##
**Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique.**
*(English)*
Zbl 1150.53001

Progress in Mathematics 266. Basel: Birkhäuser (ISBN 978-3-7643-8641-2/hbk). xiv, 282 p. (2008).

The Bochner-Weitzenböck formula, sometimes referred to as the Bochner technique, is one of the most important techniques in geometric analysis. There are many formulae which can be derived for various situations. One of the classical and standard such techniques is the original Bochner argument to estimate the first Betti number \(b^1(M)\) of a closed oriented Riemannian manifold \((M, g)\). By the Hodge-de Rham theory, \(b^1(M)\) is equal to the dimension of the space of harmonic \(1\)-forms, \(\mathcal H^1(M)\), on \(M\). A formula of this type, independently rediscovered by Bochner, states that for every harmonic \(1\)-form \(\omega\),

\[ \frac{1}{2}\Delta |\omega|^2 = |\nabla \omega|^2 + \text{Ric}(\omega^\#, \omega^\#),\tag{1} \]

where \(\Delta\) and \(\text{Ric}\) are the Laplace-Beltrami operator and the Ricci curvature of \(M\), respectively, \(\nabla\) denotes the extension of \(1\)-forms of the Levi-Civita connection, and \(\omega^\#\) is the vector field dual to \(\omega\), defined by \(\langle \omega^\#, X \rangle = \omega(X)\) for any vector field \(X\). Thus if \(\text{Ric} \geq 0\), then \(|\omega|\) is subharmonic and so is constant since \(M\) is a closed manifold. This implies that \(\omega\) is parallel and so \(b^1(M) = \dim{\mathcal H}^1(M) \leq \dim(M)\). Moreover if \(\text{Ric}(x) > 0\) for some point \(x\in M\), then \(\omega = 0\) and hence \(b^1(M) = 0\).

As the authors mention in the introduction of the book, there is a good expository work about this topic by H. Wu [The Bochner technique in differential geometry. Math. Rep. Ser. 3, No. 2, i–xii, 289–538 (1988; Zbl 0875.58014)]. The book by Wu concentrates on presenting how to use the Bochner technique by illustrating classical well-known results. Whereas, in the book under review the authors aim to present an extension of the classical Bochner technique describing a selection of results recently obtained by them. Furthermore, they also try to treat in a unified and detailed way a variety of problems whose common thread is the validity of Bochner-Weitzenböck formulae including both cases of real Riemannian manifolds and complex (Hermitian or Kählerian) manifolds and compact and noncompact manifolds. They explain the vanishing properties for harmonic functions and harmonic maps, holomorphic maps and pluriharmonic maps and finite dimensionality of the space of these maps.

Sometimes the authors omit proofs of theorems that are well-known or not essential for the aim of the book. For example, they do not give the proof for the Bochner-Weitzenböck formula for a harmonic map between two Riemannian manifolds. Instead they present a proof for an analogous formula in Kählerian geometry. However, they give detailed proofs for theorems which are considered to be very useful.

On the other hand, if at each point \(x\) in a Riemannian manifold \(M\), \(q(x)/2\) is the lowest eigenvalue of the Ricci tensor, then from (1) one has

\[ \Delta |\omega|^2 - q(x) |\omega|^2 \geq 0. \]

This implies that such type of Schrödinger operator \(L = - \Delta + q\) is very important in the theory of geometric analysis. The authors realize this and give some descriptions about spectral properties and applications for the Schrödinger operator on Riemannian manifolds. In particular, if \(M\) is compact, the existence of a nonnegative function \(\psi\) satisfying \(\Delta \psi - q(x) \psi \geq 0\) or of a positive function \(\varphi\) satisfying \(\Delta \varphi - q(x) \varphi \leq 0\) implies some geometric or topological obstructions on \(M\). Moreover, the existence of a positive function \(\varphi\) satisfying \(\Delta \varphi - q(x) \varphi \leq 0\) is related to spectral properties of the operator \(-\Delta + q(x)\).

In the noncompact case, the relevant functions (harmonic or holomorphic) may fail to be bounded, and even if they are bounded, they may not attain their maximum. So, in case of noncompact manifolds, one must assume suitable \(L^p\) integrability conditions for the relevant functions. The authors explain the \(L^2\) cohomology of a complete manifold and describe the vanishing properties (resp., finite dimensionality) of the space of \(L^2\) (resp., \(L^p\)) harmonic differential forms of a complete manifold.

In Chapter 1, Harmonic, pluriharmonic and holomorphic maps and basic Hermitian and Kählerian geometry, the authors begin a quick review of harmonic maps between Riemannian manifolds, where in particular they describe the Weitzenböck formula and derive a sharp version of Kato’s inequality. Then they introduce the basic facts on the geometry of complex manifolds, and Hermitian manifolds, concentrating on the Kähler case. The chapter ends with a derivation of Weitzenböck-type formulae for pluriharmonic and holomorphic maps.

Let \(f: (M, g) \to (N, h)\) be a smooth map between Riemannian manifolds so that the total differential \(df\) is a section of the bundle \(T^*M \otimes f^{-1}TN\). The energy density \(e(f)\) of \(f\) is defined by

\[ e(f) = \frac{1}{2} |df|^2 = \frac{1}{2} \text{tr}_g f^*h = \frac{1}{2} g^{ij} h_{\alpha\beta} \frac{\partial f^\alpha}{\partial x^i}\frac{\partial f^\beta}{\partial x^j} \]

in local coordinates \((x^i)\) and \((y^\alpha)\) of \(M\) and \(N\), respectively. Here \(f^\alpha = y^\alpha \circ f\) and \(g^{ij}\) represents the inverse of the matrix coefficient \(g_{ij} = \langle \frac{\partial }{\partial x^i}, \frac{\partial }{\partial x^j}\rangle\).

If \(\Omega \subset M\) is a compact domain, the energy \(f|_\Omega\) is defined by

\[ E_\Omega (f) = \int_\Omega d(f)\, dv_g. \]

A smooth map \(f: (M, g)\to (N, h)\) is said to be harmonic if for any compact domain \(\Omega \subset M\) it is a critical point of the energy functional \(E_\Omega: C^\infty(M, N) \to {\mathbb R}\) with respect to variations preserving \(f\) on \(\partial \Omega\). It is easy to see that a map \(f: M \to N\) is harmonic if and only if it satisfies the Euler-Lagrange equation \(\tau(f) = 0\), where \(\tau(f) = \text{tr}_g Ddf\) is called the tension field of \(f\).

