arXiv subjects

Xiaohan Cai

Publications and source records attributed to Xiaohan Cai.

7 recordsLinked to original sources

arXiv2026

Dimension of polynomial growth harmonic functions on locally conformally flat manifolds with nonnegative Ricci curvature

Let $\mathcal{H}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding $\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\mathcal{H}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\operatorname{dim}\mathcal{H}_{d}(\mathbb{R}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim \mathcal{H}_d(M)$ case by case.

math.DG↗

arXiv2026

Sharp rigidity for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature

We study the quasilinear Liouville equation \[ -\Delta_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \] then the underlying manifold is isometric to the Euclidean space and $u$ is a standard bubble solution. Both the leading coefficient and the remainder term in the assumption are sharp. The key ingredient in the proof is the connection between the logarithmic lower bound and a sharp upper bound on the total volume of the solution. We also formulate a conjecture concerning the interaction between the sub-logarithmic decay of solutions and the underlying geometry, and prove it for $n=2$, as well as for $n\ge 3$ under a strengthened assumption.

math.AP↗

arXiv2025

A modified Bakry-\'Emery $\Gamma_2$ criterion inequality and the monotonicity of the Tsallis entropy

The Bakry-\'Emery $\Gamma_2$ criterion inequality provides a method for establishing the logarithmic Sobolev inequality. We prove a one-parameter family of weighted Bakry-\'Emery $\Gamma_2$ criterion inequalities which in the limit case yields the improved constant due to Ji \cite{Ji24}. Furthermore, we establish a modified weighted $\Gamma_2$ criterion inequality which could be interpreted as a monotonicity of the Tsallis entropy under the heat flow and yields a family of sharp Sobolev inequalities.

math.DG↗

arXiv2025

Uniqueness results for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary

We prove some Liouville-type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary, thereby confirming some cases of Wang's conjecture (J. Geom. Anal. 31, 2021). We further investigate Wang's conjecture on warped product manifolds and provide a partial verification of this conjecture, which also yields an alternative proof of Gu-Li's resolution of the conjecture in the $\mathbb{B}^n$ case (Math. Ann. 391, 2025). Our approach is based on a general principle of employing the P-function method to such Liouville-type results, with particular emphasis on the role of a closed conformal vector field inherent to such manifolds.

math.AP↗

arXiv2025

A note on Wang's conjecture for harmonic functions with nonlinear boundary condition

We obtain some Liouville type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary and partially verifies Wang's conjecture (J. Geom. Anal. 31 (2021)). For the specific manifold $\mathbb{B}^n$, we present a new proof of this conjecture, which has been resolved by Gu-Li (Math. Ann. 391(2025)). Our proof is based on a general principle of applying the P-function method to such Liouville type results. As a further application of this method, we obtain some classification results for nonnegative solutions of some semilinear elliptic equations with a nonlinear boundary condition.

math.AP↗

arXiv2024

An area growth estimate of the Liouville equation

We establish an area growth estimate for solutions that are bounded from above of the Liouville equation $\Delta u+K e^{2u}=0$ with a positive pinched curvature $0<\lambda\leq K\leq\Lambda$. As an application, we provide a new proof of Eremenko-Gui-Li-Xu's result in [EGLX]. We also classify solutions with an upper bound in the half plane with the boundary having constant geodesic curvature.

math.AP↗

arXiv2023

Liouville equations on complete surfaces with nonnegative Gauss curvature

We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases: on the one end, if the solution decays not too fast, then $(M,g)$ must be isometric to the standard Euclidean plane; on the other end, if $(M,g)$ is isometric to the flat cylinder $\mathbb{S}^1\times \mathbb{R}$, then solutions must decay linearly and are completely classified.

math.AP↗