Yau's conjecture states that the volume of the nodal set of
Laplace eigenfunctions on a compact Riemannian manifold is comparable to
the square root of the corresponding eigenvalue. Donnelly and Fefferrman
proved Yau's conjecture for real analytic metrics but the conjecture stays
widely open for smooth metrics specially in dimensions n>2. Recently
Sogge-Zelditch and Colding-Minicozzi have established new lower bounds for
the volume of the nodal sets. In this talk we give a new proof of
Colding-Minicozzi's result using a different method. This is a joint work
with Christopher Sogge and Zuoqin Wang.