First, the authors present the Bochner-Weitzenböck formula for harmonic maps without proof. If \(f : (M, g) \to (N, h)\) is a harmonic map, then

\[ \tfrac{1}{2} \Delta |df|^2 = |Ddf|^2 + \sum_i \langle df(\text{Ric}^M(e_i)), df(e_i) \rangle_h- \sum_{i,j} \langle R^N(df(e_i), df(e_j))df(e_j), df(e_i)\rangle_h. \]

In particular, if \(f\) is a harmonic function, then

\[ \tfrac{1}{2} \Delta |\nabla f|^2 = |\text{Hess} f|^2 + \text{Ric}(\nabla f, \nabla f). \]

For later applications of the Bochner-Weitzenböck formulae, the authors give a detailed proof of the so-called Kato-type inequality. Let \(f: M^n \to N^m\) be a harmonic map between Riemannian manifolds of dimensions \(n\) and \(m\), respectively. Then

\[ |Ddf|^2 - |\nabla |df||^2 \geq \frac{1}{n-1} |\nabla |df||^2. \]

The Hessian of \(f\) is symmetric, and the harmonicity of \(f\) implies that the Hessian matrix of \(f\) is traceless. So, diagonalizing the Hessian matrix of \(f\), one can prove a Kato-type inequality.

Now let \((M, g, J)\) be an almost Hermitian manifold with an almost complex structure \(J\) and let \((N, h)\) be a Riemannian manifold. Given a map \(f: M \to N\), the complex linear extension \(Ddf^{\mathbb C}\) of the Hessian (second fundamental tensor) \(Ddf\) of \(f\) splits into

\[ Ddf^{\mathbb C} = Ddf^{(2,0)} +Ddf^{(1,1)} + Ddf^{(0,2)}. \]

The map \(f: (M, g, J) \to (N, h)\) is said to be pluriharmonic or \((1,1)\)-geodesic if \(Ddf^{(1,1)} = 0\). In the case \(N = {\mathbb R}\), \(Ddf^{(1,1)}\) is a Hermitian form referred to as the Levi form of \(f\). We say that the function \(f: (M, g, J) \to {\mathbb R}\) is pluriharmonic if all eigenvalues of its Levi form are nonnegative. Note that if \((M, g, J)\) is a Kähler manifold, then the notion of pluriharmonic map does not depend on the choice of the Kähler metric \(g\) on \(M\). Let \(f: (M, g, J) \to (N, h)\) be a pluriharmonic map from a Kähler manifold into a Riemannian manifold. Then

\[ \Delta |df|^2 = 16 \sum F^\alpha_{kt}\overline{F^\alpha_{kt}} - 16 \sum F^\alpha_{k}F^\beta_{t} F^\gamma_{\bar k}F^\delta_{\bar t}\left(R^M_{\alpha\beta\gamma\delta} + R^N_{\alpha\beta\gamma\delta}\right) + 16 \sum R_{l \bar k} F^\alpha_{\bar k}F^\alpha_{l}, \]

where \(f^*\omega^\alpha = F^\alpha_{i} \varphi^i + F^\alpha_{\bar i}\overline{\varphi}^i\), \(\omega^\alpha\) and \(\varphi^i\) are (local) orthonormal coframes of \(M\) and \(N\), respectively, and \(F^\alpha_{kt}\) denotes the derivative of \(F^\alpha_{k}\). In particular, if the Ricci curvature of \(M\) satisfies \({}^M\text{Ric}(x) \geq - \rho(x)\) and \(N\) has nonpositive Hermitian curvature, then

\[ |df|^2 \Delta |df|^2 \geq \left|\nabla|df|^2 \right|^2 - 2 \rho(x) |df|^4. \]

Furthermore, if \((M, g, J_M)\) and \((N, h, J_N)\) are both Kähler manifolds, \(f: M \to N\) is holomorphic and writing \(f^*\omega^\alpha = B^\alpha_i \varphi^i\), then

\[ \Delta |df|^2 = 16 \sum B^\alpha_{ik} \overline{B}^\alpha_{ik} + 16 \sum R_{i\bar j} \overline{B}^\alpha_{i}B^\alpha_{j} - 16 \sum \overline{B}^\alpha_{i}B^\beta_{i}B^\gamma_{k}K^\alpha_{\beta\gamma\delta}. \]

In particular, if \(\text{Ric}^M(x) \geq - \rho(x)\) and \(N\) has holomorphic bisectional curvature bounded by \(k(z)\), then

\[ |df|^2 \Delta|df|^2 \geq \left|\nabla|df|^2 \right|^2 - 2 \rho(x) |df|^4 - 2k(f(x)) |df|^6. \]

In Chapter 2, Comparison results, the authors give a detailed description of comparison theorems in Riemannian geometry under curvature conditions, both pointwise and integral. They begin with general comparison results for the Laplacian and the Hessian of the distance function. Adopting P. Petersen’s treatment [P. Petersen, Convergence theorems in Riemannian geometry. Comparison geometry. Cambridge: Cambridge University. Math. Sci. Res. Inst. Publ. 30, 167–202 (1997; Zbl 0898.53035)], the approach is analytic in that it only uses comparison results for ODEs avoiding the use of Jacobi fields, and it is not limited to the case where the bound on the relevant curvature is a constant, but is given in terms of a suitable function of the distance from a reference point.

