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Uniqueness of blowups and Łojasiewicz inequalities. (English) Zbl 1337.53082
In this paper, the authors study the singularities of a mean curvature flow and they prove that at each generic singularity the blow-up is unique, i.e., it does not depend on the sequence of rescalings. This is the first general uniqueness theorem of blowups for a geometric PDE at a noncompact singularity and implies the regularity of the singular set of the flow. The proof of the uniqueness result is based on two completely new infinite-dimensional Łojasiewicz-type inequalities. In real algebraic geometry, Łojasiewicz proved in the sixties some inequalities for real analytic functions defined on an open set \(U\) of \({\mathbb R}^n\). Infinite-dimensional versions of these inequalities were proven by Leon Simon for the area and related functionals and applied to the study of tangent cones with smooth cross section of minimal surfaces. The proofs of Simon are based on a reduction of the infinite-dimensional case to the classical Łojasiewicz inequality by a Ljapunov-Schmidt reduction argument.
In the paper under review, the infinite-dimensional Łojasiewicz inequalities are proved for the functional \(F\), defined on the space of hypersurfaces, obtained by integrating the Gaussian over a hypersurface \(\Sigma \subset {\mathbb R}^{n+1}\), namely \[ F(\Sigma) = (4\pi)^{-n/2}\int_{\Sigma}\mathrm{e}^{-{{|x|^2}\over 4}}\mathrm{d}\mu. \] The inequalities are directly proved on the functional \(F\), without any finite-dimensional reduction, and play a basic role in the proof of the uniqueness of blowups.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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