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-Poinca 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)
Remark: 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).
Remark: 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.
[19] Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature, Sci. China Math (2026+)
Remark: A new version has been updated on arxiv, 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.
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.
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:arxiv.
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.
22 Feb 2026 [New Paper] Stability of optimal transport on metric measure spaces, preprint at:arxiv.
We prove a quantitative stability of Kantorovich potentials on non-smooth metric measure spaces with synthetic lower Ricci curvature bound, thereby confirming a recent conjecture of Kitagawa, Letrouit and Mérigot. Our proof, which employs the heat kernel-regularized c-transform, does not rely on linear structure or sectional curvature bounds, is new even in the smooth setting. As a corollary, we get a quantitative stability of optimal transport maps on Alexandrov spaces with lower curvature bound.
27 May 2026 [New Paper] Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit, preprint at:arxiv.
The sliced Wasserstein distance SW_2 compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard computational tools in statistics, imaging, and machine learning, the rigidity behind the elementary comparison
\[
SW_2^2(\mu,\nu)\leq \frac1d W_2^2(\mu,\nu)
\]
has not been systematically studied. In this paper, we prove the rigidity for any absolutely continuous source measures.
For quantitative stability, we introduce the sliced Poincaré-Korn (SPK) constant $\kappa_{\mathrm{SPK}}(\mu)$, defined as an new spectral gap of an averaged ridge-projection quadratic form on gradient fields modulo the family $\{\lambda x+b\}$. Whenever this constant is positive, we prove a stability estimate for the sliced Wasserstein deficit, up to a one-dimensional Lipschitz scale for the projected monotone transports. We obtain the sharp SPK constant for the Gaussian measures as the most important example, and establish positive SPK bounds for bounded perturbations of the Gaussian and compact classes of gradient fields for fixed source measures.
Finally, we show that anisotropic Gaussians give a sharp obstruction: neither a Bakry-Emery lower curvature bound nor a usual Poincar\'e inequality alone can imply a global sliced Poincaré-Korn inequality.
27 May 2026 [New Paper] Quantitative Stability of Wasserstein Barycenters over Alexandrov Spaces with Lower Curvature Bounds, preprint at:arxiv.
We prove quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. The proof combines the variational strategy of Carlier-Delalande-Mérigot with heat-kernel regularization, which supplies the regularity needed for dual convexity arguments in this non-smooth curved setting. The main result is an explicit strong-convexity modulus for the barycentric variance functional. As a consequence, barycenters depend Holder-continuously on the underlying distributions with respect to the 1-Wasserstein distance on the space of probability measures. We derive empirical-barycenter consistency and entropy-based sample-complexity bounds. Our proof does not rely on linear structure; in particular, the resulting estimates appear to be new even on smooth compact Riemannian manifolds.