These estimates are then applied to obtain volume comparisons. Even though the method works both for upper and lower estimates, they concentrate on upper bounds, which hold under less stringent assumptions on the manifolds, and in particular depend on lower bounds for the Ricci curvature alone. The authors also describe volume estimates under integral Ricci curvature conditions which extend previous work of S. Gallot [Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157–158, 191–216 (1988; Zbl 0665.53041)], and, more recently, by P. Petersen and G. Wei [Geom. Funct. Anal. 7, No. 6, 1031–1045 (1997; Zbl 0910.53029)]. They then describe remarkable lower estimates for the volume of large balls on manifolds with almost nonnegative Ricci curvature obtained by P. Li and R. Schoen [Acta Math. 153, 279–301 (1984; Zbl 0556.31005)] and P. Li and M. Ramachandran [Am. J. Math. 118, No. 2, 341–353 (1996; Zbl 0865.53058)], elaborating ideas of [J. Cheeger, M. Gromov and M. Taylor, J. Differ. Geom. 17, 15–53 (1982; Zbl 0493.53035)]. These estimates in particular imply that such manifolds have infinite volume. The authors end the chapter with a version of the monotonicity formula for minimal submanifolds valid for the volume of intrinsic balls in bi-Lipschitz harmonic immersions.

Chapter 3, Review of spectral theory, is devoted to a quick review of spectral theory of self-adjoint operators on Hilbert spaces modelled after E. B. Davies’ monograph [Spectral Theory and Differential Operators, Cambridge: Cambridge University Press (1996; Zbl 0893.47004)]. In particular, the authors define the essential spectrum and index of a semibounded operator, and then apply the minimax principle to describe some of their properties and their mutual relationships. In the sequel, the authors concentrate on the spectral theory of Schrödinger operator on manifolds. For a complete Riemannian manifold \(M\) let \(q \in L^\infty_{\text{loc}}(M)\). For an open set \(\Omega \subset M\), the Schrödinger operator

\[ L = L_\Omega = -\Delta + q \]

originally defined on \(C^\infty_c(\Omega)\) is considered. It follows easily from the divergence theorem that \(L\) is a symmetric operator on \(C^\infty_c(\Omega)\), which is associated to the quadratic form

\[ Q(\varphi, \psi) = \int_\Omega \langle \nabla \varphi, \nabla \psi \rangle+ q \varphi \psi. \]

Then we have for the spectrum of \(L\)

\[ \inf \text{Spec}(L) = \inf \left\{\frac{\int_\Omega (|\nabla \varphi|^2 + q\varphi^2)}{\int_\Omega \varphi^2}: 0 \neq \varphi \in C^\infty_c(\Omega)\right\}. \]

Assume that \(\Omega\) is a bounded domain. Then \(L\) has discrete spectrum consisting of a nondecreasing sequence of eigenvalues

\[ \lambda_1(L) \leq \lambda_2(L) \leq \cdots \leq \lambda_n(L) \to \infty\quad \text{as}\quad n \to \infty. \]

Furthermore, if \(\Omega \subset \Omega'\), then for every \(n\), \(\lambda_n(L_\Omega')\leq \lambda_n(L_\Omega)\). One fact which will play an important role in the book is the equivalence of the existence of weak solutions for the Schrödinger operator \(L = - \Delta + q\). This fact is a slight improved version of a result of W. F. Moss and J. Piepenbrink [Pac. J. Math. 75, 219–226 (1978; Zbl 0381.35026)] and D. Fischer-Colbrie and R. Schoen [Commun. Pure Appl. Math. 33, 199–211 (1980; Zbl 0439.53060)]. Let \(\Omega\subset M\) be a (bounded or unbounded) domain. Then the following facts are equivalent:

\[ u \operatorname{div}(\varphi\nabla u) \geq 0, \]

where \(\varphi\) is a suitable positive function. They try to make an effort to state and prove the result under the minimal regularity assumptions that will be needed for geometric applications. As a consequence, they deduce a vanishing theorem and a finite-dimensionality result for nonnegative solutions of the Bochner-type differential inequality

\[ \psi \Delta \psi + a(x) \psi^2 + A |\nabla \psi|^2 \geq 0\tag{2} \]

assuming that there exists a positive solution \(\varphi \in \text{Lip}_{\text{loc}}(M)\) of

\[ \Delta \varphi + H a(x) \varphi \leq 0 \]

for some constant \(H\) such that \(H \geq A+1, H>0\) and \(a(x)\in L^\infty_{\text{loc}}(M)\) or \(a(x) \in C^0(M)\) together with an integral condition on \(\psi\). To prove the finite dimensionality of the solutions for (2), the authors introduce a lemma due to P. Li [Ann. Sci. Éc. Norm. Supér. (4) 13, 451–468 (1980; Zbl 0466.53023)] and Poincaré-type inequalities due to Li and Schoen [loc. cit.]. They also describe a local version of the Sobolev inequality, the Moser iteration procedure and a weak Harnack inequality.

Chapters 6 to 9 are devoted to applications in different geometric contexts. In Chapter 6, Applications to harmonic maps, the authors show the usefulness of the vanishing result for (2) by deriving a number of results on harmonic maps with finite \(L^p\) energy, results on the constancy of convergent harmonic maps, and a Schwarz-type lemma for harmonic maps of bounded dilation. One of the main results in this chapter is the following. Let \((M^n, g)\) be a complete manifold whose Ricci tensor satisfies \(\text{Ric}(x) \geq - \rho(x)\) on \(M\) for some continuous function \(\rho(x)\). Having fixed \(H \geq \frac{n-2}{n-1}\), set \(L^H = -\Delta - H\rho(x)\) and assume that \(\lambda_1(L^H) \geq 0\). Let \((N, h)\) be a manifold of nonpositive sectional curvature, \(K_N \leq 0\). Then any harmonic map \(f: M \to N\) with energy density satisfying \(|df|^2 \in L^\gamma(M)\) for some \(\frac{n-2}{n-1}\leq \gamma\leq H\), is constant. They then describe topological results by R. Schoen and S.-T. Yau [Comment. Math. Helv. 51, 333–341 (1976; Zbl 0361.53040)], concerning the fundamental group of manifolds of nonnegative Ricci curvature and of stable minimal hypersurfaces immersed in nonpositively curved ambient spaces. The chapter ends by generalizing to noncompact settings the finiteness theorems of L. Lemaire [Topology 16, 199–201 (1977; Zbl 0343.53029)], for harmonic maps of bounded dilation into a negatively curved manifold, on the assumption that the domain manifold has a finitely generated fundamental group.

