25.07.2024, 16:30
– Raum 0.14
Forschungsseminar Differentialgeometrie
Dirac eigenvalues, hyperspherical radius and applications
Christian Bär (UP)
Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n-2)-dimensional submanifolds. Hence it is countably (n-2)-rectifiable and its Hausdorff dimension is at most n-2. Moreover, it has locally finite (n-2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n-2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.