# Gluing constructions in differential geometry

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I will discuss various geometric gluing constructions. First I will discuss constructions for Constant Mean Curvature hypersurfaces in Euclidean spaces including my earlier work for two-surfaces in three-space which settled the Hopf conjecture for surfaces of genus two and higher, and recent generalizations in collaboration with Christine Breiner in all dimensions. I will then briefly mention gluing constructions in collaboration with Mark Haskins for special Lagrangian cones in Cn. A large part of my talk will concentrate on doubling and desingularization constructions for minimal surfaces and on applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the Mean Curvature flow. Finally I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on four-manifolds and related geometric objects.