Estimates for the Heat Kernel on Differential Forms on Riemannian Symmetric Spaces and Applications

We prove upper bounds estimates for the large time behavior of the heat kernel and for the resolvent of the form Laplacian on Riemannian symmetric spaces, and we obtain L 2+ǫestimates for its resolvent on locally symmetric spaces.We deduce lower bounds for the bottom of the spectrum of the form Laplacian and some results on the vanishing of the L 2 -cohomology of locally symmetric spaces.