arXiv subjects

Wen-Qi Li

Publications and source records attributed to Wen-Qi Li.

3 recordsLinked to original sources

arXiv2026

Positive Bakry-\'Emery Ricci Curvature on Homotopy Spheres

Wei and Wylie asked whether a complete weighted manifold with nonnegative Bakry--\'Emery Ricci curvature and bounded potential must admit a Riemannian metric with nonnegative Ricci curvature. We answer this question negatively in every dimension $8k+1$ and $8k+2$, where $k\geq1$. Our main geometric result is that every smooth homotopy sphere of dimension at least seven admits a weighted core metric $(g,e^{-f})$ with $\Ric_g+\Hess_g f>0$. On the other hand, in dimensions $8k+1$ and $8k+2$, where $k\geq1$, we prove that a homotopy sphere with nonzero $\alpha$-invariant admits no Riemannian metric with $\Ric\geq0$. Since homotopy spheres with nonzero $\alpha$-invariant exist in every dimension $8k+1$ and $8k+2$, and since compactness makes the potential bounded, these manifolds provide the required counterexamples.

math.DG↗

arXiv2025

Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application

In this paper, we establish a parabolic Harnack inequality for positive solutions of the $ϕ$-heat equation and prove Gaussian upper and lower bounds for the $ϕ$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\varepsilon$-range. Building on these results, we demonstrate: The $L^1_ϕ$-Liouville theorem for $ϕ$-subharmonic functions, $L^1_ϕ$-uniqueness property for solutions of the $ϕ$-heat equation and lower bounds for eigenvalues of the weighted Laplacian $Δ_ϕ$. Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted $L^p(μ)$-norm constraint on $|\nablaϕ|^2$.

math.DG↗

arXiv2023

New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost Solitons

In this paper, we establish a new volume comparison theorem for a complete manifold with a function $\rho(x)$ as the lower bound of the Bakry-Emery Ricci curvature. As applications, we obtain a new volume rigidity result of the gradient Ricci almost solitons. Furthermore, we extend the results of Cao and Zhou \cite{CZ} to shrinking gradient Ricci almost solitons and get the rigidity result with respect to the maximal volume growth.

math.DG↗