Geometry & Topology Seminar: "Jacobians with Prescribed Eigenvectors"

Abstract: Let $\Omega\subset {\mathbb R}^n$ be open and let $\mathcal R$ be a partial frame on $\Omega$, that is a set of $m$ linearly independent vector fields prescribed on $\Omega$ ($m\leq n$). We consider the problem of determining all maps $F:\Omega\to{\mathbb R}^n$ with the property that each of the vector fields in $\mathcal R$ is an eigenvector of the Jacobian matrix of $F$. Our initial motivation for considering this problem comes from the geometric study of hyperbolic conservative systems $u_t +f(u)_x=0$ in one spatial dimension. This problem is, however, of independent geometric interest and, in turn, leads to interesting overdetermined systems of PDEs, which can be studied via classical integrability theorems, such as Frobenius and Darboux theorems, and their appropriate generalizations. This is a joint work with Michael Benfield and Kris Jenssen.

Host: Xiang Tang