Extrinsic Geometry of Convex Surfaces

Key themes

Isometric Embeddings

This is the central theme, exploring the conditions under which a given intrinsic metric on a two-dimensional manifold can be realized as the metric induced by an embedding into a higher-dimensional Euclidean space, specifically for convex surfaces. Pogorelov's work provides definitive answers for the existence and uniqueness of such embeddings.

Regularity Theory

A significant focus of the book is on the smoothness (regularity) of the solutions. Pogorelov demonstrates that even when starting with generalized or weak solutions, the resulting embedded surfaces often possess a higher degree of smoothness (e.g., C^2), which is a deep and non-trivial result in geometric analysis.

Convexity in Geometry

The book rigorously explores the properties and implications of convexity for surfaces. Convexity imposes strong constraints on the geometry, leading to many of the powerful existence and uniqueness theorems. The intrinsic properties of convex surfaces, particularly their positive Gaussian curvature, are central to the analysis.

Worth discussing

Discuss the historical context of the Weyl problem and Pogorelov's unique approach to its solution compared to previous attempts.