1. Introduction & Purposes

Gluing and desingularization are important and powerful techniques in geometric analysis. These methods have been widely used to resolve difficult problems and to reveal new geometric phenomena. The general strategy involves identifying suitable local models, gluing them to initial geometric objects to construct approximate solutions with controlled properties, and then applying PDE techniques to solve highly nonlinear equations and obtain genuine solutions.

These techniques have played a central role in many developments in differential geometry. In particular, they have been used in the study of constant mean curvature (CMC) surfaces, minimal surfaces, special Lagrangian cones, self-shrinkers, free boundary minimal surfaces, and certain geometric manifolds such as Eguchi–Hanson spaces.

This course will introduce some of the main ideas and constructions in these gluing methods and explore how they are applied in geometric analysis.

2. Outline & Descriptions

In the first part of the course, I will review various constructions in Differential Geometry by PDE gluing methods, including constructions of CMC (hyper)surfaces, desingularization and doubling constructions for minimal (hyper)surfaces, and the determination of the index and nullity of some minimal surfaces.

In the second part I will discuss in some detail the gluing methodology emphasizing important ideas in the constructions presented in the first part. These lectures will be designed to be accessible to graduate students. 

3. Registration

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