Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature

Abstract: In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole, nonnegative Ricci curvature and such that the geodesic spheres centered at the pole are mean-convex (or if the square of the distance function from the pole is convex), any biharmonic function of subquadratic growth must be harmonic, and hence, any biharmonic function of sublinear growth must be constant. Our proof relies on a new local $L^2$ estimate for the Laplacian of biharmonic functions combined with a mean value inequality. Examples where our theorem applies include hypersurfaces of positive sectional curvature in $\mathbb{R}^n$, and manifolds with a pole of nonnegative Ricci curvature whose curvature decays at infinity rapidly enough.

• The paper demonstrates that in complete manifolds with a pole and nonnegative Ricci curvature, biharmonic functions showing subquadratic growth are reduced to harmonic functions.
• The methodology integrates local L2 energy estimates, a hole-filling iteration lemma, and a mean value inequality to rigorously control the Laplacian of biharmonic functions.
• The results extend classical Liouville theorems and open new pathways for investigating polyharmonic functions in varied geometric settings.

Liouville Theorem for Biharmonic Functions on Manifolds of Nonnegative Ricci Curvature

Introduction

This paper extends Yau's Liouville theorem, which specifies conditions under which harmonic functions on manifolds are constant, to encompass biharmonic functions. Specifically, it is shown that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, biharmonic functions displaying subquadratic growth are harmonic. Consequently, sublinear growth entails that these functions are constant. The proof hinges on a local $L^2$ estimate for the Laplacian of biharmonic functions, combined with a mean value inequality.

Background and Main Contributions

Traditionally, Liouville theorems have focused on harmonic functions bounded in Euclidean spaces. Extensions to biharmonic and polyharmonic functions have been limited in scope within the context of Riemannian manifolds. The paper identifies analogs to Euclidean theorems through rigorous local $L^2$ energy estimates for biharmonic functions, emphasizing the importance of geometric assumptions such as mean convexity of spheres and convexity of the square of the distance function from poles.

In Euclidean spaces, bounded polyharmonic functions, also known as biharmonic functions, reduce to harmonic polynomials of degree one. On manifolds, these results require adapting traditional approaches, particularly in scenarios involving nonnegative Ricci curvature. The geometric assumptions, such as having manifolds with a pole and certain curvature criteria, are pivotal in achieving the extended theorem.

Technical Approach and Proofs

The proof integrates three pivotal components:

1. Local Energy Inequality: A detailed energy inequality for biharmonic function Laplacians is presented, which is analogous to the classical Caccioppoli inequality. This facilitates understanding how subquadratic growth impacts the behavior of these functions.
2. Hole-Filling Iteration Lemma: This technique assists in establishing bounds over growing domains by leveraging singular factors in energy inequalities.
3. Mean Value Inequality: Applied to estimate harmonic functions arising from biharmonic scenarios, facilitating the reduction of complex biharmonic analysis to simpler harmonic cases.

The hypotheses for mean-convexity or geometric convexity were explored in depth; with explicit bounds for cutoff functions playing a crucial role, particularly using a novel application of Young’s inequality that adapts to the geometry of the manifold.

Implications and Future Work

The result constitutes a crucial step in extending Liouville-type properties to higher order elliptic operators in Riemannian settings, contributing significantly to geometric analysis. The identification of natural geometric conditions under which biharmonic functions behave as harmonic introduces new pathways for employing curvature conditions in manifold theory.

Future work could further optimize the theorem's assumptions, potentially aiming to remove geometric restrictions such as mean-convexity by exploring alternative curvature constraints or extended manifold types. Additionally, exploring extensions to polyharmonic cases beyond biharmonic functions presents intriguing prospects. The adaptation of energy-based inequalities to higher-dimensional operators in such contexts remains an open and promising domain of research.

Conclusion

This paper provides a robust framework for extending Liouville’s theorem from harmonic to biharmonic functions on manifolds exhibiting nonnegative Ricci curvature. This advances understanding of function growth behavior relative to manifold geometry and lays groundwork for additional exploration into higher-order harmonic functions across varying geometric configurations.