The sharp Weyl formula

This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) in a natural way, so that N(λ‎) = cλ‎ⁿ + O(λ‎superscript n minus 1). The chapter then details the sup-norm estimates for eigenfunctions and spectral clusters. Next, this chapter proves the sharp Weyl formula and in doing so, outlines a number of theorems, the first of which the chapter focuses on in establishing its sharpness and in obtaining improved bounds for its Weyl formula's error term. Finally, the chapter shows that improved bounds are also available for the remainder term in the Weyl formula when (M,g) has nonpositive sectional curvature.

The double cover relative to a convex domain and the relative isoperimetric inequality

We prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

Perturbations of the Harmonic Map Equation

We consider perturbations of the harmonic map equation in the case where the target manifold is a closed Riemannian manifold of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.