Geometric gradient flow in the space of smooth embeddings
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Given an embedding of a closed k-dimensional manifold M into N-dimensional Euclidean space R^N, we aim to perform negative gradient flow of a penalty function P that acts on the space of all smooth embeddings of M into R^N to find an ideal manifold embedding. We study the computation of the gradient for a penalty function that contains both a curvature and distance term. We also find a lower bound for how long an embedding will remain in the space of embeddings when moving in a fixed, normal gradient direction. Finally, we study the distance penalty function in a special case in which we can prove short time existence of the negative gradient flow using the Cauchy-Kovalevskaya Theorem.