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Price Inequality and the Growth of Harmonic Functions on Non-Positively Curved Manifolds

Luca F. Di Cerbo, Hayden Hunter, Aaron K. Thrasher

TL;DR

This work addresses the growth rate of the $L^2$-energy of harmonic functions on geodesic balls in complete simply connected manifolds with non-positive sectional curvature. It develops a double-sided Price inequality featuring a damping function $\mu(R)$ and curvature-dependent mean-curvature bounds, yielding explicit exponential bounds for $\int_{B_R} f^2$ in terms of the curvature pinching $k, k'$ and the auxiliary function. The results have implications for the analytic structure of potential degree-one counterexamples to the Singer conjecture in negatively curved spaces, by constraining harmonic functions with $L^2$-integrable gradient. Additionally, the paper provides sharp estimates for positive harmonic functions, showing that the hyperbolic Poisson kernel governs maximal $L^2$-growth, with precise bounds in both the hyperbolic and pinched-curvature settings, supported by hypergeometric-function computations and classical gradient estimates.

Abstract

We obtain effective estimates for the growth rate of the $L^2$-energy of harmonic functions on geodesic balls in complete simply connected non-positively curved Riemannian manifolds with pinched sectional curvature. Our study relies upon a double-sided Price inequality for harmonic functions. Finally, we apply this circle of ideas to study the analytical structure of a potential counterexample to the Singer conjecture in degree one.

Price Inequality and the Growth of Harmonic Functions on Non-Positively Curved Manifolds

TL;DR

This work addresses the growth rate of the -energy of harmonic functions on geodesic balls in complete simply connected manifolds with non-positive sectional curvature. It develops a double-sided Price inequality featuring a damping function and curvature-dependent mean-curvature bounds, yielding explicit exponential bounds for in terms of the curvature pinching and the auxiliary function. The results have implications for the analytic structure of potential degree-one counterexamples to the Singer conjecture in negatively curved spaces, by constraining harmonic functions with -integrable gradient. Additionally, the paper provides sharp estimates for positive harmonic functions, showing that the hyperbolic Poisson kernel governs maximal -growth, with precise bounds in both the hyperbolic and pinched-curvature settings, supported by hypergeometric-function computations and classical gradient estimates.

Abstract

We obtain effective estimates for the growth rate of the -energy of harmonic functions on geodesic balls in complete simply connected non-positively curved Riemannian manifolds with pinched sectional curvature. Our study relies upon a double-sided Price inequality for harmonic functions. Finally, we apply this circle of ideas to study the analytical structure of a potential counterexample to the Singer conjecture in degree one.
Paper Structure (6 sections, 9 theorems, 76 equations)

This paper contains 6 sections, 9 theorems, 76 equations.

Table of Contents

  1. Introduction and Organization
  2. Double-Sided Price Inequality
  3. On the Singer Conjecture in Degree One
  4. Estimates for Positive Harmonic Functions
  5. Appendix A: $\mu$-Function on Euclidean Space
  6. Appendix B: Integration of the Hyperbolic Poisson Kernel over a Geodesic Sphere

Key Result

Theorem 2

Let $f$ be a harmonic function on complete Riemannian manifold $M^{n}$. Then, $\int_{M}f^{p} = \infty$ for $p >1$ or $f$ is a constant function.

Theorems & Definitions (19)

  • Theorem 2: Yau
  • Definition 4
  • Remark 5
  • Theorem 6: Double-Sided Price Inequality
  • proof
  • Theorem 12
  • proof
  • Proposition 15
  • proof
  • Theorem 19
  • ...and 9 more