Analysis Seminar: Critical sets of harmonic functions and Almgren's frequency function

Abstract: Given a harmonic function on a nice domain in n-dimensional Euclidean space, its critical set (the set of points where its gradient vanishes) is known to have dimension at most n-2, with locally finite (n-2)-dimensional Hausdorff measure. For harmonic polynomials, the Hausdorff measure can be bounded in terms of the degree of the polynomial. For more general harmonic functions, Almgren's frequency function serves as an analog of the degree of a polynomial, measuring local growth properties of the harmonic function.

Lin conjectured that for a harmonic function with frequency at most N, the (n-2)-dimesnional Hausdorff measure of the critical set is (locally) at most CN^2. The first (and only) quantitative bound in this direction was due to Naber and Valtorta, who showed a bound of the form exp(CN^2). In this talk, I'll discuss an improvement of this bound to the nearly polynomial threshold N^(C log N) via a multiscale analysis, some linear algebra, and geometric combinatorics. This is joint work with Josep Gallegos and Eugenia Malinnikova.

Host: Alan Chang