Analysis on manifolds has become one of the most dynamic and influential areas of modern mathematics, driving breakthroughs in geometry, partial differential equations, and mathematical physics, while increasingly shaping fields such as statistics, data science, and artificial intelligence. This book offers a clear and engaging introduction to the powerful analytic and geometric techniques that have defined the subject—from Yau’s gradient estimates and the Li–Yau differential Harnack inequality to the Sacks–Uhlenbeck blow‑up method and contemporary geometric flows.

Spanning seven cohesive chapters, the book blends foundational theory with modern developments, guiding readers through heat kernel analysis, harmonic map theory, minimal surfaces, and geometric flows. The final chapters showcase new results arising from the author’s recent collaborations, highlighting cutting‑edge progress on β‑symplectic critical surfaces and mean curvature flows.

Accessible yet rigorous, this book is ideal for researchers and advanced students seeking both a solid grounding in geometric analysis and a window into current research at the forefront of the field.

Jiayu Li, a Distinguished Chair Professor at the School of Mathematical Sciences, USTC, is dedicated to research in geometric analysis. In recognition of his contributions to the field, he has received the State Natural Science Award (Second Class) and the Shiing-Shen Chern Prize from the Chinese Mathematical Society. He has been a Humboldt Fellow at the Max Planck Institute (MPI) in Germany, a Research Scientist at the Abdus Salam ICTP, and the Head of the MPI Partner Group for Mathematics at the Academy of Mathematics and Systems Science.