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Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

Michael B. Law

TL;DR

This work classifies polynomial-growth drift-harmonic functions on asymptotically paraboloidal (AP) manifolds by combining geometric analysis with spectral and parabolic techniques. It introduces the AP framework, defines drift-Laplacians \\mathcal{L}_f, and spaces \\mathcal{H}_d that capture polynomial growth, proving that drift-harmonic functions asymptotically separate variables and span a finite-dimensional space with dimension \\dim \\mathcal{H}_d = \\sum_{\\lambda_k \\le d} m_k. The authors develop a detailed inductive procedure that alternates asymptotic control (A) and construction (C) to build a basis \\mathcal{B}_d with pairwise almost-orthogonality, leveraging frequency functions, preservation of almost orthogonality, and a blowdown/parabolic framework. Central tools include a nonlinear ODE for the frequency, a three-circles principle for drift-harmonic functions, and a robust blowdown analysis to connect the AP geometry to model-parabolic behavior. The results apply to steady gradient Ricci solitons (e.g., the Bryant soliton), yield Liouville-type conclusions, and provide a precise spectral-type decomposition for drift-harmonic functions on soliton ends, enriching the analytic toolbox for geometric PDEs on noncompact manifolds.

Abstract

We construct and classify all polynomial growth solutions to certain drift-harmonic equations on complete manifolds with paraboloidal asymptotics. These encompass the natural drift-harmonic equations on certain steady gradient Ricci solitons. Specifically, we show that all drift-harmonic functions with polynomial growth asymptotically separate variables, and compute the dimensions of spaces of drift-harmonic functions with a given polynomial growth rate. The proof uses an inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.

Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

TL;DR

This work classifies polynomial-growth drift-harmonic functions on asymptotically paraboloidal (AP) manifolds by combining geometric analysis with spectral and parabolic techniques. It introduces the AP framework, defines drift-Laplacians \\mathcal{L}_f, and spaces \\mathcal{H}_d that capture polynomial growth, proving that drift-harmonic functions asymptotically separate variables and span a finite-dimensional space with dimension \\dim \\mathcal{H}_d = \\sum_{\\lambda_k \\le d} m_k. The authors develop a detailed inductive procedure that alternates asymptotic control (A) and construction (C) to build a basis \\mathcal{B}_d with pairwise almost-orthogonality, leveraging frequency functions, preservation of almost orthogonality, and a blowdown/parabolic framework. Central tools include a nonlinear ODE for the frequency, a three-circles principle for drift-harmonic functions, and a robust blowdown analysis to connect the AP geometry to model-parabolic behavior. The results apply to steady gradient Ricci solitons (e.g., the Bryant soliton), yield Liouville-type conclusions, and provide a precise spectral-type decomposition for drift-harmonic functions on soliton ends, enriching the analytic toolbox for geometric PDEs on noncompact manifolds.

Abstract

We construct and classify all polynomial growth solutions to certain drift-harmonic equations on complete manifolds with paraboloidal asymptotics. These encompass the natural drift-harmonic equations on certain steady gradient Ricci solitons. Specifically, we show that all drift-harmonic functions with polynomial growth asymptotically separate variables, and compute the dimensions of spaces of drift-harmonic functions with a given polynomial growth rate. The proof uses an inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.
Paper Structure (35 sections, 57 theorems, 362 equations)

This paper contains 35 sections, 57 theorems, 362 equations.

Table of Contents

  1. Introduction
  2. Motivation and the main result
  3. Overview of the proof
  4. Frequency functions
  5. Preservation of almost orthogonality
  6. Constructing drift-harmonic functions
  7. Geometry of asymptotically paraboloidal manifolds
  8. Definition and examples of AP manifolds
  9. Conventions
  10. The asymptotic cross-section
  11. Analytical machinery for drift-harmonic functions
  12. Frequency and related functionals
  13. An ODE lemma
  14. Another formula for $D'/D$
  15. Almost separation of variables and preservation of almost orthogonality
  16. ...and 20 more sections

Key Result

Theorem 1.2

Let $(M^n,g,r)$ be an AP manifold of dimension $n \geq 3$, and let $f \in C^\infty(M)$ satisfy Assumption assump:f. For each $d \in \mathbb{R}$ and $u \in \mathcal{H}_d$:

Theorems & Definitions (122)

  • Theorem 1.2: Main theorem
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 112 more