Seminars & Colloquia Calendar

Abstract: In this talk I will discuss several results from my thesis concerning the integrable structures on manifolds. The Frobenius-type theorems describe the necessary and sufficient conditions on a locally integrable structure for when it is equal to the span of (some real and complex) coordinate vector fields. The typical examples are:

- The Frobenius theorem, that an involutive real subbundle is spanned by some real coordinate vector fields

- The Newlander-Nirenberg theorem, that an involutive almost complex structure is spanned by some complex coordinate vector fields

- Nirenberg’s complex Frobenius theorem, which is the combination of the above two cases.

    In the talk we will introduce the involutive condition with weakest regularity assumption. We obtain the Frobenius theorem when the subbundle is only log-Lipschitz continuous. For a \(C^{k,s}\) complex Frobenius structure, we show that there is a \(C^{k,s}\) coordinate chart such that the structure is spanned by coordinate vector fields that are \(C^{k,s?\epsilon}\) for all \(\epsilon >0\), where the \(\epsilon>0\) loss in the result is optimal. This is a joint work with Brian Street.