In Chapter 7, Some topological applications, the authors describe the topology at infinity of a Riemannian manifold \(M\), and more specifically, the number of unbounded connected components (such a component is called an end) of the complement of a compact domain \(D\) in \(M\). The number of ends of a manifold will play a crucial role in the structure of both noncompact manifolds and Kähler manifolds. The chapter begins with an account of the theory relating the topology at infinity and suitable classes of harmonic functions on the manifold as developed by P. Li and L. F. Tam [Ann. Math. (2) 125, 171–207 (1987; Zbl 0622.58001), J. Differ. Geom. 35, No. 2, 359–383 (1992; Zbl 0768.53018)] and others. For instance, let \({\mathcal H}^\infty_{\mathcal D}(M)\) be the space of bounded harmonic functions with finite Dirichlet integral on \(M\), and denote by \(N(D)\) the number of non-parabolic ends of \(M\) with respect to the relatively compact domain \(D\). Then one has

\[ N(D) \leq \dim {\mathcal H}^\infty_{\mathcal D}(M). \]

It follows that if \({\mathcal H}^\infty_{\mathcal D}(M)\) is finite dimensional, then \(M\) has finitely many non-parabolic ends, whose number is bounded above by \(\dim {\mathcal H}^\infty_{\mathcal D}(M)\). Assume a complete noncompact connected Riemannian manifold \(M\) satisfies \(\text{Ric}(x) \geq - \rho(x)\) for some nonnegative continuous function \(\rho\). As a corollary of the fact mentioned above, if \(\lambda_1(L) \geq 0\) for \(L = -\Delta + \rho\), then \(M\) has at most one non-parabolic ends, and if \(\lambda_1(L_{M-K}) \geq 0\) for some compact subset \(K\), then \(M\) has at most finitely many non-parabolic ends. In particular, following the arguments due to H. Cao, Y. Shen and S. Zhu [Math. Res. Lett. 4, No. 5, 637–644 (1997; Zbl 0906.53004)], and P. Li and J. Wang [Math. Res. Lett. 9, No. 1, 95–103 (2002; Zbl 1019.53025)], one can show that when the manifold supports an \(L^1\)-Sobolev inequality, then all ends are non-parabolic. This can also be applied to submanifolds of Cartan-Hadamard manifolds, providing that the second fundamental form is small in a suitable integral norm. In the chapter, using a gluing technique of T. Napier and M. Ramachandran [Geom. Funct. Anal. 5, No. 5, 809–851 (1995; Zbl 0860.53045)], the authors provide the details of a construction sketched by Li and Ramachandran [loc. cit.] of harmonic functions with controlled \(L^2\) energy growth that will be used in the structure theorems for Kähler manifolds. The last two sections of the chapter contain further applications of these techniques to problems concerning line bundles over Kähler manifolds, and to the reduction of codimension of harmonic immersions with less than quadratic \(p\)-energy growth.

Chapter 8, Constancy of holomorphic maps and the structure of complete Kähler manifolds, begins with a detailed description of a vanishing result of P. Li and S.-T. Yau [Compos. Math. 73, No. 2, 125–144 (1990; Zbl 0701.53082)], on the constancy of holomorphic maps with values in a Hermitian manifold with suitably negative holomorphic bisectional curvature. Then two variations of the result follows, where the conclusion is obtained under different assumptions: in the first, using Poisson equation techniques, an integral growth condition on the Ricci tensor is replaced by a volume growth, while in the second one assumes a pointwise lower bound on the Ricci curvature which is not necessarily integrable, together with some spectral assumptions on a variant of the operator \(L\). They then apply this in the proof of the existence of pluri-subharmonic exhaustions due to Li and Ramachandran, which is crucial in obtaining the important structure theorem of Napier and Ramachandran [loc. cit.], and Li and Ramachandran [loc. cit.].

In Chapter 9, Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality, the authors give a detailed proof of a warped product splitting theorem due to P. Li and J. Wang [J. Reine Angew. Math. 566, 215–230 (2004; Zbl 1050.53049)]. Namely, let \((M, g)\) be a complete manifold of dimension \(n \geq 3\) satisfying \(\lambda_1 = \lambda_1(-\Delta) >0\) and suppose \(\text{Ric}\geq - \frac{m-1}{m-2} \lambda_1\). Then either \((M, g)\) has only one non-parabolic end, or \((M, g)\) splits as the warped product \({\mathbb R}\times \Sigma\) with metric \(g = dt^2 + \cosh^2 (t\sqrt{\frac{\lambda_1}{m-2}}) h_\Sigma\), where \((\Sigma, h_\Sigma)\) is a compact, isometrically embedded hypersurface of \((M, g)\) satisfying \(\text{Ric}_\Sigma \geq-\lambda_1\).

There are two main ingredients in the proof. The first is to prove that the metric splitting holds provided the manifold supports a non-constant harmonic function \(u\) for which the Bochner inequality with a sharp constant in the refined Kato inequality is in fact an equality. The second ingredient consists of energy estimates for a suitable harmonic function \(u\) on \(M\) obtained by means of an exhaustion procedure. This is the point where the Poincaré-Sobolev inequality plays a crucial role. In the second section, the authors show that when \(M\) supports an \(L^2\) Poincaré-Sobolev inequality, then a nonnegative \(L^p\) solution \(\psi\) of the differential inequality \(\psi \Delta \psi + a(x) \psi^2 + A |\nabla \psi|^2 \geq 0\) must vanish provided a suitable integral norm of the potential \(a(x)\) is small compared to the Sobolev constant.

The book contains two appendices. The first is devoted to the unique continuation property for solutions of elliptic partial differential systems on manifolds, which plays an essential role in the finite dimensionality result of Chapter 5. Apart from some minor modifications, the authors’ arguments follow the line of J. Kazdan’s paper [Commun. Pure Appl. Math. 41, No. 5, 667–681 (1988; Zbl 0632.35015)].

