I work in the field of Geometric Analysis, using analytic tools (mostly PDE methods) to study geometric problems. My research focuses on regularity, compactness, existence of critical variational problem such as 2D harmonic maps and 4D Yang-Mills.

Ricci Flow

Studying the Ricci flow on surfaces with/without conical singularities. Focus on existence, uniqueness, and asymptotic behavior of solutions.

• Ricci flow with conical singularities

• Ricci flow with rough initial data

Harmonic Maps

Investigating the existence, regularity, and blow-up behavior of harmonic maps and biharmonic maps. Developing neck analysis techniques for degenerating sequences.

• Biharmonic map regularity

• Neck analysis

• Tangent cone uniqueness

• Blow-up phenomena

Yang-Mills Theory

Exploring the Yang-Mills equations and their geometric applications. Studying blow-up behavior in dimension four and the α-flow approximation.

• Yang-Mills flow

• Blow-up in dimension four

• α-flow approximation

• Minimizing methods

Willmore Surfaces

Studying the compactness of Willmore surfaces and their geometric properties. Developing 3-circle theorems and analyzing degeneration of complex structures.

• 3-circle theorem

• Complex structure degeneration

• Variational problems

Research Interests

My research lies at the intersection of partial differential equations and differential geometry. I am particularly interested in understanding the behavior of geometric flows (such as Ricci flow) on singular spaces, and the regularity/compactness theory for harmonic maps and Yang-Mills fields. Through careful analysis of nonlinear PDEs, I aim to uncover fundamental properties of geometric variational problems.

Publications and Preprints

Preprints

• Y. Li and H. Yin, On the Ends of Willmore Surfaces with Curvature Decay, arXiv preprint, 2026.

• A. Waldron and H. Yin, The Yang-Mills equation near instanton-anti-instanton configurations, arXiv preprint, 2026.

• H. Qian and H. Yin, On the blow-up of harmonic maps from surfaces to homogeneous manifolds, arXiv preprint, 2026.

• Y. Li, H. Yin, and J. Zhou, 3-circle Theorem for Willmore surface II -- degeneration of the complex structure, arXiv preprint, 2024.

• H. Yin, On the blow-up of Yang-Mills fields in dimension four, arXiv preprint, 2023.

Published Papers

• B. Wang and H. Yin, Curvature at the infinity of asymptotically flat Einstein manifold, To appear in Trans. Amer. Math. Soc., 2025.

• Y. Li and H. Yin, 3-circle Theorem for Willmore surface I, J. London Math. Soc., 2025.

• P.M. Topping and H. Yin, Smoothing a measure on a Riemann surface using Ricci flow, Ars Inveniendi Analytica, 2024.

• P.M. Topping and H. Yin, Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces, Proc. London Math. Soc. 128, e12600, 2024.

• H. Yin, Direct minimizing method for Yang-Mills energy over SO(3) bundle, Math. Ann., 2023.

• H. Yin, Analysis aspects of Ricci flow on conical surfaces, Acta Math. Sinica (Engl. Ser.) 38, 807-857, 2022.

• H. Yin, Higher order neck analysis of harmonic maps and its applications, Ann. Global Anal. Geom. 62, 401-438, 2022.

• Y. Gui and H. Yin, Schauder estimates on smooth and singular spaces, Ann. Global Anal. Geom. 59, 457-481, 2021.

• H. Yin, Generalized neck analysis of harmonic maps from surfaces, Calc. Var. Partial Differential Equations 60, Paper No. 117, 2021.

• P.M. Topping and H. Yin, Rate of curvature decay for the contracting cusp Ricci flow, Comm. Anal. Geom. 28, 1221-1250, 2020.

• Y. Chen and H. Yin, Uniqueness of tangent cone for biharmonic map with isolated singularity, Pacific J. Math. 299, 401-426, 2019.

• K. Zheng and H. Yin, Expansion formula for complex Monge-Ampère equation along cone singularities, Calc. Var. PDE 59, art.no.50, 2019.

• P.M. Topping and H. Yin, Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces, Ann. PDE 3, Art. 6, 2017.

• W. Ai and H. Yin, Neck analysis of extrinsic polyharmonic maps, Ann. Global Anal. Geom. 52, 129-156, 2017.

• L. Liu and H. Yin, Neck analysis for biharmonic maps, Math. Z. 283, 807-834, 2016.

• L. Liu and H. Yin, On the finite time blow-up of biharmonic map flow in dimension four, J. Elliptic Parabol. Equ. 1, 363-385, 2015.

• M.C. Hong, G. Tian, and H. Yin, The Yang-Mills α-flow in vector bundles over four manifolds and its applications, Comment. Math. Helv. 90, 75-120, 2015.

• M.C. Hong and H. Yin, On the Sacks-Uhlenbeck flow of Riemannian surfaces, Comm. Anal. Geom. 21, 917-955, 2013.

• M.C. Hong and H. Yin, Partial regularity of the relaxed energy for biharmonic maps, J. Funct. Anal. 262, 681-718, 2012.

• M. Giaquinta, M.C. Hong, and H. Yin, A new approximation of relaxed energies for harmonic maps and Faddeev model, Calc. Var. PDE 41, 45-69, 2011.

• H. Li and H. Yin, On stability of hyperbolic space form under the normalized Ricci flow, Int. Math. Res. Not. 2010, 2903-2924, 2010.

• H. Yin, Ricci flow on surfaces with conical singularities, J. Geom. Anal. 20, 970-995, 2010.

• H. Yin, Normalized Ricci flow on nonparabolic surfaces, Ann. Global Anal. Geom. 36, 81-104, 2009.

• H. Yin, Boundary regularity of harmonic maps from hyperbolic space into nonpositively curved manifolds, Pacific J. Math. 232, 491-509, 2007.

• W. Ding and H. Yin, Special periodic solutions of Schrödinger flow, Math. Z. 253, 555-570, 2006.

• H. Yin, Periodic solutions of Schrödinger flow from S³ to S², Chinese Ann. Math. Ser. B 27, 401-410, 2005.

(For a complete list, please visit my personal homepage or relevant academic databases.)