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arXiv · 2609.39606

Regularity of critical immersions for scale-invariant curvature energies in even dimensions

Tian Lan
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Abstract

For immersions of a closed $2h$-dimensional manifold $Σ$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_Σ \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Following the four-dimensional approach developed in joint work with Bernard, Martino, and Rivière, we prove that every weak critical immersion in the natural Sobolev class $W^{h+1,2}$, whose induced metric and its inverse have $L^\infty$ coefficients, is real-analytic in harmonic coordinates. The proof combines geometric conservation laws, structural identities, and elliptic estimates with critical Sobolev coefficients to obtain Morrey decay and bootstrap to full regularity.

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Tian Lan. 2026-10-05. Regularity of critical immersions for scale-invariant curvature energies in even dimensions. https://arxiv.org/abs/2609.39606

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