Mathematics Colloquia and Seminars

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In this talk, we discuss the regularity and renormalized volume of minimal submanifolds, Y, of arbitrary codimension in Poincare-Einstein manifolds, M. Using geometric and microlocal techniques, we derive polyhomogeneous expansions for the minimal submanifold and variations along it. We then present formulae for the first and second variations of renormalized volume. We end with an orthogonality relationship between coefficients in the expansion of Y, for the specific case of M = H^{n+1}. In all theorems, we emphasize the Dirichlet-to-Neumann type map and its presence in formulae.