Abstract: | In this talk, we discuss the nodal set of a bi-harmonic function u on an n-dimensional $C^\infty$ Riemannian manifold $M.$ We first define the frequency function and the doubling index for the bi-harmonic function u, and then establish their monotonicity formulae and doubling conditions. With the help of establishing the smallness propagation and partitions for u, we show that, for some ball, an upper bound for the measure of nodal set of the biharmonic function u can be controlled by $N^\alpha$. Among others, we also show that the n-1 dimensional Hausdordff measures of nodal sets of such solutions are bounded by the frequency function, which is independently interesting. This is a joint work with Tian Long. |