Existence and Regularity of Extremal Metrics for the Conformal Dirichlet-to-Robin Map

2026/07/29 by Samuel Pérez-Ayala
Mathematics · #math.DG

paper · pdf

arxiv created 2026/07/29 · arxiv updated 2026/07/31

Abstract

We study the variational properties of the spectrum of the Dirichlet-to-Robin map D_g on connected compact manifolds with boundary of dimension at least three. For the first eigenvalue, we show that Type II Yamabe metrics extremize the first normalized eigenvalue functional, and we characterize all extremals. If [g] is a conformal class for which D_g has at least two negative eigenvalues, then we show the existence of a generalized metric that maximizes the second normalized eigenvalue of D_g in the conformal class. Moreover, we show that each such metric either defines a solution to an Escobar--Yamabe type equation on manifolds with boundary that changes sign along the boundary, or a weakly free-boundary harmonic map into the unit Euclidean ball.

Citations

• Sign-changing solutions to the Escobar problem on manifolds with boundary
• Notes on the uniqueness of Type II Yamabe metrics
• Maximizing higher eigenvalues in dimensions three and above
• Extremising eigenvalues of the GJMS operators in a fixed conformal class
• Conformal optimization of eigenvalues on surfaces with symmetries
• Unique continuation for Robin problems with non-smooth potentials
• Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds
• Min-max harmonic maps and a new characterization of conformal eigenvalues
• Conformally maximal metrics for Laplace eigenvalues on surfaces
• Conformal invariants from nodal sets II. Manifolds with boundary
• The qualitative behavior at the free boundary for approximate harmonic maps from surfaces
• Moser iteration applied to elliptic equations with critical growth on the boundary
• Escobar-Yamabe compactifications for Poincare-Einstein manifolds and rigidity theorems
• The Calderón problem for the conformal Laplacian
• Regularity and quantification for harmonic maps with free boundary
• Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces
• Minimal surfaces and eigenvalue problems
• Second Eigenvalue of the Yamabe Operator and Applications
• Variational aspects of Laplace eigenvalues on Riemannian surfaces
• Conformal Deformation to Scalar Flat Metrics with Constant Mean Curvature on the Boundary in Higher Dimensions
• Laplacian eigenvalue functionals and metric deformations on compact manifolds

Related

• Extremal metrics for Laplace eigenvalues in perturbed conformal classes
• Laplace and Steklov extremal metrics via n-harmonic maps
• Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"
• Extremal metrics for the eigenvalues of the Laplacian on manifolds with boundary
• On a new functional for extremal metrics of the conformal laplacian in high dimension