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Weak immersions with second fundamental form in a critical Sobolev space

Dorian Martino, Tristan Rivière

TL;DR

This work addresses the compactness of smooth immersions under a critical Sobolev bound on the second fundamental form in even dimensions n >= 4. It proves that such weak immersions induce a controlled C^1 differential structure via a harmonic-coordinate atlas and establishes a weak sequential closure for energy-bounded sequences, modulo finitely many singularities. The construction combines Hardy inequalities, Lorentz-Sobolev estimates, and a careful atlas framework to pass to the limit, yielding a bubble-tree type decomposition of singularities and a concrete model for the limit geometry. The results advance the study of conformally invariant Lagrangians for immersions in higher dimensions, including generalized Willmore energies like Graham Reichert, by providing a rigorous compactness and regularity theory for weak limits.

Abstract

We develop the analysis of Lipschitz immersions of $n$-dimensional manifolds into $\mathbb{R}^d$ having their second fundamental forms bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$ in dimension $n\geq 4$ even and any codimension. We prove that, while such a weak immersion is not necessary $C^1$, it generates a $C^1$ differential structure on the domain. More precisely, for any such an immersion, there is an atlas in which the first fundamental form is continuous and the transition maps are $C^1$. We prove that this $C^1$ structure is diffeomorphic to the original one. This result is the starting point of the analysis of the behavior of sequences of weak immersions with second fundamental forms uniformly bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$. In the second part of the paper we establish a weakly sequential closure theorem for such sequences. This analysis is motivated by the study of conformally invariant Lagrangian of immersions in dimension larger than two such as generalized Willmore energies, for instance the Graham--Reichert functional obtained in the computation of renormalized volumes of five-dimensional minimal submanifolds of the hyperbolic space $\mathbb{H}^{d+1}$.

Weak immersions with second fundamental form in a critical Sobolev space

TL;DR

This work addresses the compactness of smooth immersions under a critical Sobolev bound on the second fundamental form in even dimensions n >= 4. It proves that such weak immersions induce a controlled C^1 differential structure via a harmonic-coordinate atlas and establishes a weak sequential closure for energy-bounded sequences, modulo finitely many singularities. The construction combines Hardy inequalities, Lorentz-Sobolev estimates, and a careful atlas framework to pass to the limit, yielding a bubble-tree type decomposition of singularities and a concrete model for the limit geometry. The results advance the study of conformally invariant Lagrangians for immersions in higher dimensions, including generalized Willmore energies like Graham Reichert, by providing a rigorous compactness and regularity theory for weak limits.

Abstract

We develop the analysis of Lipschitz immersions of -dimensional manifolds into having their second fundamental forms bounded in the critical Sobolev space in dimension even and any codimension. We prove that, while such a weak immersion is not necessary , it generates a differential structure on the domain. More precisely, for any such an immersion, there is an atlas in which the first fundamental form is continuous and the transition maps are . We prove that this structure is diffeomorphic to the original one. This result is the starting point of the analysis of the behavior of sequences of weak immersions with second fundamental forms uniformly bounded in the critical Sobolev space . In the second part of the paper we establish a weakly sequential closure theorem for such sequences. This analysis is motivated by the study of conformally invariant Lagrangian of immersions in dimension larger than two such as generalized Willmore energies, for instance the Graham--Reichert functional obtained in the computation of renormalized volumes of five-dimensional minimal submanifolds of the hyperbolic space .

Paper Structure

This paper contains 38 sections, 25 theorems, 365 equations.

Table of Contents

  1. Introduction
  2. Geometric and analytic setting
  3. Notations
  4. Lorentz spaces
  5. Weak immersions and the functional $\mathcal{E}$
  6. Convergence of manifolds
  7. Extrinsic estimates
  8. Hardy inequality and consequences
  9. Volume growth of extrinsic balls
  10. Construction of one chart
  11. Second fundamental form of a slice
  12. Construction of the good slice
  13. A given good slice is a union of topological spheres, each of them being close to a round sphere
  14. Extension of the good slice
  15. The interior of a good slice is the union of topological balls
  16. ...and 23 more sections

Key Result

Theorem 1.2

Let $\Sigma$ be a closed oriented manifold of even dimension $n\geq 4$. Every $\vec{\Phi}\in \mathcal{I}_{\frac{n}{2}-1,2}(\Sigma;\mathbb{R}^d)$ induces a differential structure of class $C^1$. More precisely, there exists an atlas of harmonic coordinates in which, the metric $g_{\vec{\Phi}}$ is con

Theorems & Definitions (79)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2: Proposition 11.1.10 in petersen2016
  • Lemma 2.3: Lemma 11.1.9 in petersen2016
  • Theorem 3.1: Theorem 6.2 in MarRiv2025
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 69 more