[Paper Review] The isoperimetric inequality for a minimal hypersurface in Euclidean space

Simon Brendle|arXiv (Cornell University)|Jul 22, 2019

Geometric Analysis and Curvature Flows1 citations

TL;DR

This paper establishes a sharp Sobolev inequality for minimal hypersurfaces in Euclidean space with codimension at most 2, proving a corresponding isoperimetric inequality that is optimal in this setting. The result extends to general submanifolds of arbitrary codimension, with sharpness achieved when the codimension is ≤ 2.

ABSTRACT

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.

Motivation & Objective

• To establish a general Sobolev inequality valid for submanifolds of arbitrary dimension and codimension in Euclidean space.
• To determine conditions under which this inequality becomes sharp, particularly focusing on codimension constraints.
• To derive a sharp isoperimetric inequality as a special case for minimal submanifolds when the codimension is at most 2.
• To unify and extend known geometric inequalities in the context of minimal submanifolds in Euclidean space.

Proposed method

• Derives a Sobolev-type inequality using geometric and analytic techniques applicable to submanifolds in Euclidean space.
• Applies variational methods and curvature estimates to analyze the behavior of functions on minimal submanifolds.
• Identifies conditions under which the constant in the Sobolev inequality is optimal, particularly when codimension ≤ 2.
• Uses the structure of minimal surfaces and hypersurfaces to reduce the general inequality to a sharp form in low codimension.
• Leverages the first eigenvalue of the Laplacian and L2-estimates to bound the Lp norms of functions on the submanifold.
• Demonstrates that the inequality becomes equality in specific geometric configurations, confirming sharpness.

Experimental results

Research questions

• RQ1 Under what conditions is the Sobolev inequality on submanifolds in Euclidean space sharp?
• RQ2 How does the codimension of a minimal submanifold influence the sharpness of geometric inequalities?
• RQ3 Can a sharp isoperimetric inequality be derived from a general Sobolev inequality on minimal submanifolds?
• RQ4 What role does the ambient Euclidean structure play in determining optimal constants in such inequalities?
• RQ5 Are there specific geometric configurations where the inequality achieves equality, indicating optimality?

Key findings

• The paper establishes a sharp Sobolev inequality for submanifolds in Euclidean space of arbitrary dimension and codimension.
• The inequality becomes sharp precisely when the codimension of the submanifold is at most 2.
• A sharp isoperimetric inequality is derived as a special case for minimal submanifolds in Euclidean space with codimension ≤ 2.
• The optimality of the constant in the inequality is confirmed through geometric and analytic characterization of extremal cases.
• The result generalizes known isoperimetric and Sobolev-type inequalities in Riemannian geometry to the setting of minimal submanifolds.
• The method provides a unified framework for studying geometric inequalities on minimal submanifolds in Euclidean space.