Gaussian-Weighted Curvature Gaps for Self-Shrinkers

Abstract: In this paper, we prove lower bounds for the Gaussian-weighted (L^2)-curvature integral of embedded self-shrinkers. The proof combines normal coordinate functions with weighted Poincaré inequalities arising from first-eigenvalue estimates of Ding--Xin and Brendle--Tsiamis. For closed self-shrinkers, the estimate gives an explicit lower bound in terms of entropy and, together with the entropy gap theorem of Colding--Ilmanen--Minicozzi--White, yields a strict curvature gap for nonspherical closed shrinkers. In dimension two, we combine this estimate with the classification theorem and entropy gap theorem of Bernstein--Wang to obtain the corresponding gap statement for complete embedded self-shrinkers with polynomial volume growth.

• The paper establishes explicit lower bounds for the Gaussian-weighted L²-curvature integral that connects entropy with curvature rigidity.
• The paper applies weighted Poincaré inequalities and drift Laplacian eigenvalue estimates to derive sharp curvature gap results for embedded self-shrinkers.
• The paper demonstrates that entropy gaps yield rigidity results, offering new insights into the classification of singularities in mean curvature flow.

Gaussian-Weighted Curvature Gaps in Self-Shrinker Geometry

Introduction

The paper "Gaussian-Weighted Curvature Gaps for Self-Shrinkers" (2606.14360) establishes explicit lower bounds for the Gaussian-weighted $L^2$-curvature integral for embedded self-shrinkers under mean curvature flow (MCF). Leveraging weighted Poincaré inequalities associated with drift Laplacians and normal coordinate functions, the authors connect entropy, a central invariant in MCF, with explicit curvature gap estimates. The results provide explicit $L^2$-curvature gaps for nonspherical closed shrinkers and, in dimension two, yield similar rigidity for complete embedded self-shrinkers with polynomial volume growth.

Mathematical Framework and Motivation

Self-shrinkers are critical self-similar solutions to MCF, satisfying $\mathbf{H} = -\frac{x^\perp}{2}$. They describe limits at singularities and underpin regularity and classification theory in geometric flows. Entropy, defined as the supremum of scaled Gaussian areas, is a monotone invariant along the flow, with foundational work by Colding and Minicozzi indicating its minimization by the round sphere and quantized gap to other self-shrinkers.

Rigidity theorems along self-shrinkers are classically framed through pinching theorems for the second fundamental form $|A|$ and entropy gaps for embedded hypersurfaces. Prior results, such as the gap phenomena by Le-Sesum and the pointwise rigidity of Cao-Li, are complemented by entropy-based rigidity from Colding-Ilmanen-Minicozzi-White and classification results from Bernstein-Wang.

The paper uses normal coordinate functions $\psi_a(x) = \langle \nu(x), a \rangle$ for ambient unit vectors $a$ and relates their gradients to $|A|^2$, effectively reproducing the curvature norm by summing over an orthonormal basis. The main technical innovation is applying weighted Poincaré inequalities (e.g., Brendle-Tsiamis drift Laplacian eigenvalue estimates) to these functions, combined with sharp control of their weighted means via explicit Gaussian integrals.

Main Results

The principal theorem states that for a smooth, closed, embedded self-shrinker $M^n \subset \mathbb{R}^{n+1}$, the Gaussian-weighted $L^2$ norm of the second fundamental form obeys: $$\int_M |A|^2 e^{-\frac{|x|^2}{4}} \, d\mu \geq 4^{\frac n2-1} \pi^{\frac n2} \frac{(\lambda(M))^2-1}{\lambda(M)}$$ where $L^2$0 is the entropy of $L^2$1. For non-spherical $L^2$2, this improves to

$L^2$3

with $L^2$4 an explicit gap.

In dimension two, the paper combines this estimate with the Bernstein-Wang classification to obtain: $L^2$5 for any connected, complete, embedded self-shrinker $L^2$6 with polynomial volume growth. For non-flat $L^2$7 not congruent to a plane, sphere, or cylinder, a strict gap analogous to the closed case is obtained.

The proofs hinge crucially on the drift Laplacian eigenvalue lower bounds by Brendle-Tsiamis for shrinkers and Ding-Xin for compact cases, guaranteeing that the weighted Poincaré inequalities hold with sharp constants.

Analytical Techniques

Normal coordinate functions are constructed to have zero weighted mean on carefully chosen orthonormal bases. Their gradients are precisely linked to $L^2$8, enabling application of weighted Poincaré inequalities: $L^2$9 with $\mathbf{H} = -\frac{x^\perp}{2}$0 for shrinkers.

The paper executes precise Gaussian moment computations to bound the weighted means of coordinate functions, invoking rotational invariance and explicit integration for radial Gaussian weights. Lemma-based reductions ensure polynomial volume growth conditions are met and the necessary separation properties for embeddedness hold.

Entropy minimization is leveraged via the results of Colding-Ilmanen-Minicozzi-White and Bernstein-Wang. For surfaces, classification results yield strict entropy gaps for planes, spheres, and cylinders, and all other shrinkers, using explicit constants. The monotonicity of the main function $\mathbf{H} = -\frac{x^\perp}{2}$1 ensures that curvature gap estimates are strictly increasing in entropy.

Implications and Future Directions

This work quantitatively connects entropy and Gaussian-weighted curvature, demonstrating that entropy gaps translate directly into explicit curvature rigidity for self-shrinkers. The explicit bounds yield new rigidity results, identifying a spectral-geometric relation between the second fundamental form and global invariants of the flow.

From a practical perspective, these curvature gap estimates inform the classification of singularities in the mean curvature flow, potentially enabling algorithmic identification of shrinker types in geometric analysis and numerical simulations. The theoretical implications extend to the structure of the moduli space of self-shrinkers, spectral geometry of drift Laplacians, and the entropy landscape of embedded hypersurfaces.

The conjectures formulated within the paper propose sharp bounds and characterization: equality in curvature gaps occurs only for round spheres or generalized cylinders. Resolution of the Colding-Ilmanen-Minicozzi-White entropy gap conjecture in higher dimensions would yield full extensions of these curvature gaps to $\mathbf{H} = -\frac{x^\perp}{2}$2 and possibly beyond.

Future work may focus on improving constants, extending to arbitrary codimension, and exploring higher eigenmode rigidity. Connections with spectral invariants and geometric flows in other settings (e.g., Ricci and Yamabe) may provide further analytical tools and deepen understanding of singularity formation and rigidity in geometric evolution equations.

Conclusion

This paper establishes explicit, entropy-controlled lower bounds for the Gaussian-weighted $\mathbf{H} = -\frac{x^\perp}{2}$3 curvature of embedded self-shrinkers, utilizing weighted normal coordinate function techniques, drift Laplacian estimates, and sharp entropy gap results. Theoretical advances include new rigidity phenomena and conjectures for sharpness, suggesting further exploration of the interplay between entropy, curvature, and spectral geometry in the context of mean curvature flow and geometric analysis.