Cohomology Vanishing for Free Boundary $f$-Minimal Submanifolds in Gaussian-Weighted Euclidean Balls

Abstract: Let $M^n\subset \overline{\B_R^{{n+k}}\subset} \R<sup>{n+k}$ be a compact orientable free boundary $f$-minimal submanifold of the Gaussian-weighted Euclidean ball $\left(\overline{\B_R^{{n+k}},g_{\rm} can},e^{-f}\dd V\right), f(x)=\frac c2 |x|^2,c\ge 0.$ We prove a cohomology vanishing theorem under the pointwise pinching condition $ |A|^2\le \frac{n-p}{R^2},1\le p&lt;n.$ More precisely, the space of tangential $f$-harmonic $p$-forms vanishes, and hence$H<sup>p(M;\R)=0.$ The proof is based on three elementary ingredients in the Gaussian-weighted ball: a weighted Hardy inequality obtained from the identity $\divf(x<sup>T)=n-c|x|<sup>2$, a cancellation in the weighted Weitzenböck curvature operator, and a boundary reduction showing that tangential $f$-harmonic forms satisfy the same local absolute-boundary algebra as in the unweighted case. The constant pinching threshold is independent of the Gaussian parameter $c$, and the argument also includes the unweighted case $c=0$; the strict interior positivity comes from the full Hardy--Weitzenböck coefficient rather than from the sign of $c$ alone.

• The paper establishes that under a precise pinching condition, tangential f-harmonic forms vanish, leading to H^p(M; R)=0 for free boundary f-minimal submanifolds.
• It employs weighted Hardy inequalities and Weitzenböck formula cancellations to control curvature terms, making the pinching threshold independent of the Gaussian parameter.
• The findings impose topological rigidity, notably classifying two-dimensional f-minimal surfaces as homeomorphic to disks under the curvature bounds.

Cohomology Vanishing for Free Boundary $f$-Minimal Submanifolds in Gaussian-Weighted Euclidean Balls

Introduction and Context

The paper "Cohomology Vanishing for Free Boundary $f$-Minimal Submanifolds in Gaussian-Weighted Euclidean Balls" (2606.21379) investigates the interplay between geometric curvature conditions and the topology of free boundary $f$-minimal submanifolds within Gaussian-weighted Euclidean balls. The analysis focuses on $f$-minimal submanifolds endowed with a density function $e^{-f}$, specifically where $f(x) = c|x|^2/2$, $c \geq 0$, and extends classical Bochner-type vanishing theorems to this weighted setting under a sharp extrinsic pinching condition.

The underlying theme is leveraging analytic identities from submanifold geometry, such as weighted versions of the Hardy inequality and the Weitzenböck formula, to deduce substantial topological conclusions—namely, the vanishing of certain de Rham cohomology groups. The work connects with both index-topology comparisons in minimal and $f$-minimal hypersurfaces, and develops new geometric measure-theoretic tools in the field of weighted geometry.

Main Results

The core result establishes the vanishing of tangential $f$-harmonic $p$-forms on compact orientable free boundary $f$0-minimal submanifolds $f$1, under the precise pinching condition $f$2 for $f$3. This yields $f$4 for absolute cohomology. Notably, the pinching threshold is independent of the Gaussian parameter $f$5, a claim substantiated by two analytic cancellations arising from the structure of the weighted Euclidean ball and the Gaussian drift.

In two dimensions, the theorem specializes further: a compact connected orientable free boundary $f$6-minimal surface with $f$7 is homeomorphic to the disk, indicating that higher-genus surfaces are excluded under these geometric constraints.

Technical Framework

Weighted Gaussian Setup and $f$8-Minimality

The ambient geometry is the Euclidean ball equipped with the Gaussian weight $f$9, where $f$0. The $f$1-minimal equation for a submanifold $f$2 is $f$3, with $f$4 the mean curvature vector. The free boundary condition ensures that $f$5 meets the boundary of the ball orthogonally.

Analytic Ingredients

Three fundamental analytic mechanisms drive the proof:

1. Weighted Hardy Inequality: Using the identity $f$6, a boundary integral controls interior terms, with the mean curvature term eliminated due to $f$7-minimality.
2. Weighted Weitzenböck Formula Cancellation: The curvature term in the Bochner formula is modified by the Hessian of $f$8, yielding a cancellation that produces a lower bound independent of $f$9:

$f$0

with $f$1 the shape operators.

1. Boundary Reduction: The absolute boundary condition for tangential $f$2-harmonic forms, together with the weighted codifferential, reduces the weighted boundary term to its unweighted analog.

The proof integrates these tools with the Kato inequality and a careful choice of parameters in the Hardy identity, obtaining positivity strictly in the interior, which suffices for vanishing.

Pinching Condition and Interior Positivity

The crucial estimate arises in bounding the coefficient $f$3 controlling the integrand of the Bochner formula:

$f$4

where $f$5 is the squared norm of the second fundamental form. The pinching assumption ensures the first term is nonnegative everywhere, and the second term guarantees strict positivity in the interior for all $f$6.

Implications and Extensions

Practical and Theoretical Consequences

The result demonstrates that the topology of $f$7-minimal free boundary submanifolds is rigidly constrained by extrinsic curvature bounds, even in the presence of a Gaussian weight. The formal independence from $f$8 is nontrivial and arises from delicate cancellations, while the actual pinching threshold and spatial relaxation of the condition demonstrate that positive $f$9 admits greater interior curvature, yet the same topological vanishing holds.

Topological rigidity—particularly disc-type classification in the surface case—has implications for the moduli space of such $e^{-f}$0-minimal submanifolds and their applications in geometric analysis and probability (e.g., self-shrinkers in mean curvature flow).

Future Directions

Several open questions are highlighted:

• Refined Kato Inequalities: The paper conjectures that weighted refined Kato inequalities (involving the weighted codifferential $e^{-f}$1) could potentially improve the pinching threshold. Establishing such inequalities could lead to sharper rigidity results for weighted harmonic forms.
• Classification in the Equality Case: The geometric characterization of submanifolds saturating the pinching threshold remains unresolved. Constructing or ruling out non-flat examples at the threshold would further elucidate the rigidity landscape.
• Extension to Relative Cohomology and Normal Forms: The absolute cohomology vanishing treated here leaves the case of normal $e^{-f}$2-harmonic forms and relative boundary conditions open, as boundary terms become more intricate.

Conclusion

This paper rigorously advances the understanding of the relationship between extrinsic geometric pinching and topological vanishing for free boundary $e^{-f}$3-minimal submanifolds in Gaussian-weighted balls. Through analytic cancellations and integrand estimates, it establishes vanishing theorems for tangential $e^{-f}$4-harmonic forms, confirming that strong geometric pinching dictates topological simplicity. The formal independence of the curvature threshold from the Gaussian parameter and the robust handling of the weight in both analytic and geometric terms underscore the depth of the result. The implications for rigidity, index, and moduli spaces, as well as potential for further improvement via refined analytic inequalities, make this work valuable for researchers in geometric analysis and submanifold theory.