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Course Program

Zhang Huichun

Alexandrov geometry, RCD spaces, and metric-valued analysis

Zhang Huichun

Sun Yat-sen University · School of Mathematics

Zhang Huichun is a professor at Sun Yat-sen University. He received his B.S. from Wuhan University in 2002 and his Ph.D. from Sun Yat-sen University in 2011, where he has worked since then and became a professor in 2018. His research focuses on geometric analysis on nonsmooth spaces, Alexandrov geometry, geometry-related PDEs, and geometric rigidity.

Lecture 1

Introduction to Alexandrov geometry

Lecture 2

A brief introduction to RCD spaces

Lecture 3

Bochner inequality and linear geometric analysis on Alexandrov/RCD spaces

Lecture 4

Sobolev spaces with metric targets

Lecture 5

Regularity of harmonic maps and heat flows into CAT(0) spaces

Detailed contents

Lecture outlines

Lecture 1

Introduction to Alexandrov geometry

Basic concepts and examples of Alexandrov space, the local structure near regular points, gradient exponential maps, the second variation of arc length, and DC-structure.

References.

  1. D. Burago, Y. Burago, S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33, American Mathematical Society.
  2. Y. Burago, M. Gromov, G. Perelman. A. D. Aleksandrov spaces with curvature bounded below. Uspekhi Mat. Nauk, 47 (1992), 3–51.
  3. A. Petrunin. Parallel transportation for Alexandrov space with curvature bounded below. Geometric and Functional Analysis, 8 (1998), 123–148.
  4. A. Petrunin. Semiconcave functions in Alexandrov's geometry. In: Surveys in Differential Geometry, vol. XI.
  5. G. Perelman. DC structure on Alexandrov space. Preprint, 1994.
  6. L. Ambrosio, J. Bertrand. DC calculus. Mathematische Zeitschrift, 288 (2018), 1037–1080.

Lecture 2

A brief introduction to RCD spaces

The concepts of RCD(K,N) spaces, heat flows, and the W2-contractions.

References.

  1. N. Gigli. De Giorgi and Gromov working together. Preprint, arXiv:2306.14604.
  2. L. Ambrosio. Calculus, heat flow and curvature-dimension bounds in metric measure spaces. In: Proceedings of the International Congress of Mathematicians (ICM) 2018.

Lecture 3

Bochner inequality and linear geometric analysis on Alexandrov/RCD spaces

Bochner inequality, good cut-off functions and test functions, Yau's gradient estimate and Li-Yau's estimate, heat kernel bounds.

References.

  1. H.-C. Zhang, X.-P. Zhu. Yau's gradient estimates on Alexandrov spaces. Journal of Differential Geometry, 91 (2012), 445–522.
  2. H.-C. Zhang, X.-P. Zhu. Local Li-Yau's estimates on RCD*(K,N) metric measure spaces. Calculus of Variations and Partial Differential Equations, 55 (2016), Art. 93.
  3. R. Jiang, H. Li, H.-C. Zhang. Heat kernel bounds on metric measure spaces and some applications. Potential Analysis, 44 (2016), 601–627.
  4. Y. Peng, H.-C. Zhang, X.-P. Zhu. Weyl's lemma on RCD(K,N) metric measure spaces. Calculus of Variations and Partial Differential Equations, 64 (2025), 1–23.

Lecture 4

Sobolev spaces with metric targets

Korevaar–Schoen's Sobolev spaces with metric targets, harmonic maps, and CAT(0) spaces.

References.

  1. N. Korevaar, R. Schoen. Sobolev spaces and harmonic maps for metric space targets. Communications in Analysis and Geometry, 1, 3–4 (1993), 561–659.
  2. J. Jost. Equilibrium maps between metric spaces. Calculus of Variations and Partial Differential Equations, 2, 2 (1994), 173–204.
  3. N. Gigli, A. Tyulenev. Korevaar–Schoen's energy on strongly rectifiable spaces. Calculus of Variations and Partial Differential Equations, 60, 6 (2021), Paper No. 235.

Lecture 5

Regularity of harmonic maps and heat flows into CAT(0) spaces

The idea of the proof of Lipschitz interior regularity of harmonic maps from an Alexandrov domain into a CAT(0) space, and then the same regularity results for harmonic maps and their heat flows from RCD spaces into CAT(0) spaces. If time is sufficient, the boundary regularity will also be included.

References.

  1. H.-C. Zhang, X.-P. Zhu. Lipschitz continuity of harmonic maps between Alexandrov spaces. Inventiones Mathematicae, 211, 3 (2018), 863–934.
  2. H.-C. Zhang, X.-P. Zhu. Boundary regularity of harmonic maps from RCD(K,N)-space to CAT(0)-space. Science China Mathematics, 54 (2024), 2041–2058.
  3. H.-C. Zhang, X.-P. Zhu. Optimal boundary regularity of harmonic maps from RCD(K,N)-space to CAT(0)-space, 2025. To appear in Communications in Analysis and Geometry.
  4. H.-C. Zhang, X.-P. Zhu. Lipschitz regularity of harmonic map heat flows into CAT(0) spaces. arXiv:2601.20579, 2026.
  5. B. X. Han, H.-C. Zhang, X.-P. Zhu. Lipschitz regularity of harmonic map heat flows from RCD spaces into CAT(0) spaces. arXiv:2608.03442, 2026.
  6. Y. Liu, H.-C. Zhang, X.-P. Zhu. Boundary regularity of harmonic map heat flows from RCD spaces into CAT(0) spaces. arXiv:2609.13652.