Prof. Melanie Rupflin

Geometric analysis @ University of Oxford

Prof. Melanie Rupflin

Quantitative estimates for geometric variational problems

Many interesting geometric objects are characterised as minimisers or critical points of natural geometric quantities such as the length of a curve, the area of a surface or the energy of a map. For the corresponding variational problems it is important to not only analyse critical points, but to obtain a more general understanding of the energy landscape in particular near points that are "almost critical" of the energy or that "almost minimise" the energy. To be more precise, one needs to understand whether a map u whose energy E(u) is close to the minimal possible energy must necessarily have be close to a minimiser of the energy, and similarly whether a map u for which the gradient ∇ E(u) is small must be close to a map u* for which ∇ E(u*)=0.In this talk we discuss estimates, such as Lojasiewicz inequalities, that not only give a positive answer to these questions, but indeed do so in a quantitative way, e.g. by establishing that the distance to the set of critical points is controlled by a power of the norm of ∇ E(u), and furthermore discuss how such Lojasiewicz estimates play a crucial role in the asymptotic behaviour of gradient flows.

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