Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

Abstract: We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding.

• The paper establishes new integral and pointwise gradient estimates for the Green kernel under spectral Ricci bounds.
• It introduces a weighted analytic framework that refines classical stability inequalities using novel subsolution constructions.
• It provides a streamlined analytical proof of the stable Bernstein theorem for minimal hypersurfaces in R⁴ through gradient estimates and cutoff techniques.

Gradient Estimates for Green Kernels under Spectral Ricci Bounds and the Stable Bernstein Theorem in $\mathbb{R}^4$

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Introduction and Context

The paper "Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$" (2604.14393) addresses two pivotal aspects in the theory of minimal hypersurfaces and geometric analysis. Firstly, it establishes new integral and pointwise estimates for the Green kernel on Riemannian manifolds under spectral Ricci curvature bounds, extending earlier results by Colding. Secondly, it provides a streamlined analytic proof of the stable Bernstein theorem for complete two-sided stable minimal hypersurfaces in $^4$, confirming their rigidity as hyperplanes.

The stable Bernstein problem is central in minimal hypersurface theory, with historical roots in geometric measure theory and links to the classification of minimal cones and hypersurfaces. For dimensions $n \leq 5$, the theorem asserts that every connected, two-sided, complete, stable minimally immersed hypersurface in $^{n+1}$ is necessarily a hyperplane. Previous proofs for $n=4$ leveraged geometric input such as volume growth and the Gauss-Bonnet theorem, but persistent obstacles, especially in dimension $n=6$, have limited the extension of these techniques.

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Stable Minimal Hypersurfaces: Analytical Methods and Weighted Integral Estimates

The authors develop an analytic framework to prove integral inequalities for stable minimal hypersurfaces, circumventing geometric dependencies by incorporating weight functions. The classical stability inequality is

$\int_M |A|^2 \psi^2 \leq \int_M |\nabla\psi|^2 \quad \forall \psi \in \Lip_c(M),$

where $A$ denotes the second fundamental form.

Their method introduces weights satisfying

$\Delta u + |A|^2 u \le 0,$

with $\mathbb{R}^4$0, guaranteed by spectral stability. By refining the integration-by-parts argument and utilizing Simons' equation,

$\mathbb{R}^4$1

the authors derive, for parameters $\mathbb{R}^4$2 dependent on $\mathbb{R}^4$3,

$\mathbb{R}^4$4

This generalizes and relaxes previous algebraic restrictions (between $\mathbb{R}^4$5 and $\mathbb{R}^4$6) arising in the classical approach.

For the proof of the stable Bernstein theorem in $\mathbb{R}^4$7, the authors choose cutoff functions $\mathbb{R}^4$8, where $\mathbb{R}^4$9 is the Green kernel. The convergence of the right-hand side of the weighted inequality to zero as $^4$0—together with pointwise gradient estimates on $^4$1—enables them to conclude that $^4$2, i.e., the hypersurface is flat.

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Spectral Ricci Bounds and Gradient Estimates for Green Kernels

A core innovation is the extension of sharp pointwise gradient estimates for Green kernels to the setting of spectral Ricci bounds. The key structural assumption is that the manifold $^4$3 satisfies

$^4$4

with $^4$5 non-negative or subcritical in the spectral sense.

Under these conditions, the authors establish:

• If $^4$6 has finite index and $^4$7 is a positive solution to $^4$8 in $^4$9, then

$n \leq 5$0

where $n \leq 5$1 is the minimal positive Green kernel and $n \leq 5$2 is determined by boundary values.

• If $n \leq 5$3 is subcritical, and $n \leq 5$4 is the positive Green kernel for $n \leq 5$5,

$n \leq 5$6

with equality if and only if $n \leq 5$7 and $n \leq 5$8 is isometric to $n \leq 5$9.

These results generalize Colding's classical gradient estimates for Green kernels on manifolds with $^{n+1}$0 and bridge the gap between spectral Ricci bounds and analytic control of harmonic functions.

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Key Lemma: Constructing Sub-solutions via Powers and Weights

A central technical result is the lemma enabling the construction of sub-solutions for linear equations by combining suitable powers of sub- and super-solutions. Specifically, if $^{n+1}$1 and $^{n+1}$2 satisfy

$^{n+1}$3

then for each $^{n+1}$4 the function $^{n+1}$5 is a subsolution for $^{n+1}$6 up to an explicit positive gradient term. This lemma underpins the iterative and weighted integration arguments in the paper and provides a tool for potential further extensions to other criticality and elliptic problems.

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Stable Bernstein Theorem in $^{n+1}$7: Proof and Implications

Employing the above analytic machinery, the authors furnish a concise proof of the stable Bernstein theorem for $^{n+1}$8 (hypersurfaces in $^{n+1}$9). The proof hinges on:

• Integral inequalities with weighted powers involving the Green kernel,
• Pointwise and integral gradient estimates for $n=4$0,
• Iterative cut-off techniques leveraging the vanishing of $n=4$1 at infinity.

A critical algebraic condition restricts the method to $n=4$2, aligning with prior observations that extensions to $n=4$3 and higher require substantial new ideas due to breakdowns in key inequalities.

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Practical and Theoretical Implications

Practically, these results strengthen the analytic toolkit available for studying stable minimal hypersurfaces, especially in dimensions up to $n=4$4. The methodology sidesteps geometric prerequisites like volume growth or specific structural theorems, facilitating potential adaptations to other contexts where analytical control using spectral curvature bounds is available.

Theoretically, the pointwise gradient estimates for Green kernels provide new monotonicity-type tools and rigidity statements, enabling further exploration of flatness and stability phenomena in geometric analysis. The authors signal the possible extension of these techniques to flatness results in low dimensions for broader classes of elliptic equations.

The rigidity in the case of equality characterizes Euclidean space, with criticality theory playing a decisive role in confirming the uniqueness of ground states and minimal solutions. This intersection of spectral theory, geometric analysis, and PDEs may inform developments in the analysis of geometric flows and the theory of canonical metrics.

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Conclusion

The paper establishes a new analytic route to the stable Bernstein theorem in $n=4$5 and enhances the theory of gradient estimates for harmonic functions and Green kernels on manifolds with spectral Ricci bounds (2604.14393). The approach generalizes previous work, relaxes algebraic constraints via weighted inequalities, and provides a scalable technical lemma for generating subsolutions. The results consolidate the rigidity of stable minimal hypersurfaces in Euclidean four-space and lay analytic foundations for possible future extensions to higher dimensions and more general geometric settings.