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Regularity of critical immersions for scale-invariant curvature energies in even dimensions
Abstract
For immersions of a closed 2h-dimensional manifold Σ into ℝm, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term ∫Σ |∇(h−1) I I⃗|g2 dvolg 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 Wh+1,2, whose induced metric and its inverse have L∞ 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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BibTeX
@misc{lan2026regularity,
title = {{Regularity of critical immersions for scale-invariant curvature energies in even dimensions}},
author = {Tian Lan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39606},
eprint = {2609.39606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39606},
}
RIS
TY - UNPB
TI - Regularity of critical immersions for scale-invariant curvature energies in even dimensions
AU - Tian Lan
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.39606
UR - https://arxiv.org/abs/2609.39606
AB - For immersions of a closed 2h-dimensional manifold Σ into ℝ^m, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term ∫_Σ |∇^(h−1) I I⃗|_g^2 dvol_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^∞ 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.
ER -