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John E. Bravo

Publications and source records attributed to John E. Bravo.

4 recordsLinked to original sources
arXiv2025

On Liouville's theorem and the Strong Liouville Property

We explore Liouville's theorem and the Strong Liouville Property (SLP) for harmonic functions on Riemannian cones and surfaces. Our approach recasts the classical Liouville property in terms of the growth of radial eigenfunctions (in the case of manifolds with rotational symmetry), allowing us to recover and sharpen known results under minimal assumptions. We provide explicit estimates for the slowest-growing nonconstant harmonic functions on cones and surfaces, and construct examples where doubling fails but Liouville and SLP still hold. Finally, we prove a nonlinear Liouville theorem for $p$-subharmonic functions, $p\geq 2$, under curvature bounds, in complete Riemannian surfaces with a pole which simultaneously recover Milnor's and Cheng--Yau's theorems as particular cases. This result appears to be new and suggests a unified geometric perspective on linear and nonlinear Liouville phenomena.

math.AP↗
arXiv2025

A Polyharmonic Liouville Hierarchy on Complete Manifolds of Nonnegative Ricci Curvature

In this paper, we establish a complete Liouville--type hierarchy for polyharmonic functions on Riemannian manifolds with nonnegative Ricci curvature. Extending Yau's classical result for harmonic functions and our recent biharmonic Liouville theorem, we prove that on any complete manifold of nonnegative Ricci curvature, every $k$--polyharmonic function of growth $o(r^{2(k-1)})$ must in fact be $(k-1)$--polyharmonic. Iterating this procedure yields the result that all polyharmonic functions of sublinear growth are constant.The key innovation is a new $L^{2}$ estimate for the Laplacian of a polyharmonic function, obtained by induction through a delicate cutoff construction combined with a hole--filling argument. This provides the first sharp geometric extension of the Euclidean classification of polyharmonic functions to manifolds of nonnegative Ricci curvature, and completes a natural hierarchy of Yau--type Liouville theorems for iterates of the Laplacian.

math.DG↗
arXiv2025

Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature

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 and nonnegative Ricci curvature, 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.

math.DG↗