The stable Bernstein theorem in $\mathbb{R}^{7}$

Mon, 14 Sep 2026 15:19:59 +0000

Main theorem

Stable Bernstein theorem

Flatness in $\mathbb R^7$

Every smooth, connected, complete, two-sided stable minimal immersion $F:M^6\to\mathbb R^7$ has $A\equiv0$, and its image is a hyperplane. No properness, volume-growth, or curvature bound is assumed.

Proof architecture

The proof first establishes all subcritical Green-function moments, then uses critical energy identities and the terminal Green flux to force flatness. Write $A$ for the second fundamental form and $G$ for the minimal positive Green function with a unit pole. For a fixed small $t_0>0$, set

$$ J(t)=\int_{{G=t}}|\nabla G|^3,dA_g, \qquad \mathcal M_\sigma=\int_0^{t_0}J(t)t^{-\sigma-1},dt. $$

Part I · All subcritical moments

Step 1 · Preparation

Geometric reduction and Green functions

A blow-up argument reduces the problem to bounded curvature. Unit Green flux and the estimate $|\nabla G|\leq CG$ give the initial integrability range $\mathcal M_\sigma<\infty$ for $\sigma<2$. The target is every $\sigma<5/2$, the critical exponent in dimension six.

Step 2 · Initial improvement

From $2$ to $2.46$

Bochner's formula, Simons' identity, stability, and the divergence-free tensor $\tfrac12|A|^2g-A^2$ supply the integral identities. The full Codazzi constraint sharpens the directional estimates, while a Schoen–Simon–Yau inequality retaining its Laplacian term controls cutoff errors. Moment continuation yields $\mathcal M_{2.46}<\infty$.

Step 3 · Nonlinear stability

From $2.46$ to $2.4962$

Apply stability to a nonlinear norm combining a curvature test function and a Green-gradient test function. Differentiating this norm retains an extra nonnegative gradient term. Together with the Green and curvature identities, it gives coercive estimates on fifteen rational parameter intervals and proves $\mathcal M_{2.4962}<\infty$.

Step 4 · Spectral coercivity

Every exponent below $5/2$

Additional divergence identities for homogeneous curvature tensors involving $A^4$, $\operatorname{tr}(A^3)A$, and $\operatorname{tr}(A^4)g$ strengthen the coercive estimate near the critical exponent. They continue the moment range from $2.4962$ to every $\sigma<5/2$.

Part II · Critical energies and rigidity

Step 5 · Critical energies

Remove the exterior cutoffs

All subcritical moments are now finite. An elementary spectral inequality, stability, and Simons' identity give finite critical curvature and derivative energies. The argument uses each fixed subcritical moment, with no uniform bound required as $\sigma\uparrow5/2$.

Step 6 · Rigidity

Terminal flux forces flatness

The energy bounds give finite limits of the normalized Green flux $t^{-5/2}J(t)$ at both ends. Brendle's sharp isoperimetric inequality determines the sign of the terminal Bochner boundary term. Combining this sign with the critical spectral inequality forces the critical curvature energy to vanish, so $A\equiv0$.

Stable Bernstein Theorem | Gaoming Wang