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A fourth-order Cherrier-Escobar problem with prescribed corner behavior on the half-ball

Jeffrey S. Case, Yueh-Ju Lin, Stephen E. McKeown, Cheikh Birahim Ndiaye, Paul Yang

TL;DR

The paper analyzes a fourth-order Cherrier–Escobar-type problem on the half-ball $B^4_+$, showing that a conformal change can concentrate the interior Gauss–Bonnet contribution at the corner. It reduces the problem to solving a biharmonic equation $\Delta^2\omega=0$ on $B^4_+$ with boundary data given by the boundary operators $P_3^M$, $P_3^N$ and a corner operator $P_2$, under a compatibility constraint at the corner; it proves existence of solutions with prescribed normal data along $M$ and $N$ and constant corner value, with uniqueness when the corner value is fixed. The analysis interweaves explicit biharmonic constructions on the ball, spherical-harmonic expansions on $S^3$, elliptic regularity, and the conformal group of the half-ball, including a conformal inversion that swaps boundary components. The work also demonstrates the abundance of non-$S^2$-invariant solutions produced by the conformal group and provides a blueprint toward the general problem on manifolds with corners, emphasizing how corner geometry governs the distribution of topological information.

Abstract

We show that the half-ball in $\mathbb{R}^4$ can be conformally changed so that the only contribution to the Gauss--Bonnet formula is a constant term at the corner. This may be seen as a fourth-order Cherrier--Escobar-type problem on the half-ball.

A fourth-order Cherrier-Escobar problem with prescribed corner behavior on the half-ball

TL;DR

The paper analyzes a fourth-order Cherrier–Escobar-type problem on the half-ball , showing that a conformal change can concentrate the interior Gauss–Bonnet contribution at the corner. It reduces the problem to solving a biharmonic equation on with boundary data given by the boundary operators , and a corner operator , under a compatibility constraint at the corner; it proves existence of solutions with prescribed normal data along and and constant corner value, with uniqueness when the corner value is fixed. The analysis interweaves explicit biharmonic constructions on the ball, spherical-harmonic expansions on , elliptic regularity, and the conformal group of the half-ball, including a conformal inversion that swaps boundary components. The work also demonstrates the abundance of non--invariant solutions produced by the conformal group and provides a blueprint toward the general problem on manifolds with corners, emphasizing how corner geometry governs the distribution of topological information.

Abstract

We show that the half-ball in can be conformally changed so that the only contribution to the Gauss--Bonnet formula is a constant term at the corner. This may be seen as a fourth-order Cherrier--Escobar-type problem on the half-ball.
Paper Structure (4 sections, 8 theorems, 85 equations, 1 table)

This paper contains 4 sections, 8 theorems, 85 equations, 1 table.

Table of Contents

  1. Setup
  2. Analysis on the sphere
  3. Construction
  4. Action of the conformal group

Key Result

Theorem A

Let $M = S^3_+$ and $N = B^3$, with $\Sigma = S^2 = M \cap N$. Let $\psi \in C^{\infty}(M)$ and $\varphi \in C^{\infty}(N)$ satisfy Then there exists $\omega \in C^{3}(B^4_+)$ such that $\omega|_{\Sigma}$ is constant, $\omega$ solves the boundary value problem sphereconds, and With the additional condition the solution is unique.

Theorems & Definitions (16)

  • Theorem A
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 6 more