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Chapter 1. Harmonic, pluriharmonic, holomorphic maps and basic Hermitian and Kählerian geometry

The aim of the chapter is to review some basic facts of Riemannian and complex geometry, in order to compute, for instance, some Bochner-type formulas that we shall need in the sequel. In doing so, we do not aim at giving a detailed treatment of the subject, but only to set down notation and relevant results, illustrating some of the computational techniques involved in the proofs.

Chapter 2. Comparison Results

In this section we describe some comparison results for the Hessian and the Laplacian of the distance function and for the volume of geodesic balls under curvature conditions. In some cases, the results we are going to describe improve on classical results.

Chapter 3. Review of spectral theory

In this chapter we collect the results from spectral theory that will be used in the sequel, concentrating our attention on the spectral theory of Schrödinger operators on Riemannian manifolds. The vanishing and finite-dimensionality results which we will present in the next chapters are based on assumptions on the spectrum of suitable Schrödinger operators. And some of the geometric conditions which we will encounter in later chapters can be interpreted in spectral sense.

Chapter 4. Vanishing results

As we mentioned in the introduction, the aim of this book is to present a unified approach to different geometrical questions such as the study of the constancy of harmonic maps, the topology at infinity of submanifolds, the L2-cohomology, and the structure and rigidity of Riemannian and Kählerian manifolds (see Sections 6.1, 7.4, 7.5, 7.6, 8.1, and Appendix B).

Chapter 5. A finite-dimensionality result

As briefly mentioned at the beginning of the previous chapter, typical geometric applications of Theorem 4.5 are obtained by applying it when the function ψ is the norm of the section of a suitable vector bundle. In appropriate circumstances, the theorem guarantees that certain vector subspaces of such sections are trivial, the main geometric assumption being the existence of a positive solution ϕ of the differential inequality
$$ \Delta \varphi + Ha\left( x \right)\varphi \leqslant 0 weakly on M, $$
where a(x) is a lower bound for the relevant curvature term. According to Lemma 3.10 this amounts to requiring that the bottom of the spectrum of the Schrödinger operator −Δ − Ha(x) is non-negative.

Chapter 6. Applications to harmonic maps

In this section we show the usefulness of Theorem 4.5 by deriving a number of results on harmonic maps. We begin by establishing a Liouville-type theorem which compares with classical work by Schoen and Yau, [146]. Direct inspection shows that our result, emphasizing the role of a suitable Schrödinger operator related to the Ricci curvature of the domain manifold, unifies in a single statement the situations considered in [146]; see Remark 6.22 below. We also give a version of this result in case the domain manifold is Kähler and see how this allows weaker integrability conditions on the energy density of the map. From this, we derive a number of geometric conclusions. We then provide a sharp upper estimate on the growth of the energy of a harmonic map. We close the section with a Schwarz-type lemma for harmonic maps with bounded dilation, and some applications to the fundamental group which extend results by Schoen and Yau and Lemaire ([93]).

Chapter 7. Some topological applications

Given a compact set K in M, an end E(K) with respect to K is an unbounded connected component of M\K. By a compactness argument, it is readily seen that the number of ends with respect to K is finite,

Chapter 8. Constancy of holomorphic maps and the structure of complete Kähler manifolds

The aim of this section is to prove three versions of a vanishing result by Li and Yau for holomorphic maps. The first theorem is a strengthening of the original formulation and its proof follows the lines of [107].

Chapter 9. Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality

Up to now, we have been using Theorem 4.5 to show that solutions of a differential problem of the type
$$ \left\{ \begin{gathered} \psi \Delta \psi + a\left( x \right)\psi ^2 \geqslant - A\left| {\left. {\nabla \psi } \right|^2 ,} \right. \hfill \\ \psi \geqslant 0 \hfill \\ \end{gathered} \right. $$
have to be identically zero. The aim of this section is to present a geometrical problem in which the second alternative of Theorem 4.5 does actually occur, that is, ψ becomes a positive solution of the linear equation
$$ \Delta \psi + a\left( x \right)\psi = 0 $$


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