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Monotonicity formulas for minimal submanifolds involving Möbius transformations

Doanh Pham

TL;DR

This work establishes monotonicity formulas for minimal submanifolds under Möbius transformations by analyzing the images of concentric balls through the isometric sphere associated with a Möbius map. The authors first prove precise volume and weighted monotonicity results for the special case of reflections, using the coarea formula and a divergence argument, and then extend these results to general Möbius transformations via a standard decomposition φ = ψσ. The main contributions include explicit monotone quantities J(r) and I(r) and their rigidity characterizations, which yield sharp prescribed-point volume estimates in the Möbius-transformed setting and connect to classical results for pointwise area bounds. These results provide a comprehensive framework for understanding how minimal submanifolds behave under the broad class of Möbius maps and have potential implications for geometric measure theory and geometric analysis in Euclidean spaces.

Abstract

For a minimal submanifold of the Euclidean space, we prove monotonicity formulas for its (weighted) volume within images of concentric balls under Möbius transformations.

Monotonicity formulas for minimal submanifolds involving Möbius transformations

TL;DR

This work establishes monotonicity formulas for minimal submanifolds under Möbius transformations by analyzing the images of concentric balls through the isometric sphere associated with a Möbius map. The authors first prove precise volume and weighted monotonicity results for the special case of reflections, using the coarea formula and a divergence argument, and then extend these results to general Möbius transformations via a standard decomposition φ = ψσ. The main contributions include explicit monotone quantities J(r) and I(r) and their rigidity characterizations, which yield sharp prescribed-point volume estimates in the Möbius-transformed setting and connect to classical results for pointwise area bounds. These results provide a comprehensive framework for understanding how minimal submanifolds behave under the broad class of Möbius maps and have potential implications for geometric measure theory and geometric analysis in Euclidean spaces.

Abstract

For a minimal submanifold of the Euclidean space, we prove monotonicity formulas for its (weighted) volume within images of concentric balls under Möbius transformations.

Paper Structure

This paper contains 5 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of monotonicity formulas
  4. The case of reflections
  5. General case

Key Result

Theorem 1.1

Fix $a \in B^n_1$ and denote the family of balls On the half space $\{x \in \mathop{\mathrm{\mathbb{R}}}\nolimits^n: |a|^2 + 1 - 2\langle a,x \rangle > 0\}$, define the function Suppose that $\Sigma$ is a $k$-dimensional minimal submanifold of $E_{\bar{s}}$ with $\partial\Sigma \subset E_{\bar{s}}$ for some $\bar{s} > 0$. For $s \in (0, \bar{s})$, define Then for $0 < s < t < \bar{s}$, we have

Theorems & Definitions (12)

  • Theorem 1.1: Zhu_JFA2018Naff-Zhu_Monotonicity for minimal in constant cruvature_arXiv2022
  • Corollary : Brendle-Hung_GAFA2017
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2: see e.g. Beardon_Geometry of Discrete
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['Proposition: main result for reflections']}
  • ...and 2 more