In the second appendix, the authors review some basic facts concerning the \(L^p\) cohomology of complete noncompact Riemannian manifolds. They start with describing the basic definitions of the \(L^p\) de Rham complex and discussing some simple, but significant examples. They then give some classical results like Hodge, de Rham, Kodaira decomposition, and briefly consider the role of \(L^p\) harmonic differential forms. Finally, the authors illustrate some of the relationships between \(L^p\) cohomology and the geometry and the topology of the underlying manifold both for \(p = 2\) and \(p \neq 2\).

\[ \frac{1}{2}\Delta |\omega|^2 = |\nabla \omega|^2 + \text{Ric}(\omega^\#, \omega^\#),\tag{1} \]

where \(\Delta\) and \(\text{Ric}\) are the Laplace-Beltrami operator and the Ricci curvature of \(M\), respectively, \(\nabla\) denotes the extension of \(1\)-forms of the Levi-Civita connection, and \(\omega^\#\) is the vector field dual to \(\omega\), defined by \(\langle \omega^\#, X \rangle = \omega(X)\) for any vector field \(X\). Thus if \(\text{Ric} \geq 0\), then \(|\omega|\) is subharmonic and so is constant since \(M\) is a closed manifold. This implies that \(\omega\) is parallel and so \(b^1(M) = \dim{\mathcal H}^1(M) \leq \dim(M)\). Moreover if \(\text{Ric}(x) > 0\) for some point \(x\in M\), then \(\omega = 0\) and hence \(b^1(M) = 0\).

As the authors mention in the introduction of the book, there is a good expository work about this topic by H. Wu [The Bochner technique in differential geometry. Math. Rep. Ser. 3, No. 2, i–xii, 289–538 (1988; Zbl 0875.58014)]. The book by Wu concentrates on presenting how to use the Bochner technique by illustrating classical well-known results. Whereas, in the book under review the authors aim to present an extension of the classical Bochner technique describing a selection of results recently obtained by them. Furthermore, they also try to treat in a unified and detailed way a variety of problems whose common thread is the validity of Bochner-Weitzenböck formulae including both cases of real Riemannian manifolds and complex (Hermitian or Kählerian) manifolds and compact and noncompact manifolds. They explain the vanishing properties for harmonic functions and harmonic maps, holomorphic maps and pluriharmonic maps and finite dimensionality of the space of these maps.

Sometimes the authors omit proofs of theorems that are well-known or not essential for the aim of the book. For example, they do not give the proof for the Bochner-Weitzenböck formula for a harmonic map between two Riemannian manifolds. Instead they present a proof for an analogous formula in Kählerian geometry. However, they give detailed proofs for theorems which are considered to be very useful.

On the other hand, if at each point \(x\) in a Riemannian manifold \(M\), \(q(x)/2\) is the lowest eigenvalue of the Ricci tensor, then from (1) one has

\[ \Delta |\omega|^2 - q(x) |\omega|^2 \geq 0. \]

This implies that such type of Schrödinger operator \(L = - \Delta + q\) is very important in the theory of geometric analysis. The authors realize this and give some descriptions about spectral properties and applications for the Schrödinger operator on Riemannian manifolds. In particular, if \(M\) is compact, the existence of a nonnegative function \(\psi\) satisfying \(\Delta \psi - q(x) \psi \geq 0\) or of a positive function \(\varphi\) satisfying \(\Delta \varphi - q(x) \varphi \leq 0\) implies some geometric or topological obstructions on \(M\). Moreover, the existence of a positive function \(\varphi\) satisfying \(\Delta \varphi - q(x) \varphi \leq 0\) is related to spectral properties of the operator \(-\Delta + q(x)\).

In the noncompact case, the relevant functions (harmonic or holomorphic) may fail to be bounded, and even if they are bounded, they may not attain their maximum. So, in case of noncompact manifolds, one must assume suitable \(L^p\) integrability conditions for the relevant functions. The authors explain the \(L^2\) cohomology of a complete manifold and describe the vanishing properties (resp., finite dimensionality) of the space of \(L^2\) (resp., \(L^p\)) harmonic differential forms of a complete manifold.

In Chapter 1, Harmonic, pluriharmonic and holomorphic maps and basic Hermitian and Kählerian geometry, the authors begin a quick review of harmonic maps between Riemannian manifolds, where in particular they describe the Weitzenböck formula and derive a sharp version of Kato’s inequality. Then they introduce the basic facts on the geometry of complex manifolds, and Hermitian manifolds, concentrating on the Kähler case. The chapter ends with a derivation of Weitzenböck-type formulae for pluriharmonic and holomorphic maps.

Let \(f: (M, g) \to (N, h)\) be a smooth map between Riemannian manifolds so that the total differential \(df\) is a section of the bundle \(T^*M \otimes f^{-1}TN\). The energy density \(e(f)\) of \(f\) is defined by

\[ e(f) = \frac{1}{2} |df|^2 = \frac{1}{2} \text{tr}_g f^*h = \frac{1}{2} g^{ij} h_{\alpha\beta} \frac{\partial f^\alpha}{\partial x^i}\frac{\partial f^\beta}{\partial x^j} \]

in local coordinates \((x^i)\) and \((y^\alpha)\) of \(M\) and \(N\), respectively. Here \(f^\alpha = y^\alpha \circ f\) and \(g^{ij}\) represents the inverse of the matrix coefficient \(g_{ij} = \langle \frac{\partial }{\partial x^i}, \frac{\partial }{\partial x^j}\rangle\).

If \(\Omega \subset M\) is a compact domain, the energy \(f|_\Omega\) is defined by

\[ E_\Omega (f) = \int_\Omega d(f)\, dv_g. \]

A smooth map \(f: (M, g)\to (N, h)\) is said to be harmonic if for any compact domain \(\Omega \subset M\) it is a critical point of the energy functional \(E_\Omega: C^\infty(M, N) \to {\mathbb R}\) with respect to variations preserving \(f\) on \(\partial \Omega\). It is easy to see that a map \(f: M \to N\) is harmonic if and only if it satisfies the Euler-Lagrange equation \(\tau(f) = 0\), where \(\tau(f) = \text{tr}_g Ddf\) is called the tension field of \(f\).

