Hayden Hunter | Searcharxiv

arXiv2026

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

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.

math.DG↗

arXiv2023

On Higher Dimensional Milnor Frames

A classic result of Milnor shows that any 3-dimensional unimodular metric Lie algebra admits an orthonormal frame with at most three nontrivial structure constants. These frames are referred to as Milnor frames. We define extensions of Milnor frames into higher dimensions and refer to these higher dimensional analogues as Lie algebras with Milnor frames. We determine that $n$-dimensional, $n \geq 4$, Lie algebras with Milnor frames are isomorphic to the direct sum of 3-dimensional Heisenberg Lie algebras $\mathfrak{h}^3$, 4-dimensional 3-step nilpotent Lie algebras $\mathfrak{h}^4$, and an abelian Lie algebra $\mathfrak{a}$. Moreover, for any Lie algebra $\mathfrak{g}\not\cong \mathfrak{h}^3 \oplus \mathfrak{a}$ with a Milnor frame, there exists an inner product structure $g$ on $\mathfrak{g}$ such that $(\mathfrak{g}, g)$ does not admit an orthonormal Milnor frame.

math.DG↗