AI 中文总结
研究具有非负Ricci曲率流形上的拟线性Liouville方程，通过解的最优对数下界得出流形与欧氏空间等距且解为标准泡解，证明中关键是对数下界与总体积上界联系，还提出并证明了关于解与几何相互作用的猜想。
AI 中文摘要
我们研究了在具有非负Ricci曲率的完备非紧黎曼流形上的拟线性Liouville方程\[ -\Delta_n u=e^u \]。我们的第一个结果表明，如果一个解\(u\)满足最优对数下界\[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{当 }r(x)\to+\infty \]，那么基础流形与欧几里得空间等距且\(u\)是标准泡解。假设中的首项系数和余项都是尖锐的。证明中的关键要素是对数下界与解的总体积的尖锐上界之间的联系。我们还提出了一个关于解的次对数衰减与基础几何之间相互作用的猜想，并在\(n = 2\)以及\(n\ge 3\)在一个强化假设下证明了它。
英文摘要
We study the quasilinear Liouville equation \[ -Δ_n u=e^u \] 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.
Comments18 pages. Comments are welcome
