Overview

Dr. Li is an Assistant Professor in the Department of Mathematics, Statistics and Physics at 黑洞社区. He earned his Ph.D. from University of California, San Diego in 2017, under the guidance of Professors Lei Ni and Ben Chow. Prior to his current position, Dr. Li was a Visiting Assistant Professor at the University of California, Irvine from 2017 to 2022, where he worked with Professor Richard Schoen, and he also held a Britton Postdoctoral Fellowship at McMaster University from 2020 to 2021.

Dr. Li specializes in Geometric Analysis, employing partial differential equations to investigate the geometric and topological properties of spaces such as Riemannian manifolds. His research interests include Ricci flow, Ricci solitons, 碍盲丑濒别谤-Ricci flow, curvature and topology, the curvature operator of the second kind, Einstein four-manifolds, Gauss curvature flow, modulus of continuity estimates, eigenvalue problems, heat kernels, Robin boundary conditions, matrix Li-Yau-Hamilton estimates, and parabolic frequencies.

Information

Areas of Research Interest

Geometric Analysis

Differential Geometry

Partial Differential Equations

Areas of Teaching Interest

Analysis

Geometry

Algebra

Publications
  • Li, Xiaolong. New sphere theorems under curvature operator of the second kind,
  • Li, Xiaolong; Liu, Hao-Yue; Ren, Xin-An. Matrix Li-Yau-Hamilton estimates under 碍盲丑濒别谤-Ricci flow,
  • Li, Xiaolong; Wang, Kui. Robin heat kernel comparison on manifolds,
  • Li, Xiaolong; Wang, Kui; Wu, Haotian. The second Robin eigenvalue in non-compact rank-1 symmetric spaces,
  • Li, Xiaolong, Wang, Kui; Wu, Haotian. An upper bound for the first nonzero Steklov eigenvalue,
  • Li, Xiaolong. Product manifolds and the curvature operator of the second kind, Pacific J. Math, to appear.
  • Li, Xiaolong; Tu, Yucheng; Wang, Kui. On a class of quasilinear operators on smooth metric measure spaces, Comm. Anal. Geom., to appear.
  • Fluck, Harry; Li, Xiaolong. The curvature operator of the second kind in dimension three, J. Geom. Anal. 34 (2024), no.6, Paper No. 187, 19pp.  
  • Li, Xiaolong; Zhang, Yongjia. 碍盲丑濒别谤ity of Einstein four-manifolds, Math. Z. 307 (2024), no. 1, Paper No. 4, 10pp. 
  • Li, Xiaolong. Manifolds with nonnegative curvature operator of the second kind, Commun. Contemp. Math. 26 (2024), no.3 Paper No. 2350003, 26pp.
  • Li, Xiaolong; Zhang, Qi S. Matrix Li-Yau-Hamilton estimates under Ricci flow and parabolic frequency, Calc. Var. Partial Differential Equations. 63 (2024), no.3, Paper No. 63, 38pp. 
  • Li, Xiaolong, Wang, Kui; Wu, Haotian. On the second Robin eigenvalue of the Laplacian, Calc. Var. Partial Differential Equations. 62 (2023), no. 9, Paper No. 256, 17 pp.
  • Li, Xiaolong. 碍盲丑濒别谤 manifolds and the curvature operator of the second kind. Math. Z. 303 (2023), no. 4, Paper No. 101, 26 pp.
  • Li, Xiaolong. 碍盲丑濒别谤 surfaces with six-positive curvature operator of the second kind, Proc. Amer. Math. Soc. 151 (2023), no. 11, 4909-4922. 
  • Li, Xiaolong; Wang, Kui. Eigenvalue estimates on quaternion-碍盲丑濒别谤 manifolds. J. Geom. Anal. 33 (2023), no. 3, Paper No. 85, 20 pp.
  • Li, Xiaolong; Zhang, Yongjia. Ancient solutions to the Ricci flow in higher dimensions, Comm. Anal. Geom. 30 (2022), no. 9, 2011-2048. 
  • Li, Xiaolong, Manifolds with 4.5-positive curvature operator of the second kind. J. Geom. Anal. 32 (2022), no. 11, Paper No. 281, 14 pp.
  • Li, Xiaolong; Wang, Kui. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian II. Math. Res. Lett. 28 (2021), no. 5, 1459鈥�1479.
  • Li, Xiaolong; Wang, Kui. Lower bounds for the first eigenvalue of the Laplacian on 碍盲丑濒别谤 manifolds. Trans. Amer. Math. Soc. 374 (2021), no. 11, 8081-8099.
  • Li, Xiaolong; Wang, Kui. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian I. J. Geom. Anal. 31 (2021), no. 8, 8686-8708.
  • Li, Xiaolong. Modulus of continuity estimates for fully nonlinear parabolic equations. Calc. Var. Partial Differential Equations 60 (2021), no. 5, Paper No. 182, 23 pp.
  • Li, Xiaolong; Wang, Kui. First Robin eigenvalue of the p-Laplacian on Riemannian manifolds. Math. Z. 298 (2021), no. 3-4, 1033鈥�1047.
  • Li, Xiaolong; Ni, Lei. 碍盲丑濒别谤-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature. J. Math. Pures Appl. (9) 138 (2020), 28鈥�45.
  • Li, Xiaolong; Wang, Kui. Parabolic frequency monotonicity on compact manifolds. Calc. Var. Partial Differential Equations. 58 (2019), no. 6, Paper No. 189, 18 pp.
  • Li, Xiaolong; Ni, Lei; Wang, Kui. Four-dimensional gradient shrinking solitons with positive isotropic curvature. Int. Math. Res. Not. IMRN 2018, no. 3, 949鈥�959.
  • Li, Xiaolong; Wang, Kui. Nonparametric hypersurfaces moving by powers of Gauss curvature. Michigan Math. J. 66 (2017), no. 4, 675鈥�682.
  • Li, Xiaolong; Wang, Kui. Moduli of continuity for viscosity solutions on manifolds. J. Geom. Anal. 27 (2017), no. 1, 557鈥�576.
  • Li, Xiaolong. Moduli of continuity for viscosity solutions. Proc. Amer. Math. Soc. 144 (2016), no. 4, 1717鈥�1724.
Grants

