Monotonicity of the convexity-inspired frequency in rotational symmetry

Determine whether the convexity-inspired frequency function \(\widetilde N(r)=\phi(r)\phi'(r)E(r)/F(r)\), associated with the normalized spherical \(L^2\)-mass of harmonic functions on a rotationally symmetric Cartan–Hadamard manifold, is monotonic without imposing the additional geometric conditions stated in Proposition 4.1.

Background

The paper introduces the frequency function $\widetilde N(r)=\phi(r)\phi'(r)E(r)/F(r)$ because it is the logarithmic derivative of the normalized spherical $L^2$-mass $\mathcal H_\phi(r)=F(r)/\phi(r)^{n-1}$ with respect to the radial variable $\tau_\phi$ defined by $d\tau_\phi/dr=1/(\phi\phi')$. Monotonicity of $\widetilde N$ would imply logarithmic convexity of the normalized spherical mass, extending the Agmon and Korevaar–Meyers theory from Euclidean space to rotationally symmetric Cartan–Hadamard manifolds.

Proposition 4.1 proves monotonicity only under two additional assumptions: $(\phi'/\phi)'\leq 0$ and $$\phi''(r)\phi(r)^{n-2}\geq (n-2)\phi'(r)\int_0^r\phi''(s)\phi(s)^{n-3}\,ds$$. The paper verifies these conditions for $\phi(r)=\sinh r$, obtaining the hyperbolic three-sphere theorem, but does not establish monotonicity in the unrestricted rotationally symmetric setting.