Stable Bernstein Theorems

The classical Bernstein theorem asks when an entire minimal graph in \(\mathbb{R}^{n+1}\) must be affine. From the variational point of view, an entire minimal graph is automatically stable, so a natural strengthening is to ask whether one can replace the graph assumption by stability alone. This leads to the stable Bernstein problem: classify complete stable minimal hypersurfaces in Euclidean space.

The statement below is the form relevant to the recent theory of stable immersions. The simply connected assumption rules out possible topological complications coming from immersions, while the two-sided assumption allows one to write the stability inequality with a globally defined normal field.

Given \(2\leq n\leq 6\), suppose \(M^n\to \mathbb{R}^{n+1}\) is a complete, two-sided, simply connected, stable minimal immersion. Then \(M\) is an affine hyperplane.

Recall that Simons’ cone

\[ \mathbf C:=\left\{ x_1^2+x_2^2+x_3^2+x_4^2=x_5^2+x_6^2+x_7^2+x_8^2 \right\} \]

is a stable minimal hypersurface in \(\mathbb{R}^8\). Thus the dimension range in the conjecture is sharp: the analogous statement is false for \(n\geq 7\).

Here are some landmarks in the history of the problem.

  1. In dimension \(n=2\), do Carmo–Peng [dCP79], Fischer–Colbrie–Schoen [FCS80], and Pogorelov [Pog81] proved the stable Bernstein theorem.

  2. Schoen–Simon–Yau [SSY75] proved the result for \(2\leq n\leq 5\) under an area-growth hypothesis. A key step in their proof is an \(L^p\) curvature estimate of the form

\[ \int_{ } \left|A_\Sigma\right|^{2p} u^{2p} d\Sigma\leq \int_{ } \left|\nabla u\right|^{2p}d\Sigma \]

for \(2p\lt{} 4+\sqrt{\frac{8}{n}}\).

  1. Schoen–Simon [SS81] proved the corresponding classification for properly embedded stable minimal hypersurfaces with area growth.

  2. Chodosh–Li [CL24] proved the case \(n=3\). Other approaches to this dimension were later given by Chodosh–Li [CL23] and by Catino–Mastrolia–Roncoroni [CMR24].

  3. Chodosh–Li–Minter–Stryker [CLMS25] proved the case \(n=4\), and Mazet [Maz24] proved the case \(n=5\).

  4. Cabré–Catino–Mari–Mastrolia–Roncoroni gave a Green-kernel proof of the \(n=3\) case and proved a sharp gradient estimate for Green kernels under spectral Ricci bounds [CCM+26].

  5. The case \(n=6\) under an area-growth hypothesis follows from Bellettini [Bel25], together with the intrinsic–extrinsic area equivalence of Florit–Simon [FS26]; see Corollary 4.1.5. The unconditional \(n=6\) case is not addressed by these area-growth arguments and remains open. A main difficulty is that the present \(\mu\)-bubble and spectral-Ricci volume-control methods do not yet seem strong enough to produce the needed area-growth input in this borderline dimension.

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Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.