Mathematics Colloquia and Seminars

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We will discuss some topics related to Sobolev maps between compact Riemannian manifolds. First we will show that the [p]-1 homotopy class of a map in W^{1,p}(M,N), defined previously by White, characterizes the path connected components of W^{1,p}(M,N). This gives a clear picture of path connected components of W^{1,p}(M,N) and helps in understanding the existence of multiple solutions. Next we discuss the solution of the density problem for Sobolev maps (a question raised by Eells and Lemaire)and its relations to the existence of smooth maps in a path connected components. Finally we describe a result which identifies all the weak limits of smooth maps under the Dirichlet energy. Similar problems for higher power energy remain open.