arXiv subjects

Qiaoling Xia

Publications and source records attributed to Qiaoling Xia.

3 recordsLinked to original sources

arXiv2026

Sharp $L^p$-uncertainty principles on Finsler measure spaces

In this paper, we prove the $L^p(p>1)$-uncertainty principles for any $1<p<n$, including the classical Heisenberg-Pauli-Weyl inequality, Caffarelli-Kohn-Nirenberg interpolation inequality and Hardy inequality in $\mathbb R^n$ as special cases, on $n(\geq 2)$-dimensional forward complete and noncompact Finsler measure spaces $(M, F, \mathfrak{m})$ with curvatures bounded from above or below by constants. Further, we characterize the sharpness of $L^p$-uncertainty principles in terms of the reversibility of $F$ and the bounds of flag (or Ricci) curvature and S-curvature induced by the measure $\mathfrak m$ and obtained some rigidity results, which generalize the related ones in [HKZ] in Finslerian case and [KKPZ] in Riemannian case.

math.DG↗

arXiv2024

Almost Ricci solitons on Finsler spaces

In this paper, (gradient) almost Ricci solitons on Finsler measure spaces $(M, F, m)$ are introduced and investigated. We prove that $(M, F, m)$ is a gradient almost Ricci soliton if and only if the infinity-Ricci curvature Ric$_\infty$ is a scalar function on $M$ when $M$ is compact. Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics $F=α+β$, which implies that every Randers (gradient) almost Ricci soliton is of isotropic S$_{BH}$-curvature. Based on this and the navigation technique, we further classify Randers almost Ricci solitons (resp. gradient almost Ricci solitons) up to classifications of Randers Einstein metrics $F$ (resp. Riemannian gradient almost Ricci solitons) and the homothetic vector fields of $F$ (resp. solutions of the equation which the weight function $f$ of $m$ satisfies) when $F$ has isotropic S$_{BH}$-curvature. As applications, we obtain some rigidity results for compact Randers (gradient) Ricci solitons and construct several Randers gradient Ricci solitons, which are the first nontrivial examples of gradient Ricci solitons in Finsler geometry.

math.DG↗

arXiv2023

Linearized heat semigroups on Finsler measure spaces and some applications

It is known that the Finsler heat flow is a nonlinear flow. This leads to the study of linearized heat semigroups for the Finsler heat flow. In this paper, we give the properties of linearized heat semigroups and prove that the semigroup is conservative on complete Finsler measure spaces $(M, F, m)$ with weighted Ricci curvature Ric$_N$ bounded from below. As applications, we give new proofs of Li-Yau's inequalities established in \cite{Xia2} and \cite{OS2} respectively in the compact case and extend them to the complete Finsler measure spaces with Ric$_N\geq K$ for $K\in \mathbb R$. Finally we give several equivalent characterizations of Ric$_\infty\geq K (K\in \mathbb R)$ via the linearized heat semigroup approach and their applications.

math.DG↗