We give a Green-function proof of the stable Bernstein theorem in $\mathbb{R}^{7}$ for smooth, connected, complete, two-sided minimal hypersurface, thus resolving the last case in stable Bernstein problem.
Stable Bernstein theorem
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.
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
$$ J(t)=\int_{{G=t}}|\nabla G|^3,dA_g, \qquad \mathcal M_\sigma=\int_0^{t_0}J(t)t^{-\sigma-1},dt. $$
Step 1 · Preparation
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
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
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
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 $\sigma<5/2$.
Step 5 · Critical energies
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
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 $A\equiv0$.
My research interests include Geometric Analysis and Partial Differential Equations.