My research is focused on optimal transport and its applications on functional inequalities, metric geometry, metric measure spaces etc., I am particularly interested in mathematics objects related to both metrics and measures.
So far, I have worked mainly on the following topics:
1. Optimal transport, analysis and geometry of Wasserstein space
2. Metric measure space with synthetic curvature bound (A la Lott-Sturm-Villani, Bakry-Emery, Alexandrov, Busemann etc.)
3. Sobolev space on metric measure space
4. Geometric and functional inequalities
[0] (Ph.D thesis) Topics on calculus on metric measure spaces，defense on 23 June 2015 at Paris, thesis-on-line: archives-ouvertes
[1] (with N. Gigli) The continuity equation on metric measure spaces, Cal. Var. PDE. (2015)
[2] (with N. Gigli) Independence on p of weak upper gradients on RCD spaces, J. Funct. Anal. (2016)
[3] (with A. Mondino) Angles between curves in metric measure spaces, Anal. Geom. Metr. Spaces. (2017)
[4] Ricci tensor on RCD(K,N) space, J. Geom. Anal. (2018)
[5] (with N. Gigli) Sobolev Spaces on Warped Products, J. Funct. Anal. (2018)
[6] Conformal transformation on metric measure spaces, Potential. Anal. (2019)
[7] New characterizations of Ricci curvature on RCD metric measure spaces, Disc. Cont. Dyn. Sist. A (2018)
[8] Characterizations of monotonicity of vector fields on metric measure space, Cal. Var. PDE. (2018)
[9] Sharp p-Poincaré inequality under Measure Contraction Property, Manuscripta Math. (2020)
[10] Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds, Adv. Math. (2020)
Remark: Part of this article, detailed computation for the measure-valued Ricci tension can be found in "Ricci tensor on smooth metric measure space with boundary" at: arxiv.
[11] (with E. Milman) Sharp Poincaré inequality under Measure Contraction Property, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2021)
[12] Rigidity of some functional inequalities on RCD spaces, J. Math. Pures Appl. (2021)
[13] (with K-T. Sturm) Curvature-dimension conditions for diffusions under time change, Ann. Math. Pura Appl . (2022)
[14] (with Z-F. Xu) Sharp uncertainty principles on metric measure spaces, Cal. Var. PDE. (2024)
[15] On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach, J. Funct. Anal. (2024).
[16] (with A. Pinamonti, Z-F.Xu, K.Zambanini) Maz’ya–Shaposhnikova Meet Bishop–Gromov, Potential. Anal. (2025).
Remark: This is a replacement & improvement of the paper I wrote with Andrea Pinamonti: On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity, preprint at: arxiv.
[17] (with D.-Y. Liu, Z.-N. Zhu) Barycenter curvature-dimension condition for extended metric measure spaces, Indag. Math (2025)
This is a survey about a new curvature-dimension condition for extended metric measure spaces, called Barycenter-Curvature Dimension condition, introduced in our preprint "On the geometry of Wasserstein barycenter I" see arxiv.
[18] ABP estimate on metric measure spaces via optimal transport, J. Diff. Equ. (2026).
We estabish an Alexandroff-Bakelman-Pucci (ABP) type estimate on metric measure spaces, by dealing with an inverse problem of optimal transport, and Otto's calculus.
12 Oct 2021 (with A. Pinamonti) On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited, preprint at: arxiv.
See paper [15] above for a more simple & direct version, for spaces with nice tangent spaces.
20 Aug 2024 [Update] Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature, preprint at: arxiv.
A new version has been updated on arxiv (08.20.2024), we find a totally new proof to the rigidity theorem, and the assumption on the existence of a needle decomposition has been removed. Though, the previous version, which includes several quantitative estimate for log-concave measures, has its own interest.
A short version without the proof of rigidity is here: A sharp isoperimetric inequality in metric measure spaces with non-negative Ricci curvature, preprint at: arxiv.
2 Dec 2024 [New Paper] On the geometry of Wasserstein barycenter I, preprint at: arxiv, see also a survey about BCD condition at: arxiv.
We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein barycenter, and prove Jensen's inequality in an abstract framework.
This generalized several results on Euclidean space, Riemannian manifolds and Alexandrov spaces, to metric measure spaces satisfying Riemannian Curvature-Dimension condition à la Lott--Sturm--Villani, and some extended metric measure spaces including abstract Wiener spaces and configuration spaces over Riemannian manifolds.
We also introduce a new curvature-dimesion condition, we call Barycenter-Curvature-Dimension condition BCD. We prove its stability under measured-Gromov-Hausdorff convergence and prove the existence of the Wasserstein barycenter under this new condition. In addition, we get some geometric inequalities including a multi-marginal Brunn-Minkowski inequality and a functional Blaschke-Santaló type inequality.
17 Aug 2025 [New Paper] On the structure of Busemann spaces with non-negative curvature, preprint at:cvgmt.
We study the sturcture of Busemann spaces with non-negative curvature, or Busemann concave metric spaces. This family of space are generalization of strictly convex Banach spaces. Our work enriches the theory of synthetic sectional curvature lower bound for metric spaces, and provides some useful tools and examples to study Finslerian metric measure spaces.