[Paper Review] Higher codimension relative isoperimetric inequality outside a convex set

Brian Krummel|arXiv (Cornell University)|Oct 13, 2017

Point processes and geometric inequalities13 references3 citations

TL;DR

This paper establishes a sharp higher codimension relative isoperimetric inequality for $(m+1)$-dimensional area-minimizing integral currents lying outside a convex set $Σ \subset \mathbb{R}^{n+1}$, proving that the infimum of the isoperimetric ratio is achieved precisely by a flat half-disk in a half-space. The result extends prior codimension-one results and confirms a conjecture by Choe, using tools from geometric measure theory, monotonicity formulas, and compactness theorems for integral currents.

ABSTRACT

We consider an isoperimetric inequality for $(m+1)$-dimensional area minimizing submanifolds of arbitrary codimension which lie outside a convex set $\mathcal{K} \subset \mathbb{R}^{n+1}$ and are bounded by a submanifold of $\mathbb{R}^{n+1} \setminus \mathcal{K}$ and the convex set $\mathcal{K}$. We show that the least value of the isoperimetric ratio is attained for an $(m+1)$-dimensional flat half-disk of $\mathbb{R}^{n+1}_+$. This extends prior work of Choe, Ghomi, and Ritoré in codimension one and proves a conjecture of Choe in the case of relative area minimizers.

Motivation & Objective

• To establish a sharp relative isoperimetric inequality for $(m+1)$-dimensional area-minimizing integral currents in $\mathbb{R}^{n+1}\setminus\mathcal{K}$, where $\mathcal{K}$ is a closed convex set with nonempty interior.
• To extend previous codimension-one results by Choe, Ghomi, and Ritoré to arbitrary codimension.
• To confirm a conjecture by Choe regarding relative area minimizers in higher codimensions.
• To analyze the structure and regularity of area-minimizing currents near the boundary of a convex set $\mathcal{K}$, including singularities and boundary behavior.
• To prove that the optimal isoperimetric ratio is attained only by flat half-disks in a half-space, using compactness and blow-up techniques.

Proposed method

• Uses the framework of integral currents in $\mathbb{R}^{n+1}\setminus\mathcal{K}$ to generalize smooth submanifolds and allow for singularities and multiplicities.
• Applies the Federer-Fleming compactness theorem to establish existence of area-minimizing currents $R$ with $\partial R = T$ in $\mathbb{R}^{n+1}\setminus\mathcal{K}$, where $T$ is an $m$-current with boundary on $\partial\mathcal{K}$.
• Employs a slicing argument with radius $\rho^*$ to localize the problem to a bounded convex set $\widetilde{\mathcal{K}} = \mathcal{K} \cap \overline{B_{2\rho^*}(0)}$, enabling the use of the bounded case.
• Applies a monotonicity formula and tangent cone analysis to study the asymptotic behavior of minimizing sequences and rule out mass concentration at infinity.
• Uses reflection and comparison with the standard isoperimetric inequality in $\mathbb{R}^{n+1}$ to derive the sharp constant $2^{-1/m}$ in the half-space case.
• Establishes equality in the isoperimetric ratio if and only if $R$ is a multiplicity-one flat half-disk in $\mathbb{R}^{n+1}_+$, using the area-mean curvature characterization of hemispheres and stability arguments.

Experimental results

Research questions

• RQ1 What is the sharp lower bound for the relative isoperimetric ratio $\frac{\mathbf{M}(T)^{\frac{m+1}{m}}}{\mathbf{M}(R)}$ for $(m+1)$-dimensional area-minimizing integral currents $R$ lying outside a convex set $\mathcal{K}$?
• RQ2 Does the minimizer of this ratio occur only for flat half-disks in a half-space, as conjectured by Choe, in arbitrary codimension?
• RQ3 How does the presence of singularities and non-rectifiable boundaries affect the existence and regularity of relative area minimizers?
• RQ4 Can the relative isoperimetric inequality be extended from codimension one to higher codimensions using integral current theory?
• RQ5 What is the role of the convexity of $\mathcal{K}$ in ensuring the existence of minimizing currents and the validity of the sharp inequality?

Key findings

• The sharp relative isoperimetric inequality holds: $\frac{\mathbf{M}(T)^{\frac{m+1}{m}}}{\mathbf{M}(R)} \geq 2^{-1/m} \frac{\mathcal{H}^m(\partial \mathbb{B}^{m+1})^{\frac{m+1}{m}}}{\mathcal{H}^{m+1}(\mathbb{B}^{m+1})}$ for any $m$-dimensional integral current $T$ and $(m+1)$-dimensional area-minimizing current $R$ in $\mathbb{R}^{n+1}\setminus\mathcal{K}$ with $\partial R = T$ and $\partial R \subset \partial\mathcal{K}$.
• Equality holds in the inequality if and only if $R$ is a multiplicity-one flat half-disk in $\mathbb{R}^{n+1}_+$, i.e., a half-ball in a half-space.
• The infimum of the isoperimetric ratio is attained only by such flat half-disks, proving the sharpness of the bound.
• The proof relies on a localization argument using a large ball $B_{\rho^*}(0)$, reducing the unbounded case to a bounded convex set $\widetilde{\mathcal{K}}$, where the inequality is known to hold.
• The method uses a blow-up argument and compactness to rule out mass concentration at infinity, ensuring the limit of almost minimizers satisfies the inequality.
• The result confirms Choe’s conjecture in the case of relative area minimizers, extending prior work in codimension one to arbitrary codimension via integral currents.