Lectures on Differential Geometry by Richard Schoen and Shing-Tung Yau is a seminal work in the field of geometric analysis, originating from a series of lectures delivered at the Institute for Advanced Study
in Princeton between 1984 and 1985. While first published in Chinese in 1989, the authoritative English translation (1994) remains a cornerstone reference for postgraduate and professional mathematicians. Key Features and Content
The text is renowned for bridging the gap between classical differential geometry and modern nonlinear partial differential equations (PDEs). sites.lsa.umich.edu Submanifold Theory:
Detailed exploration of submanifolds in Euclidean space and their differential calculus. Geometric Flows:
Advanced chapters cover the uniformization of surfaces via heat flow and geometric flows of curves in the plane. Minimal Surfaces: schoen yau lectures on differential geometry pdf new
Extensive discussion on elliptic equations as they pertain to the geometry of minimal surfaces. Open Problem Lists:
A unique and highly valued feature of the book is its massive collection of over 200 open research problems (compiled in 1982 and 1991), which have guided research for decades. Editions and Availability Lectures on Differential Geometry
The classic text Lectures on Differential Geometry by Richard Schoen and Shing-Tung Yau is widely considered a cornerstone of modern geometric analysis. Originally based on lectures given at the Institute for Advanced Study in 1984–1985, it has been a definitive reference for researchers for decades. Core Content & Structure
The book is structured into three distinct pedagogical levels, making it more than just a typical textbook: Lectures on Differential Geometry by Richard Schoen and
Part I: Submanifolds of Euclidean Space: An intuitive introduction to geometry through classical theory, focusing on submanifolds and differential calculus.
Part II: Riemannian Geometry: A comprehensive "first course" covering smooth manifolds, connections, curvature, and foundational formulas like Chern-Gauss-Bonnet.
Part III: Geometric Analysis (Advanced Topics): This is where the authors' expertise shines, delving into elliptic and parabolic equations, minimal surfaces, and geometric flows like Ricci flow. Key Highlights for Advanced Readers
The Problem Lists: One of the most famous features of the book is its extensive lists of open problems (nearly 220 in total). These provide a roadmap for the research programme of using curvature to understand topology. Piece: On the Hunt for the “New” Schoen–Yau
PDE-Driven Approach: Unlike some purely formal geometry texts, this work emphasizes the interplay between differential equations and geometry, reflecting Yau’s influential "analyst's geometer" style.
Historical Impact: The text was instrumental in training a generation of mathematicians and is considered an essential tool for anyone studying major 20th-century achievements in the field. Critical Reception
In the world of geometric analysis, few names carry as much weight as Richard Schoen and Shing-Tung Yau. Their collaborative work on minimal surfaces, positive mass theorem, and scalar curvature rigidity has shaped modern differential geometry. Over the years, lecture notes from courses they taught — often titled something like “Lectures on Differential Geometry” — have circulated in various forms, some typed, some scanned, some updated.