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Quantitative Stability for Yamabe minimizers on manifolds with boundary

Benjamín Borquez, Rayssa Caju, Hanne Van Den Bosch

TL;DR

This work develops a quantitative stability theory for Escobar's Yamabe-type energy on compact manifolds with boundary by reducing to a boundary functional via harmonic extension and Lyapunov–Schmidt reduction. It proves that the energy deficit controls the distance to the minimizer set with a power 2+γ, where γ≥0 and may be positive under Adams–Simon-type nondegeneracy, and it establishes a conformally invariant stability bound as a corollary. The analysis combines local boundary stability with a global compactness argument to obtain a robust stability inequality for the Escobar energy, and it verifies the sharp γ=0 ball case while clarifying the role of the boundary in the stability landscape. The framework links sharp trace Sobolev inequalities, Dirichlet-to-Neumann maps, and Adams–Simon positivity to illuminate stability phenomena for scalar-flat metrics with prescribed boundary mean curvature.

Abstract

This paper addresses the quantitative stability for a Yamabe-type functional on compact manifolds with boundary introduced by Escobar. Minimizers of the functional correspond to scalar-flat metrics with constant mean curvature on the boundary. We prove that the deficit controls the distance to the minimizing set to a suitable power by reducing the problem to the analogous question for an effective functional on the boundary.

Quantitative Stability for Yamabe minimizers on manifolds with boundary

TL;DR

This work develops a quantitative stability theory for Escobar's Yamabe-type energy on compact manifolds with boundary by reducing to a boundary functional via harmonic extension and Lyapunov–Schmidt reduction. It proves that the energy deficit controls the distance to the minimizer set with a power 2+γ, where γ≥0 and may be positive under Adams–Simon-type nondegeneracy, and it establishes a conformally invariant stability bound as a corollary. The analysis combines local boundary stability with a global compactness argument to obtain a robust stability inequality for the Escobar energy, and it verifies the sharp γ=0 ball case while clarifying the role of the boundary in the stability landscape. The framework links sharp trace Sobolev inequalities, Dirichlet-to-Neumann maps, and Adams–Simon positivity to illuminate stability phenomena for scalar-flat metrics with prescribed boundary mean curvature.

Abstract

This paper addresses the quantitative stability for a Yamabe-type functional on compact manifolds with boundary introduced by Escobar. Minimizers of the functional correspond to scalar-flat metrics with constant mean curvature on the boundary. We prove that the deficit controls the distance to the minimizing set to a suitable power by reducing the problem to the analogous question for an effective functional on the boundary.

Paper Structure

This paper contains 12 sections, 13 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. The case of the ball
  4. General manifolds: notation and preliminary constructions.
  5. The boundary functional Q .
  6. Local analysis near a minimizer.
  7. General manifolds: Proof of the main theorems.
  8. Local stability estimate.
  9. Compactness and global stability
  10. Proof of Proposition \ref{['prop:ASpint']}
  11. Details on the Lyapunov-Schmidt reduction
  12. Compactness of the set of minimizers

Key Result

Theorem 1.1

Let $(M,g)$ be a compact Riemannian manifold with boundary of dimension $n\geqslant 3$, such that $Q(M,\partial M) < Q(B^{n},\partial B^{n})$. Assume also that the reference metric $g$ has $R_{g}\geqslant 0$. Then there exist constants $C >0$ and $\gamma \geqslant 0$, depending on $(M,g)$, such that

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Proposition 1.4: $\text{AS}_{p}$ implies $\gamma >0$
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 18 more