First, the authors present the Bochner-Weitzenböck formula for harmonic maps without proof. If \(f : (M, g) \to (N, h)\) is a harmonic map, then

\[ \tfrac{1}{2} \Delta |df|^2 = |Ddf|^2 + \sum_i \langle df(\text{Ric}^M(e_i)), df(e_i) \rangle_h- \sum_{i,j} \langle R^N(df(e_i), df(e_j))df(e_j), df(e_i)\rangle_h. \]

In particular, if \(f\) is a harmonic function, then

\[ \tfrac{1}{2} \Delta |\nabla f|^2 = |\text{Hess} f|^2 + \text{Ric}(\nabla f, \nabla f). \]

For later applications of the Bochner-Weitzenböck formulae, the authors give a detailed proof of the so-called Kato-type inequality. Let \(f: M^n \to N^m\) be a harmonic map between Riemannian manifolds of dimensions \(n\) and \(m\), respectively. Then

\[ |Ddf|^2 - |\nabla |df||^2 \geq \frac{1}{n-1} |\nabla |df||^2. \]

The Hessian of \(f\) is symmetric, and the harmonicity of \(f\) implies that the Hessian matrix of \(f\) is traceless. So, diagonalizing the Hessian matrix of \(f\), one can prove a Kato-type inequality.

Now let \((M, g, J)\) be an almost Hermitian manifold with an almost complex structure \(J\) and let \((N, h)\) be a Riemannian manifold. Given a map \(f: M \to N\), the complex linear extension \(Ddf^{\mathbb C}\) of the Hessian (second fundamental tensor) \(Ddf\) of \(f\) splits into

\[ Ddf^{\mathbb C} = Ddf^{(2,0)} +Ddf^{(1,1)} + Ddf^{(0,2)}. \]

The map \(f: (M, g, J) \to (N, h)\) is said to be pluriharmonic or \((1,1)\)-geodesic if \(Ddf^{(1,1)} = 0\). In the case \(N = {\mathbb R}\), \(Ddf^{(1,1)}\) is a Hermitian form referred to as the Levi form of \(f\). We say that the function \(f: (M, g, J) \to {\mathbb R}\) is pluriharmonic if all eigenvalues of its Levi form are nonnegative. Note that if \((M, g, J)\) is a Kähler manifold, then the notion of pluriharmonic map does not depend on the choice of the Kähler metric \(g\) on \(M\). Let \(f: (M, g, J) \to (N, h)\) be a pluriharmonic map from a Kähler manifold into a Riemannian manifold. Then

\[ \Delta |df|^2 = 16 \sum F^\alpha_{kt}\overline{F^\alpha_{kt}} - 16 \sum F^\alpha_{k}F^\beta_{t} F^\gamma_{\bar k}F^\delta_{\bar t}\left(R^M_{\alpha\beta\gamma\delta} + R^N_{\alpha\beta\gamma\delta}\right) + 16 \sum R_{l \bar k} F^\alpha_{\bar k}F^\alpha_{l}, \]

where \(f^*\omega^\alpha = F^\alpha_{i} \varphi^i + F^\alpha_{\bar i}\overline{\varphi}^i\), \(\omega^\alpha\) and \(\varphi^i\) are (local) orthonormal coframes of \(M\) and \(N\), respectively, and \(F^\alpha_{kt}\) denotes the derivative of \(F^\alpha_{k}\). In particular, if the Ricci curvature of \(M\) satisfies \({}^M\text{Ric}(x) \geq - \rho(x)\) and \(N\) has nonpositive Hermitian curvature, then

\[ |df|^2 \Delta |df|^2 \geq \left|\nabla|df|^2 \right|^2 - 2 \rho(x) |df|^4. \]

Furthermore, if \((M, g, J_M)\) and \((N, h, J_N)\) are both Kähler manifolds, \(f: M \to N\) is holomorphic and writing \(f^*\omega^\alpha = B^\alpha_i \varphi^i\), then

\[ \Delta |df|^2 = 16 \sum B^\alpha_{ik} \overline{B}^\alpha_{ik} + 16 \sum R_{i\bar j} \overline{B}^\alpha_{i}B^\alpha_{j} - 16 \sum \overline{B}^\alpha_{i}B^\beta_{i}B^\gamma_{k}K^\alpha_{\beta\gamma\delta}. \]

In particular, if \(\text{Ric}^M(x) \geq - \rho(x)\) and \(N\) has holomorphic bisectional curvature bounded by \(k(z)\), then

\[ |df|^2 \Delta|df|^2 \geq \left|\nabla|df|^2 \right|^2 - 2 \rho(x) |df|^4 - 2k(f(x)) |df|^6. \]

In Chapter 2, Comparison results, the authors give a detailed description of comparison theorems in Riemannian geometry under curvature conditions, both pointwise and integral. They begin with general comparison results for the Laplacian and the Hessian of the distance function. Adopting P. Petersen’s treatment [P. Petersen, Convergence theorems in Riemannian geometry. Comparison geometry. Cambridge: Cambridge University. Math. Sci. Res. Inst. Publ. 30, 167–202 (1997; Zbl 0898.53035)], the approach is analytic in that it only uses comparison results for ODEs avoiding the use of Jacobi fields, and it is not limited to the case where the bound on the relevant curvature is a constant, but is given in terms of a suitable function of the distance from a reference point.

These estimates are then applied to obtain volume comparisons. Even though the method works both for upper and lower estimates, they concentrate on upper bounds, which hold under less stringent assumptions on the manifolds, and in particular depend on lower bounds for the Ricci curvature alone. The authors also describe volume estimates under integral Ricci curvature conditions which extend previous work of S. Gallot [Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157–158, 191–216 (1988; Zbl 0665.53041)], and, more recently, by P. Petersen and G. Wei [Geom. Funct. Anal. 7, No. 6, 1031–1045 (1997; Zbl 0910.53029)]. They then describe remarkable lower estimates for the volume of large balls on manifolds with almost nonnegative Ricci curvature obtained by P. Li and R. Schoen [Acta Math. 153, 279–301 (1984; Zbl 0556.31005)] and P. Li and M. Ramachandran [Am. J. Math. 118, No. 2, 341–353 (1996; Zbl 0865.53058)], elaborating ideas of [J. Cheeger, M. Gromov and M. Taylor, J. Differ. Geom. 17, 15–53 (1982; Zbl 0493.53035)]. These estimates in particular imply that such manifolds have infinite volume. The authors end the chapter with a version of the monotonicity formula for minimal submanifolds valid for the volume of intrinsic balls in bi-Lipschitz harmonic immersions.

