Research

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: 

1. Moduli of continuity for viscosity solutions,

Proc. Amer. Math. Soc., 144, no. 4, 1717–1724, 2016. arXiv:1511.02184.

2. Moduli of continuity for viscosity solutions on manifolds (with Kui Wang),

J. Geom. Anal, 27, no. 1, 557-576, 2017  arXiv:1511.02185.

3. Nonparametric hypersurfaces moving by powers of Gauss curvature (with Kui Wang),

Michigan Math. J. 66, no.4, 675-682, 2017 arXiv:1606.01287.

4. Four-dimensional gradient shrinking solitons with positive isotropic curvature (with Lei Ni and Kui Wang),

IMRN,  no. 3, 949-959, 2018 arXiv:1603.05264.

5. Parabolic frequency monotonicity on compact manifolds (with Kui Wang),

 Calc. Var. Partial Differential Equations,  no. 6, Art. 189, 18 pp, 2019  arXiv:1804.09806.

6. Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature (with Lei. Ni), 

 Jour. Math. Pures  Appl, (9) 138, 28-45, 2020, arXiv:1903.02615.

7. First Robin eigenvalue of the p-Laplacian on Riemannian manifolds (with Kui Wang), 

Math. Z. 298, no.3-4, 1033-1047, 2021,  arXiv:2002.06472.

8. Modulus of continuity estimates for fully nonlinear parabolic equations,

Calc. Var. Partial Differential Equations, 60, no. 5, Paper No. 182, 23 pp, 2021, arXiv:2006.16631,

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, arXiv:1910.02295.

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, arXiv:2010.12792.

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,  arXiv:1911.04596.

12. Ancient solutions to the Ricci flow in higher dimensions (with Yongjia Zhang),

Comm. Anal. Geom., 30, no. 9, 2011-2048, 2022,  arXiv:1812.04156.

13. Manifolds with 4.5-positive curvature operator of the second kind,

J. Geom. Anal, no. 11, 281, 2022, arXiv:2206.15011.

14. Eigenvalue estimates on quaternion-Kähler manifolds (with Kui Wang),

J. Geom. Anal, 33, no. 3, 85, 2023, arXiv:2105.06303.

15. Kähler manifolds and the curvature operator of the second kind,

Math. Z., 303, no. 4, 101, 2023, arXiv:2208.14505.

16. Manifolds with nonnegative curvature operator of the second kind,

Commun. Contemp. Math., to appear, arXiv:2112.08465.

17. Kähler surfaces with six-positive curvature operator of the second kind,

Proc. Amer. Math. Soc., 151, no.11, 4909-4922, 2023, arXiv:2207.00520.

18. On the second Robin eigenvalue of the Laplacian (with Kui Wang and Haotian Wu).

Calc. Var. Partial Differential Equations., 62, 256, 2023, arXiv:2003.03087.

19. On a class of quasilinear operators on smooth metric measure spaces (with Yuchen Tu and Kui Wang),

Comm. Anal. Geom., accepted, arXiv:2009.10418.

20. Matrix Li-Yau-Hamilton estimates under Ricci flow and parabolic frequency (with Qi S. Zhang),

Calc. Var. Partial Differential Equations., 63, 63, 2024, arXiv:2306.10143.

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, arXiv:2303.17663.

22. Kählerity of Einstein four-manifolds (with Yongjia Zhang),

Math. Z., 307, no.1, Paper No. 4, 10 pp, 2024, arXiv:2206.04870.

23. Product manifolds and the curvature operator of the second kind,

Pacific J. Math, 332, no. 1, 167-193, 2024, arXiv:2209.02119.

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, arXiv:2003.03093.

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, arXiv2307.10920.

Preprints: 

26. The second Robin eigenvalue in non-compact rank-1 symmetric spaces (with Kui Wang and Haotian Wu),

arXiv:2208.07546, submitted.

27. Robin heat kernel comparison on manifolds (with Kui Wang),

arXiv:2208.13402, submitted.

28. New sphere theorem under curvature operator of the second kind,

arXiv:2407.13847, submitted.

Collaborators: 

Harry Fluck (Graduate student at Cornell University)

Hao-Yue Liu (Graduate student at China University of Mining and Technology)

Lei Ni (Professor at UC San Diego)

Xin-An Ren (Professor at China University of Mining and Technology)

Yucheng Tu (Amazon, Obtained Ph.D. at UC San Diego)

Kui Wang (Associate Professor at Soochow University)

Haotian Wu (Senior Lecturer at University of Sydney)

Qi S. Zhang (Professor at UC Riverside)

Yongjia Zhang (Tenure-track Associate Professor at Shanghai Jiao Tong University)