[Paper Review] A Liouville-type theorem for stable minimal hypersurfaces

Leon Simon|arXiv (Cornell University)|Jan 16, 2021

Geometric Analysis and Curvature Flows5 references4 citations

TL;DR

This paper establishes a Liouville-type theorem for strictly stable minimal hypersurfaces in $ ^{n+1+ u}$ that lie on one side of a cylindrical cone $ _{0} \times \r^\ell$, where $ _{0}$ is a strictly stable, area-minimizing hypercone in $ ^{n+1}$ with isolated singularity at the origin. Under finite density at infinity, the hypersurface must be cylindrical—i.e., $M = S \times \r^\ell$ for some smooth, strictly stable minimal hypersurface $S \subset \r^{n+1}$, proving rigidity under geometric and stability constraints.

ABSTRACT

We prove that if $M$ is a strictly stable complete minimal hypersurface in Euclidean space with finite density at infinity and which lies on one side of a minimal cylinder with cross-section a strictly stable area minimizing hypercone, then $M$ must be cylindrical. Applications will be given in the references [Sim20a], [Sim20b].

Motivation & Objective

• To establish a Liouville-type rigidity result for stable minimal hypersurfaces in higher codimension.
• To address the structure of complete, strictly stable minimal hypersurfaces lying on one side of a cylindrical cone with singular vertex.
• To prove that such hypersurfaces must be cylindrical under finite density at infinity and strict stability.
• To provide a key technical tool for constructing examples of strictly stable minimal hypersurfaces with arbitrary singular sets in high codimension.
• To extend the understanding of minimal hypersurfaces in non-compact, singular ambient spaces using stability and growth estimates.

Proposed method

• Derives $L^2$ growth estimates for solutions of the Jacobi equation on the hypersurface $M$ using the spectral gap of the Jacobi operator on the link $ _{0} \cap \s^{n}$.
• Analyzes the normal component $\nu_y = (e_{n+2} \cdot \nu, \ldots, e_{n+1+\ell} \cdot \nu)$ of the unit normal to $M$ in the $y$-directions, showing its $L^2$ norm decays like $R^{-2 + \gamma + \alpha}$ for large $R$, with $\gamma = \ell + 2 + \beta_1$.
• Uses distance function $d(x) = \text{dist}((x,y), \r)$ to the cylinder $\r = \r_0 \times \r^\ell$ and derives $L^2$ growth bounds for $d|M$.
• Applies unique continuation and elliptic regularity to show that the $L^2$-expansion of functions on $M$ in terms of homogeneous $ $-harmonic polynomials converges smoothly and in $L^2$, establishing completeness of the basis.
• Combines the growth estimates to derive a contradiction unless $\nu_y \equiv 0$, implying $M$ is cylindrical.
• Uses the characteristic exponents $\gamma_1^{\pm}$ from the Jacobi operator on $\r_0$ to control the decay rate of normal components, relying on the strict stability condition $\lambda_1 > -(n-2)^2/4$.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions does a strictly stable minimal hypersurface lying on one side of a cylindrical cone in $\r^{n+1+\ell}$ necessarily become cylindrical?
• RQ2 Can finite density at infinity and strict stability force a minimal hypersurface to inherit the product structure of its ambient cylinder?
• RQ3 What role does the spectral gap of the Jacobi operator on the link of the cone play in controlling the decay of normal components of the hypersurface?
• RQ4 How do $L^2$ growth estimates for Jacobi fields constrain the possible geometry of stable minimal hypersurfaces in singular ambient spaces?
• RQ5 Is it possible to rule out non-cylindrical stable minimal hypersurfaces in $\r^{n+1+\ell}$ that lie on one side of a singular cone?

Key findings

• If $M$ is a strictly stable, complete, minimal hypersurface in $\r^{n+1+\ell}$ lying on one side of $\r = \r_0 \times \r^\ell$, with $\r_0$ a strictly stable, area-minimizing hypercone in $\r^{n+1}$ with isolated singularity at the origin, then $M$ must be cylindrical: $M = S \times \r^\ell$ for some smooth, strictly stable minimal hypersurface $S \subset \r^{n+1}$.
• The $L^2$ norm of the normal component $\nu_y$ on $M \cap B_R$ decays like $R^{-2 + \gamma + \alpha}$ for $\gamma = \ell + 2 + \beta_1$, where $\beta_1 = 2\left(\left(\frac{n-2}{2}\right)^2 + \lambda_1\right)^{1/2}$, and $\lambda_1$ is the first eigenvalue of the Jacobi operator on $\Sigma = \r_0 \cap \s^n$.
• The contradiction in the decay rate $R^{-2 + \gamma + \alpha}$ for $R > 1$ implies $\nu_y \equiv 0$, forcing $M$ to be tangent to the $y$-directions and hence cylindrical.
• The proof relies on $L^2$ growth estimates for Jacobi fields and unique continuation, showing that the $L^2$-expansion of functions on $M$ in terms of homogeneous $ $-harmonic polynomials converges smoothly and in $L^2$, establishing completeness of the basis.
• The result implies that no non-cylindrical, strictly stable minimal hypersurfaces can exist under the stated conditions, even in high codimension.
• The key technical step is showing that the $L^2$ norm of $\nu_y$ cannot grow too slowly, contradicting the assumed decay unless $\nu_y \equiv 0$, which forces the cylindrical structure.