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A spherical flatness index and a stability inequality for harmonic pseudospheres

Andrea Buffoni, Giovanni Cupini, Ermanno Lanconelli

Abstract

We introduce a new flatness index for the boundary of an open subset $Ω$ of $\mathbb{R}^n$, $n\ge 2$. This index provides a necessary condition for $\partialΩ$ to be a harmonic pseudosphere and sufficient conditions for a harmonic pseudosphere to be a Euclidean sphere. These conditions will follow from a stability inequality formulated in terms of a harmonic invariant, the Kuran gap, recently introduced by the last two authors.

A spherical flatness index and a stability inequality for harmonic pseudospheres

Abstract

We introduce a new flatness index for the boundary of an open subset of , . This index provides a necessary condition for to be a harmonic pseudosphere and sufficient conditions for a harmonic pseudosphere to be a Euclidean sphere. These conditions will follow from a stability inequality formulated in terms of a harmonic invariant, the Kuran gap, recently introduced by the last two authors.

Paper Structure

This paper contains 7 sections, 9 theorems, 181 equations.

Table of Contents

  1. Introduction and main results
  2. Harmonic pseudospheres
  3. The Kuran gap
  4. The spherical flatness index
  5. Main results
  6. Computation of Some Flatness Indices
  7. Proof of the main results

Key Result

Proposition 1.5

Let $\Omega\subseteq \mathbb{R}^n$ be a bounded open set and let $x_0\in \Omega$. If $\partial\Omega$ is Lipschitz flat at a point $z\in T(\partial\Omega,x_0)$, then

Theorems & Definitions (22)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4: Lipschitz flatness
  • Proposition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9: Local sufficient condition for a harmonic pseudosphere to be a sphere
  • Corollary 1.10
  • ...and 12 more