arXiv subjects

Mingxiang Li

Publications and source records attributed to Mingxiang Li.

23 records · Page 2Linked to original sources

arXiv2024

The total Q-curvature, volume entropy and polynomial growth polyharmonic functions (II)

This is a continuation of our previous work (Advances in Mathematics 450 (2024), Paper No. 109768). In this paper, we characterize complete metrics with finite total Q-curvature as normal metrics for all dimensional cases. Secondly, we introduce another volume entropy to provide geometric information regarding complete non-normal metrics with finite total Q-curvature. In particular, we show that if the scalar curvature is bounded from below, the volume growth of such complete metrics is controlled.

math.DG↗

arXiv2024

Higher order Bol's inequality and its applications

In the conformal class of Euclidean space, we give some volume comparison theorems with help of Q-curvature. Meanwhile, for compact four dimensional manifolds with non-negative scalar curvature, we give a volume rigidity theorem with respect to Q-curvature. Finally, we make use of these results to give some sufficient and necessary conditions for the existence of solutions to some conformally invariant equations which answers an open problem raised by Hyder-Martinazzi (2021, JDE).

math.DG↗

arXiv2023

Asymptotic behavior of conformal metrics with null Q-curvature

We describe the asymptotic behavior of conformal metrics related to the GJMS operator in the null case, as the prescribed Q-curvature $f_0(x) + λ$ gradually changes. We show that if one of the maximum points of $f_0$ is flat up to order $n-1$, the normalized conformal metrics in the lowest energy level will form exactly one spherical bubble as $λ$ approaches zero using higher order Bol's inequality. This generalizes the result of Struwe (JEMS, 2020) in the two-dimensional case to higher dimensions and helps rule out the slow bubble case discussed by Ngô and Zhang (arXiv:1903.12054) to some degree.

math.DG↗

arXiv2023

The total Q-curvature, volume entropy and polynomial growth polyharmonic functions

In this paper, we investigate a conformally flat and complete manifold $(M,g)=(\mathbb{R}^n,e^{2u}|dx|^2)$ with finite total Q-curvature. We introduce a new volume entropy, incorporating the background Euclidean metric, and demonstrate that the metric $g$ is normal if and only if the volume entropy is finite. Furthermore, we establish an identity for the volume entropy utilizing the integrated Q-curvature. Additionally, under normal metric assumption, we get a result concering the behavior of the geometric distance at infinity compared with Euclidean distance. With help of this result, we prove that each polynomial growth polyharmonic function on such manifolds is of finite dimension. Meanwhile, we prove several rigidity results by imposing restrictions on the sign of the Q-curvature. Specifically, we establish that on such manifolds, the Cohn-Vossen inequality achieves equality if and only if each polynomial growth polyharmonic function is a constant.

math.DG↗

arXiv2023

A Liouville-type theorem in conformally invariant equations

Given a smooth function $K(x)$ satisfying a polynomially cone condition and $x\cdot\nabla K\leq 0$, we prove that there is no solution $u\in C^\infty(\mathbb{R}^2)$ of the equation $$-Δu=K(x)e^{2u}\quad \mathrm{on}\;\mathbb{R}^2$$ with $u\leq C$ and $\int_{\mathbb{R}^2}|K(x)|e^{2u}d x<+\infty$. As a consequence, there is no such solution if $K(x)$ is a non-constant polynomial with $x\cdot\nabla K\leq 0$. The latter result already includes a result of Struwe(JEMS 2020) as a particular case. Higher order cases are set up with additional assumption on the behavior of $Δu$ near infinity.

math.AP↗