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Some new functionals related to free boundary minimal submanifolds

Tianyu Ma, Vladimir Medvedev

TL;DR

This paper generalizes the variational characterization of metrics induced by free boundary minimal immersions (FBMI) from surfaces to higher-dimensional FBMI in geodesic balls of $\\mathbb{S}^m_+$ and $\\mathbb{H}^m$, via functionals $\\Theta_{r,i}$ and $\\Omega_{r,i}$ defined on metric spaces using Steklov eigenvalues. It introduces two new functionals $\\Xi_{r,i}^+$ and $\\Xi_{r,i}^-$ tied to Laplace eigenvalues with Robin conditions, showing that extremals of $\\Xi_{r,i}^+$ (and not $\\Xi_{r,i}^-$) correspond to FBMI into geodesic balls, and that these functionals are bounded above (with explicit disk bounds) while $\\Xi_{r,i}^-$ lacks extremals. The sphere and hyperbolic analyses establish that extremal metrics lead to FBMI or FBHM via eigenfunction maps, with the spherical-coefficient analysis revealing a unique consistent form across dimensions. The work also proves hyperbolic analogs and provides concrete bounds and canonical maximal metrics for special topologies, thereby widening the variational toolkit for free boundary minimal geometry and its connections to spectral theory.

Abstract

The metrics induced on free boundary minimal surfaces in geodesic balls in the upper unit hemisphere and hyperbolic space can be characterized as critical metrics for the functionals $Θ_{r,i}$ and $Ω_{r,i}$, introduced recently by Lima, Menezes and the second author. In this paper, we generalize this characterization to free boundary minimal submanifolds of higher dimension in the same spaces. We also introduce some functionals of the form different from $Θ_{r,i}$ and show that the critical metrics for them are the metrics induced by free boundary minimal immersions into a geodesic ball in the upper unit hemisphere. In the case of surfaces, these functionals are bounded from above and not bounded from below. Moreover, the canonical metric on a geodesic disk in a 3-ball in the upper unit hemisphere is maximal for this functional on the set of all Riemannian metric of the topological disk.

Some new functionals related to free boundary minimal submanifolds

TL;DR

This paper generalizes the variational characterization of metrics induced by free boundary minimal immersions (FBMI) from surfaces to higher-dimensional FBMI in geodesic balls of and , via functionals and defined on metric spaces using Steklov eigenvalues. It introduces two new functionals and tied to Laplace eigenvalues with Robin conditions, showing that extremals of (and not ) correspond to FBMI into geodesic balls, and that these functionals are bounded above (with explicit disk bounds) while lacks extremals. The sphere and hyperbolic analyses establish that extremal metrics lead to FBMI or FBHM via eigenfunction maps, with the spherical-coefficient analysis revealing a unique consistent form across dimensions. The work also proves hyperbolic analogs and provides concrete bounds and canonical maximal metrics for special topologies, thereby widening the variational toolkit for free boundary minimal geometry and its connections to spectral theory.

Abstract

The metrics induced on free boundary minimal surfaces in geodesic balls in the upper unit hemisphere and hyperbolic space can be characterized as critical metrics for the functionals and , introduced recently by Lima, Menezes and the second author. In this paper, we generalize this characterization to free boundary minimal submanifolds of higher dimension in the same spaces. We also introduce some functionals of the form different from and show that the critical metrics for them are the metrics induced by free boundary minimal immersions into a geodesic ball in the upper unit hemisphere. In the case of surfaces, these functionals are bounded from above and not bounded from below. Moreover, the canonical metric on a geodesic disk in a 3-ball in the upper unit hemisphere is maximal for this functional on the set of all Riemannian metric of the topological disk.

Paper Structure

This paper contains 8 sections, 12 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Paper organization
  3. Acknowledgments
  4. FBMI in spheres
  5. Critical metrics of sphere FBMI functional
  6. Coefficients of the sphere FBMI functional
  7. FBMI in hyperbolic spaces
  8. Alternative forms of FBMI characterizing functionals

Key Result

Theorem 1.2

Let $\Sigma^k$ be a $k$-dimensional compact smooth manifold with boundary. For $r<\dfrac{\pi}{2}$ and $i\geqslant 1$. Let $\mathbb B^m(r)$ denote a ball of radius $r$ in $\mathbb{S}^m$ centered at the point $(1,0,\ldots,0)$ and $V_i(g)$ the eigenspace of the Steklov eigenvalue $\theta_i$ with freque

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['2.2']}
  • proof : Proof of Theorem \ref{['2.3']}
  • Proposition 2.2
  • Theorem 3.1
  • ...and 10 more