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My research interests lie in Geometric Analysis, Differential Geometry, and Partial Differential Equations.
Differential Geometry:
Ricci flow and Ricci solitons.
Riemannian Geometry: curvature and topology; fundamental groups and holonomy; comparison geometry.
Complex Geometry: curvature and topology of Kähler manifolds;
Extrinsic Geometric Flows: Mean curvature flow; Gauss curvature flow;
Geometry of Curves, Surfaces, and Hypersurfaces;
Partial Differential Equations:
Heat flow method and its applications in differential geometry.
The two-point maximum principle and its applications in geometric analysis.
Fully nonlinear elliptic and parabolic PDEs: the theory of viscosity solutions; existence, uniqueness, and regularity theory; qualitative and asymptotic behavior.
Eigenvalue problems: lower bound estimates; isoperimetric inequality of eigenvalues of the Laplacian;
Robin boundary condition; Robin heat kernel, Robin eigenvalues.
Applications:
Physics, particularly quantum mechanics, general relativity, condensed matters, and string theory;
Image processing, quantum computing/representation theory, and information geometry;
Moving interfaces, phase changes, and related problems;
Reaction-diffusion systems in mathematical biology.
Publications and Preprints:
28. New sphere theorem under curvature operator of the second kind,
arXiv:2407.13847, submitted.
27. Robin heat kernel comparison on manifolds (with Kui Wang),
arXiv:2208.13402, submitted.
26. The second Robin eigenvalue in non-compact rank-1 symmetric spaces (with Kui Wang and Haotian Wu),
arXiv:2208.07546, submitted.
25. Matrix Li-Yau-Hamilton estimates under Kähler-Ricci flow (with Hao-Yue Liu and Xin-An Ren),
J. Geom. Anal, 35, no. 113, 2025
24. An upper bound for the first nonzero Steklov eigenvalue (with Kui Wang and Haotian Wu),
ESAIM Control, Optim. Cal. Var, 31, Paper No. 5, 2025
23. On a class of quasilinear operators on smooth metric measure spaces (with Yuchen Tu and Kui Wang),
Comm. Anal. Geom, 32, no. 10, 2757–2803, 2024.
22. Product manifolds and the curvature operator of the second kind,
Pacific J. Math, 332, no. 1, 167-193, 2024
21. The curvature operator of the second kind in dimension three (with Harry Fluck),
J. Geom. Anal, 34, no. 6, Paper No. 187, 19pp, 2024
20. Kählerity of Einstein four-manifolds (with Yongjia Zhang),
Math. Z., 307, no.1, Paper No. 4, 10 pp, 2024
19. Manifolds with nonnegative curvature operator of the second kind,
Commun. Contemp. Math, 26, no. 3, Paper No. 2350003, 26 pp, 2024
18. Matrix Li-Yau-Hamilton estimates under Ricci flow and parabolic frequency (with Qi S. Zhang),
Calc. Var. Partial Differential Equations., 63, no.3, Paper No. 63, 38pp, 2024
17. On the second Robin eigenvalue of the Laplacian (with Kui Wang and Haotian Wu).
Calc. Var. Partial Differential Equations., 62, no. 9, Paper No. 256, 17pp, 2023
16. Kähler surfaces with six-positive curvature operator of the second kind,
Proc. Amer. Math. Soc., 151, no.11, 4909-4922, 2023
15. Kähler manifolds and the curvature operator of the second kind,
Math. Z., 303, no. 4, Paper No. 101, 26pp, 2023
14. Eigenvalue estimates on quaternion-Kähler manifolds (with Kui Wang),
J. Geom. Anal, 33, no. 3, Paper No. 85, 20pp, 2023
13. Ancient solutions to the Ricci flow in higher dimensions (with Yongjia Zhang),
Comm. Anal. Geom., 30, no. 9, 2011-2048, 2022
12. Manifolds with 4.5-positive curvature operator of the second kind
J. Geom. Anal, 32, no. 11, Paper No. 281, 14pp, 2022
11. Sharp lower bounds for the first eigenvalue of the weighted p-Laplacian II (with Kui Wang),
Math. Res. Lett. 28, no. 5, 1459–1479, 2021
10. Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds (with Kui Wang),
Trans. Amer. Math. Soc., 374, no. 11,8081-8099, 2021
9. Sharp lower bounds for the first eigenvalue of the weighted p-Laplacian I (with Kui Wang),
J. Geom. Anal, 31, no. 8, 8686-8708, 2021
8. Modulus of continuity estimates for fully nonlinear parabolic equations,
Calc. Var. Partial Differential Equations, 60, no. 5, Paper No. 182, 23 pp, 2021
7. First Robin eigenvalue of the p-Laplacian on Riemannian manifolds (with Kui Wang),
Math. Z. 298, no.3-4, 1033-1047, 2021
6. Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature (with Lei. Ni),
Jour. Math. Pures Appl, (9) 138, 28-45, 2020
5. Parabolic frequency monotonicity on compact manifolds (with Kui Wang),
Calc. Var. Partial Differential Equations, no. 6, Art. 189, 18 pp, 2019
4. Four-dimensional gradient shrinking solitons with positive isotropic curvature (with Lei Ni and Kui Wang),
Int. Math. Res. Not. IMRN, no. 3, 949-959, 2018
3. Nonparametric hypersurfaces moving by powers of Gauss curvature (with Kui Wang),
Michigan Math. J. 66, no.4, 675-682, 2017
2. Moduli of continuity for viscosity solutions on manifolds (with Kui Wang),
J. Geom. Anal, 27, no. 1, 557-576, 2017
1. Moduli of continuity for viscosity solutions,
Proc. Amer. Math. Soc., 144, no. 4, 1717–1724, 2016.
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