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$Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps

Giacomo Canevari, Van Phu Cuong Le, Ramon Oliver-Bonafoux, Giandomenico Orlandi

TL;DR

The paper proves a Γ-convergence result for the p-Dirichlet energy D_p of maps from a bounded domain Ω into a smooth closed manifold 𝒩, in the regime p→k^− with Dirichlet boundary data v. It introduces topological singular sets as n-dimensional flat chains with coefficients in the Abelian group π_{k−1}(𝒩) and uses flat-chain methods, retraction maps, and a ball-construction to connect asymptotic energy to the mass of a limiting chain S solving the homological Plateau problem. The main results provide compactness and lower bounds that yield convergence of (k−p)D_p to the mass of S, and an upper bound via a dipole insertion/dense-chain construction, establishing Γ-convergence with respect to the topology of flat chains. As a consequence, energy-minimizing p-harmonic maps converge, after subsequence extraction, to a mass-minimizing chain S representing the topological singularities, thereby describing the asymptotic structure of singularities in terms of Plateau-type minimization in homology classes. The framework extends existing Ginzburg–Landau and Jacobian-type analyses to arbitrary n-scalar singular sets for manifold-valued maps and provides a no-boundary Γ-convergence variant, emphasizing the intrinsic geometric-topological nature of the limiting objects.

Abstract

We prove a $Γ$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $Ω\subseteq \mathbb{R}^{n+k}$ to $\mathscr{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold with Abelian fundamental group, where $n$ and $k$ are integers, $n \geq 0$, $k \geq 2$. We focus on the regime $p \to~k^-$ under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the $\textit{topological singular sets}$ for families of $\mathscr{N}$-valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are $n$-dimensional flat chains with coefficients in $π_{k-1}(\mathscr{N})$ endowed with a suitable norm. As a consequence of our main result, it follows that the topological singular sets of energy minimizing $p$-harmonic maps converge to a $n$-dimensional flat chain $S$ with coefficients in $π_{k-1}(\mathscr{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.

$Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps

TL;DR

The paper proves a Γ-convergence result for the p-Dirichlet energy D_p of maps from a bounded domain Ω into a smooth closed manifold 𝒩, in the regime p→k^− with Dirichlet boundary data v. It introduces topological singular sets as n-dimensional flat chains with coefficients in the Abelian group π_{k−1}(𝒩) and uses flat-chain methods, retraction maps, and a ball-construction to connect asymptotic energy to the mass of a limiting chain S solving the homological Plateau problem. The main results provide compactness and lower bounds that yield convergence of (k−p)D_p to the mass of S, and an upper bound via a dipole insertion/dense-chain construction, establishing Γ-convergence with respect to the topology of flat chains. As a consequence, energy-minimizing p-harmonic maps converge, after subsequence extraction, to a mass-minimizing chain S representing the topological singularities, thereby describing the asymptotic structure of singularities in terms of Plateau-type minimization in homology classes. The framework extends existing Ginzburg–Landau and Jacobian-type analyses to arbitrary n-scalar singular sets for manifold-valued maps and provides a no-boundary Γ-convergence variant, emphasizing the intrinsic geometric-topological nature of the limiting objects.

Abstract

We prove a -convergence result for the -Dirichlet energy functional defined on maps from a smooth bounded domain to , a -connected and smooth closed Riemannian manifold with Abelian fundamental group, where and are integers, , . We focus on the regime under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the for families of -valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are -dimensional flat chains with coefficients in endowed with a suitable norm. As a consequence of our main result, it follows that the topological singular sets of energy minimizing -harmonic maps converge to a -dimensional flat chain with coefficients in which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.

Paper Structure

This paper contains 24 sections, 43 theorems, 277 equations, 2 figures.

Table of Contents

  1. Notations and preliminary results
  2. Norms on the homotopy group of N
  3. Retraction maps
  4. Flat chains with coefficients in a normed Abelian group
  5. Topological singular sets
  6. Lower p-energy bounds via the ball construction
  7. Lower bounds on the p-energy of maps in the critical dimension
  8. Dependence on the parameter p
  9. Proof of the compactness and lower bound statement in Theorem \ref{['MainThm']}
  10. A local statement
  11. Proof of the statement \ref{['MainThm1']} in Theorem \ref{['MainThm']} completed
  12. Proof of the upper bound statement in Theorem \ref{['MainThm']}
  13. Notations and preliminary results
  14. Reduction to a dense class of chains
  15. Insertion of dipoles
  16. ...and 9 more sections

Key Result

Theorem 1

Assume that hp:N holds and let $v \in W^{1-1/k,k}(\Omega, \, \mathscr{N})$. Then, we have:

Figures (2)

  • Figure 1: Maps have singularities that are uniformly distributed in the critical dimension with $k=2$. This figure serves as an illustration of how to obtain a lower bound for the $p$-Dirichlet energy of maps with uniformly distributed singularities, via a suitable rescaling. In the general case, one must use ball constructions to prove the lower bound.
  • Figure 2: An illustration of a singular set and a dual grid within a cube.

Theorems & Definitions (84)

  • Theorem 1: $\Gamma$-convergence of the $p$-Dirichlet energies and topological singular sets
  • Theorem 2: Compactness and convergence for $p$-harmonic maps as $p \to k^-$
  • Proposition 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • Remark 1.1
  • Lemma 1.5
  • ...and 74 more