Chapter 3, Review of spectral theory, is devoted to a quick review of spectral theory of self-adjoint operators on Hilbert spaces modelled after E. B. Davies’ monograph [Spectral Theory and Differential Operators, Cambridge: Cambridge University Press (1996; Zbl 0893.47004)]. In particular, the authors define the essential spectrum and index of a semibounded operator, and then apply the minimax principle to describe some of their properties and their mutual relationships. In the sequel, the authors concentrate on the spectral theory of Schrödinger operator on manifolds. For a complete Riemannian manifold \(M\) let \(q \in L^\infty_{\text{loc}}(M)\). For an open set \(\Omega \subset M\), the Schrödinger operator

\[ L = L_\Omega = -\Delta + q \]

originally defined on \(C^\infty_c(\Omega)\) is considered. It follows easily from the divergence theorem that \(L\) is a symmetric operator on \(C^\infty_c(\Omega)\), which is associated to the quadratic form

\[ Q(\varphi, \psi) = \int_\Omega \langle \nabla \varphi, \nabla \psi \rangle+ q \varphi \psi. \]

Then we have for the spectrum of \(L\)

\[ \inf \text{Spec}(L) = \inf \left\{\frac{\int_\Omega (|\nabla \varphi|^2 + q\varphi^2)}{\int_\Omega \varphi^2}: 0 \neq \varphi \in C^\infty_c(\Omega)\right\}. \]

Assume that \(\Omega\) is a bounded domain. Then \(L\) has discrete spectrum consisting of a nondecreasing sequence of eigenvalues

\[ \lambda_1(L) \leq \lambda_2(L) \leq \cdots \leq \lambda_n(L) \to \infty\quad \text{as}\quad n \to \infty. \]

Furthermore, if \(\Omega \subset \Omega'\), then for every \(n\), \(\lambda_n(L_\Omega')\leq \lambda_n(L_\Omega)\). One fact which will play an important role in the book is the equivalence of the existence of weak solutions for the Schrödinger operator \(L = - \Delta + q\). This fact is a slight improved version of a result of W. F. Moss and J. Piepenbrink [Pac. J. Math. 75, 219–226 (1978; Zbl 0381.35026)] and D. Fischer-Colbrie and R. Schoen [Commun. Pure Appl. Math. 33, 199–211 (1980; Zbl 0439.53060)]. Let \(\Omega\subset M\) be a (bounded or unbounded) domain. Then the following facts are equivalent:

- (i)
- There exists \(w \in C^1(\Omega)\), \(w >0\) weak solution of \(\Delta w - q(x) w = 0\) on \(\Omega\).
- (ii)
- There exists \(\varphi \in H^1_{\text{loc}}(\Omega), \varphi >0\) weak solution of \(\Delta \varphi - q(x) \varphi \leq 0\) on \(\Omega\).
- (iii)
- \(\lambda_1(L_\Omega) \geq 0\).

\[ u \operatorname{div}(\varphi\nabla u) \geq 0, \]

where \(\varphi\) is a suitable positive function. They try to make an effort to state and prove the result under the minimal regularity assumptions that will be needed for geometric applications. As a consequence, they deduce a vanishing theorem and a finite-dimensionality result for nonnegative solutions of the Bochner-type differential inequality

\[ \psi \Delta \psi + a(x) \psi^2 + A |\nabla \psi|^2 \geq 0\tag{2} \]

assuming that there exists a positive solution \(\varphi \in \text{Lip}_{\text{loc}}(M)\) of

\[ \Delta \varphi + H a(x) \varphi \leq 0 \]

for some constant \(H\) such that \(H \geq A+1, H>0\) and \(a(x)\in L^\infty_{\text{loc}}(M)\) or \(a(x) \in C^0(M)\) together with an integral condition on \(\psi\). To prove the finite dimensionality of the solutions for (2), the authors introduce a lemma due to P. Li [Ann. Sci. Éc. Norm. Supér. (4) 13, 451–468 (1980; Zbl 0466.53023)] and Poincaré-type inequalities due to Li and Schoen [loc. cit.]. They also describe a local version of the Sobolev inequality, the Moser iteration procedure and a weak Harnack inequality.

Chapters 6 to 9 are devoted to applications in different geometric contexts. In Chapter 6, Applications to harmonic maps, the authors show the usefulness of the vanishing result for (2) by deriving a number of results on harmonic maps with finite \(L^p\) energy, results on the constancy of convergent harmonic maps, and a Schwarz-type lemma for harmonic maps of bounded dilation. One of the main results in this chapter is the following. Let \((M^n, g)\) be a complete manifold whose Ricci tensor satisfies \(\text{Ric}(x) \geq - \rho(x)\) on \(M\) for some continuous function \(\rho(x)\). Having fixed \(H \geq \frac{n-2}{n-1}\), set \(L^H = -\Delta - H\rho(x)\) and assume that \(\lambda_1(L^H) \geq 0\). Let \((N, h)\) be a manifold of nonpositive sectional curvature, \(K_N \leq 0\). Then any harmonic map \(f: M \to N\) with energy density satisfying \(|df|^2 \in L^\gamma(M)\) for some \(\frac{n-2}{n-1}\leq \gamma\leq H\), is constant. They then describe topological results by R. Schoen and S.-T. Yau [Comment. Math. Helv. 51, 333–341 (1976; Zbl 0361.53040)], concerning the fundamental group of manifolds of nonnegative Ricci curvature and of stable minimal hypersurfaces immersed in nonpositively curved ambient spaces. The chapter ends by generalizing to noncompact settings the finiteness theorems of L. Lemaire [Topology 16, 199–201 (1977; Zbl 0343.53029)], for harmonic maps of bounded dilation into a negatively curved manifold, on the assumption that the domain manifold has a finitely generated fundamental group.

