigma,p)\in[\omega_d,\infty], \]

where \(\omega_d=|B_1^{\mathbb{R}^d}|\). In particular, \(\Sigma\) has bounded intrinsic area density if and only if it has bounded extrinsic area density; in either case, the immersion is proper.

Corollary 4.1.5

Let \(2\leq n\leq6\), and let \(\Sigma^n\hookrightarrow \mathbb{R}^{n+1}\) be a complete, connected, two-sided, stable minimal immersion. If \(\Sigma\) has Euclidean area growth, equivalently

\[ |B_R^\Sigma(p)|\leq \Lambda R^n \quad\text{for some }p\in\Sigma,\ \Lambda\lt{}\infty,\text{ and all }R\gt{}0, \]

\[ \mathrm{Area}(\Sigma\cap B_R^{\mathbb{R}^{n+1}}(p))\leq \Lambda R^n \quad\text{for some }p\in\mathbb{R}^{n+1},\ \Lambda\lt{}\infty,\text{ and all }R\gt{}0, \]

then \(\Sigma\) is an affine hyperplane.

For \(2\leq n\leq5\), this is the Schoen–Simon–Yau area-growth stable Bernstein theorem. The new borderline case is \(n=6\): Bellettini proves the classification under extrinsic Euclidean area growth, and Theorem 4.1.4 converts intrinsic area growth into the same extrinsic hypothesis. Thus the area-growth version of the immersed stable Bernstein theorem is settled in the full sharp range \(2\leq n\leq6\).

A natural question is whether the compactness conclusion in Theorem 4.1.2 remains true if we assume \(\mathcal{H}^{n-1}(\mathrm{sing}M_k)=0\) instead of \(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\). The answer is yes, by the following deep result of Wickramasekera.

Theorem 4.1.6

Suppose \(V_i\) is a sequence of stationary integral \(n\)-varifolds in \(B_2^{n+1}(0)\) and \(V_i\) also satisfies the following conditions:

\(0\in \mathrm{spt}\|V_i\|\).
\(\|V_i\|(B_2^{n+1}(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(i\).
(Stability) Each \(V_i\) is stable in \(B_2^{n+1}(0)\) on its regular set, i.e., for any \(\phi \in C_c^1(\mathrm{reg}V_i)\),

\[ \int_{\mathrm{reg}V_i} |A_i|^2 \phi^2 d\|V_i\| \leq \int_{\mathrm{reg}V_i} |\nabla \phi|^2 d\|V_i\|. \]

(Alpha-Structural Hypothesis) There exists \(\alpha\in (0,1)\) such that for each \(i\), no point of \(\mathrm{spt}\|V_i\|\cap B_1^{n+1}(0)\) has a neighborhood in which \(\mathrm{spt}\|V_i\|\) is the union of three or more embedded \(C^{1,\alpha}\) hypersurfaces-with-boundary meeting only along their common boundary.

Then, up to a subsequence, \(V_i\) converges in the varifold sense to a stationary integral \(n\)-varifold \(V_\infty\) in \(B_2^{n+1}(0)\), which is stable and whose singular set in \(B_2^{n+1}(0)\) has Hausdorff dimension at most \(n-7\).

The theorem is formulated for stationary integral varifolds, so no orientability assumption is part of the statement. In applications to stable immersions, two-sidedness is imposed separately when one writes the stability inequality on the regular set.

The next conjecture concerns compactness for stable minimal immersions with singular sets.

The class of branched two-sided stable minimal \(n\)-dimensional immersions with the singular set of locally finite \((n-2)\)-measure is compact under varifold convergence.

The first result in this direction is the following density-\(2\) regularity theorem.

Theorem 4.1.7

Let \(\delta \in (0,1)\). Suppose \(M_k\) is a sequence of orientable stable minimal hypersurfaces immersed in \(B_2^{n+1}(0)\) such that:

\(0\in \bar{M}_k\).
\(\|M_k\|(B_2^{n+1}(0))\leq (3-\delta)\omega_n 2^n\).
\(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\).

The general case, with non-optimal singular set dimension, is proved by Hong-Li-Wang [HLW24].

Theorem 4.1.8

Suppose \(M_k\) is a sequence of orientable stable minimal hypersurfaces immersed in \(B_2^{n+1}(0)\) such that:

\(0\in \bar{M}_k\).
\(\|M_k\|(B_2^{n+1}(0))\leq \Lambda\).
\(\sup_k\operatorname{dim}_{\mathcal{H}}(\mathrm{sing}M_k)\lt{} n-4+\frac{4}{n}\).

Minter–Xiao subsequently proved the optimal non-branched version of this regularity and compactness theorem.

Theorem 4.1.9

Let \(n\geq2\). Suppose \(M_k\) is a sequence of two-sided stable minimal hypersurfaces smoothly and properly immersed in \(B_1^{n+1}(0)\) such that:

\(0\in \bar M_k\) for each \(k\);
\(\sup_k\mathcal H^n(M_k\cap B_1^{n+1}(0))\lt{}\infty\);
\(\mathcal H^{n-2}(\mathrm{sing}M_k)=0\) for each \(k\), where

\[ \mathrm{sing}M_k:=B_1^{n+1}(0)\cap(\bar M_k\setminus M_k). \]

Then, after passing to a subsequence, \(M_k\) converges as varifolds to a stationary integral varifold \(V\) in \(B_1^{n+1}(0)\). Moreover, there is a relatively closed set

\[ S\subset \mathrm{spt}\,\|V\|\cap B_1^{n+1}(0), \qquad \dim_{\mathcal H}S\leq n-7, \]

such that \(V\) is represented on \(B_1^{n+1}(0)\setminus S\) by a proper, two-sided, stable minimal immersion, and \(M_k\) converges locally smoothly to this immersion away from \(S\). In particular, \(S=\emptyset\) for \(2\leq n\leq6\), while \(S\) is discrete for \(n=7\).

The hypothesis \(\mathcal H^{n-2}(\mathrm{sing}M_k)=0\) is the optimal non-branched assumption. It improves Theorem 4.1.8, where the non-immersed singular set is required to have Hausdorff dimension strictly smaller than \(n-4+\frac4n\). The branch point case is not included here: when the non-immersed singular set has positive \(\mathcal H^{n-2}\)-measure, branch points may occur and the corresponding compactness theory remains a separate problem.

Regularity and Compactness Results for Stable Minimal Hypersurfaces

Gaoming Wang

Assistant Professor