Quantitative stratification for harmonic maps

The HADES seminar on Tuesday, May 2nd will be at 3:30 pm in Room 740.

Speaker: Jason Zhao

Abstract: It is well-known that stationary harmonic maps are singular on a set of at least codimension $2$. We will exposit the work of Cheeger and Naber which improves the result by establishing effective volume estimates of tubular neighborhoods of the singular set. The primary purpose of the talk is to highlight the two key ingredients in the proof,

  • quantitative differentiation; functions in a given class cannot be far away from the infinitesimal behavior except at finitely many scales,
  • cone-splitting; lesser symmetries can be combined to form a greater symmetry,

which have proven extremely robust in the fields of geometric PDE and metric geometry. Combined with $\epsilon$-regularity theorems, one can pass to a priori estimates, e.g. for minimizing harmonic maps in $W^{1, p} \cap W^{2, p/2}$ in the sub-critical regime $p < 3$.

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