The Regularity of Critical Points to the Dirichlet Energy of the Mean Curvature in Dimension 4

TL;DR

This work proves analytic regularity for weak critical points of the Dirichlet energy of the mean curvature in dimension four. By recasting the Euler–Lagrange equation into a divergence form and exploiting invariances via Noether currents, the authors develop a conservation-law framework in the presence of rough metrics $g\in L^{\infty}\cap W^{1,4}$, together with a Hodge-decomposition-based analysis. A sequence of structural identities and estimates on currents $\vec{L},R,S$ controls nonlinear interactions and enables a bootstrap that ultimately yields real-analyticity in any $g_{\vec{\Phi}}$-harmonic coordinates. The combination of divergence-form methods, conservation laws, and compensation techniques advances the regularity theory for four-dimensional curvature energies and paves the way for generalized Willmore-type functionals in higher codimension. The results have potential implications for geometric analysis and mathematical physics where scaling-invariant curvature energies arise.

Abstract

We prove that weak immersions of four dimensional manifolds in $\mathbb{R}^5$ which are critical points to the Dirichlet Energy of the Mean Curvature are analytic in any given local harmonic chart. This variational problem is a model case for the large family of scaling invariant Lagrangians, hence critical, of curvature energies in 4 dimensions depending on the first and the second fundamental form. Because of the criticality of this variational problem, the regularity result is obtained through an abundant use of methods from integrability by compensation theory such as interpolation spaces estimates.

Theorems & Definitions (56)

• Definition I.1
• Theorem I.2
• Lemma II.1
• proof
• Definition II.2
• Definition II.3: Lorentz spaces
• Lemma II.4: Hunt
• Definition II.5
• Lemma II.6
• Lemma II.7: Triebel95