2024-2027: NSF-DMS, National Science Foudation, $219,854.

2023-2025: LEAPS-MPS, National Science Foudation, $250,000.

2022-2027: Travel Support for Mathematicians, Simons Foundation, $42,000.

Areas of Service
  • 2022 - present: Organzier (with T. Davis. R. Heckman, W. Ingle) of 黑洞社区 Math Circle.
  • 2022 - present: AMC 8 Competition Manager.
  • October 2024: Organzier (with L. Wang and Q. Zhang) of a special session at the AMS Fall Western Sectional Meeting at UC Riverside.
  • April 2024: Organizer (with L. Ni) of the Kansas Geometric Analysis Conference at 黑洞社区.
  • March 2023: Organizer (with L. Amorim, D. Auckly, M. Jablonski, Y.J. Lin, X.H. Nguyen, and C. Searle) of the Midwest Geometry Conference at Kansas State University.
  • March 2022: Organzier (with D. Auckly, I. Blank, Y.J. Lin, X.H. Nguyen, C. Searle, and P.R. Stinga) of the Midwest Geometry Conference at 黑洞社区.
  • 2022 - present: Organizer (with R. Fraser, D. Grady, Y.J. Lin and C. Searle) of the Geometry, Topology, and Analysis Seminar at 黑洞社区.
  • 2021: Organizer (with Y.J. Lin and C. Searle) of the Geometry and Topology Seminar at 黑洞社区.
  • 2021: Organizer (with L. Ni, K. Wang, Z Zhang) of International Conference on Recent Developments in Geometric Analysis (with 15 speakers)