In Chapter 7, Some topological applications, the authors describe the topology at infinity of a Riemannian manifold \(M\), and more specifically, the number of unbounded connected components (such a component is called an end) of the complement of a compact domain \(D\) in \(M\). The number of ends of a manifold will play a crucial role in the structure of both noncompact manifolds and Kähler manifolds. The chapter begins with an account of the theory relating the topology at infinity and suitable classes of harmonic functions on the manifold as developed by P. Li and L. F. Tam [Ann. Math. (2) 125, 171–207 (1987; Zbl 0622.58001), J. Differ. Geom. 35, No. 2, 359–383 (1992; Zbl 0768.53018)] and others. For instance, let \({\mathcal H}^\infty_{\mathcal D}(M)\) be the space of bounded harmonic functions with finite Dirichlet integral on \(M\), and denote by \(N(D)\) the number of non-parabolic ends of \(M\) with respect to the relatively compact domain \(D\). Then one has

\[ N(D) \leq \dim {\mathcal H}^\infty_{\mathcal D}(M). \]

It follows that if \({\mathcal H}^\infty_{\mathcal D}(M)\) is finite dimensional, then \(M\) has finitely many non-parabolic ends, whose number is bounded above by \(\dim {\mathcal H}^\infty_{\mathcal D}(M)\). Assume a complete noncompact connected Riemannian manifold \(M\) satisfies \(\text{Ric}(x) \geq - \rho(x)\) for some nonnegative continuous function \(\rho\). As a corollary of the fact mentioned above, if \(\lambda_1(L) \geq 0\) for \(L = -\Delta + \rho\), then \(M\) has at most one non-parabolic ends, and if \(\lambda_1(L_{M-K}) \geq 0\) for some compact subset \(K\), then \(M\) has at most finitely many non-parabolic ends. In particular, following the arguments due to H. Cao, Y. Shen and S. Zhu [Math. Res. Lett. 4, No. 5, 637–644 (1997; Zbl 0906.53004)], and P. Li and J. Wang [Math. Res. Lett. 9, No. 1, 95–103 (2002; Zbl 1019.53025)], one can show that when the manifold supports an \(L^1\)-Sobolev inequality, then all ends are non-parabolic. This can also be applied to submanifolds of Cartan-Hadamard manifolds, providing that the second fundamental form is small in a suitable integral norm. In the chapter, using a gluing technique of T. Napier and M. Ramachandran [Geom. Funct. Anal. 5, No. 5, 809–851 (1995; Zbl 0860.53045)], the authors provide the details of a construction sketched by Li and Ramachandran [loc. cit.] of harmonic functions with controlled \(L^2\) energy growth that will be used in the structure theorems for Kähler manifolds. The last two sections of the chapter contain further applications of these techniques to problems concerning line bundles over Kähler manifolds, and to the reduction of codimension of harmonic immersions with less than quadratic \(p\)-energy growth.

Chapter 8, Constancy of holomorphic maps and the structure of complete Kähler manifolds, begins with a detailed description of a vanishing result of P. Li and S.-T. Yau [Compos. Math. 73, No. 2, 125–144 (1990; Zbl 0701.53082)], on the constancy of holomorphic maps with values in a Hermitian manifold with suitably negative holomorphic bisectional curvature. Then two variations of the result follows, where the conclusion is obtained under different assumptions: in the first, using Poisson equation techniques, an integral growth condition on the Ricci tensor is replaced by a volume growth, while in the second one assumes a pointwise lower bound on the Ricci curvature which is not necessarily integrable, together with some spectral assumptions on a variant of the operator \(L\). They then apply this in the proof of the existence of pluri-subharmonic exhaustions due to Li and Ramachandran, which is crucial in obtaining the important structure theorem of Napier and Ramachandran [loc. cit.], and Li and Ramachandran [loc. cit.].

In Chapter 9, Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality, the authors give a detailed proof of a warped product splitting theorem due to P. Li and J. Wang [J. Reine Angew. Math. 566, 215–230 (2004; Zbl 1050.53049)]. Namely, let \((M, g)\) be a complete manifold of dimension \(n \geq 3\) satisfying \(\lambda_1 = \lambda_1(-\Delta) >0\) and suppose \(\text{Ric}\geq - \frac{m-1}{m-2} \lambda_1\). Then either \((M, g)\) has only one non-parabolic end, or \((M, g)\) splits as the warped product \({\mathbb R}\times \Sigma\) with metric \(g = dt^2 + \cosh^2 (t\sqrt{\frac{\lambda_1}{m-2}}) h_\Sigma\), where \((\Sigma, h_\Sigma)\) is a compact, isometrically embedded hypersurface of \((M, g)\) satisfying \(\text{Ric}_\Sigma \geq-\lambda_1\).

There are two main ingredients in the proof. The first is to prove that the metric splitting holds provided the manifold supports a non-constant harmonic function \(u\) for which the Bochner inequality with a sharp constant in the refined Kato inequality is in fact an equality. The second ingredient consists of energy estimates for a suitable harmonic function \(u\) on \(M\) obtained by means of an exhaustion procedure. This is the point where the Poincaré-Sobolev inequality plays a crucial role. In the second section, the authors show that when \(M\) supports an \(L^2\) Poincaré-Sobolev inequality, then a nonnegative \(L^p\) solution \(\psi\) of the differential inequality \(\psi \Delta \psi + a(x) \psi^2 + A |\nabla \psi|^2 \geq 0\) must vanish provided a suitable integral norm of the potential \(a(x)\) is small compared to the Sobolev constant.

The book contains two appendices. The first is devoted to the unique continuation property for solutions of elliptic partial differential systems on manifolds, which plays an essential role in the finite dimensionality result of Chapter 5. Apart from some minor modifications, the authors’ arguments follow the line of J. Kazdan’s paper [Commun. Pure Appl. Math. 41, No. 5, 667–681 (1988; Zbl 0632.35015)].

In the second appendix, the authors review some basic facts concerning the \(L^p\) cohomology of complete noncompact Riemannian manifolds. They start with describing the basic definitions of the \(L^p\) de Rham complex and discussing some simple, but significant examples. They then give some classical results like Hodge, de Rham, Kodaira decomposition, and briefly consider the role of \(L^p\) harmonic differential forms. Finally, the authors illustrate some of the relationships between \(L^p\) cohomology and the geometry and the topology of the underlying manifold both for \(p = 2\) and \(p \neq 2\).

Reviewer: Gabjin Yun (Yongin)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

53C43 | Differential geometric aspects of harmonic maps |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

35J60 | Nonlinear elliptic equations |