From the introduction: "While it is relatively easy to write down examples of positively curved complete Riemannian metrics on Rn, it is much more difficult to write down (explicitly or simply to show the existence) positively curved complete Kähler metrics on Cn when n>1. In fact, prior to the present article, there were only three examples in this direction, by P. F. Klembeck [Proc. Amer. Math. Soc. 64 (1977), no. 2, 313–316; MR0442290] in 1977, by H.-D. Cao [in Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), 1–16, A K Peters, Wellesley, MA, 1996; MR1417944; J. Differential Geom. 45 (1997), no. 2, 257–272; MR1449972] in 1995 and 1997, respectively. Klembeck's example is explicit, while Cao's are given as solutions to some first or second order ODEs. All three are U(n) invariant metrics on Cn.''
   In Theorem 1, the authors exhibit necessary and sufficient conditions for U(n) invariant metrics on Cn to have positive bisectional curvature. This (with some further work) in particular re-proves that the above-mentioned three metrics have positive bisectional curvature. They also exhibit conditions for positive sectional curvature, and prove that the latter two examples satisfy them, while the former constructed by Klembeck does not. Using these results, the authors exhibit new families of U(n) invariant metrics that have positive sectional or bisectional curvature.
   Among the authors' motivations in writing down these examples are the conjectures by Greene-Wu and Yau concerning uniformization of complete non-compact Kähler manifolds with positive sectional or bisectional curvature (we refer the reader to the introduction for more details on these conjectures).

This is an excellent, very well written survey on the Gauss-Bonnet theorem and its contemporary developments. The author succeeds in explaining the fundamental ideas and techniques in remarkably few pages.

This article discusses the uniformization problem in the high-dimensional case, for which a major problem is to understand complex manifolds of dimension at least two which admit the complete Kähler metric with nonnegative or nonpositive bisectional curvature, especially the compact ones. In this article two theorems of this kind are asserted and a sketch of proofs is given. For the nonpositive case, there is a conjecture of Yau asserting that for a compact Kähler manifold M with nonpositive bisectional curvature, there exists a finite cover M′ of M, such that M′ is a holomorphic and metric fiber bundle over N which is a compact Kähler manifold with nonpositive bisectional curvature and c1(N)<0, and that the fiber is a (flat) complex torus. For the nonnegative case, the authors propose a conjecture asserting that for a complete Kähler manifold Mn with nonnegative bisectional curvature, its universal covering manifold M~ is holomorphically isometric to Cn−r×Nr, where r is the Ricci rank, i.e. the maximum rank of the Ricci form. Guided by the two conjectures, the authors are able to prove the following theorems. Theorem 1 asserts that Yau's conjecture is true if one assumes that the metric on Mn is real analytic. Theorem 2 asserts the following. For a complete Kähler manifold Mn with nonnegative bisectional curvature and Ricci rank r=2, suppose that its metric is real analytic. Then its universal covering manifold M~ is holomorphically isometric to Cn−2×N2, where N2 is a complete Kähler manifold with quasi-positive Ricci tensor.
   The first major step in the proof of Theorem 1 states that when M is complete and the bisectional curvature is nonpositive or nonnegative, the Ricci kernel foliation L is always a holomorphic foliation. The second major step in the proof of Theorem 1 is to use the holomorphicity of L to derive a splitting result in the compact case, with bisectional curvature being nonnegative or nonpositive. For the proof of Theorem 2, firstly, a similar idea as step 1 above is explored. If the maximum Ricci rank is two, then in the open subset U⊂M where the Ricci form has rank two, one has the holomorphic, totally geodesic foliation L whose leaves are complete and flat of codimension 2. To prove that U is locally isometrically and holomorphically a product, one must show that a certain open subset denoted by V⊂U is empty. This set V is defined via the use of the so-called conullity operators CT which are also used in step 1 of the proof of Theorem 1, and studied in some work of K. Abe, and M. Dajczer and L. Rodriguez. Suppose that V is nonempty. Derived from CT and L, there is a distribution L~ associated with L, such that L~ is a totally geodesic, holomorphic foliation with flat leaves. The second step is to study the orthogonal distribution of L in L~, which is also totally geodesic and holomorphic. Fix a point p∈V and let Y be the leaf of this distribution passing through p. Let D be the maximal star-shaped domain around the origin for which the exponential map expp:D⊂Tp(Y)≅C→Y is defined. The desired contradiction to the assumption that V is nonempty can be reached by proving that D is actually all of C. This step is again nontrivial, which consists in showing that at the end point of a unit speed geodesic contained in Y, the rank of the Ricci tensor is still 2.

There is a conjecture by S. T. Yau which states: Let (Mn,g) be a compact Kähler manifold with nonpositive bisectional curvature. Then there exists a finite cover M′ of M such that M′ is a holomorphic and metric fiber bundle over a compact Kähler manifold N with nonpositive bisectional curvature and c1(N)<0, and the fiber is a flat complex torus. In the paper under review the authors confirm the conjecture under an additional assumption that g is real analytic (Theorem E). The Kodaira dimension of M is equal to its Ricci rank r, the maximum of the rank of the Ricci tensor ρ at each point. Let U⊂M be the open set where the rank of ρ is r and L be the distribution in U given by the kernel of ρ. One key step in the proof of Theorem E is to show Theorem A: If (Mn,g) is a complete Kähler manifold with nonpositive bisectional curvature, then L is a holomorphic foliation. Another key step is to show that leaves of L in the universal cover of M close up. The authors propose a generalized Yau conjecture: Let (Mn,g) be a compact Kähler manifold with nef cotangent bundle in the sense of Demailly and κ be its Kodaira dimension. Then there exists a finite cover M′ of M such that M′ is a holomorphic fibration without singular fibers over a projective manifold N of dimension κ with c1(N)<0, and each fiber is a complex torus.

The authors study immersed, complete, complex submanifolds Mn in CN which they call developable, which means that the image of such a submanifold's Gauss map Γ: Mn→GC(n,N) into the complex Grassmannian has dimension r<n. Recall that Γ(x) is the subspace of CN parallel to the tangent space TxM. The level sets of Γ are (n−r)-dimensional linear subvarieties of CN. If r=1 they are all parallel to each other, and hence Mn is a cylinder. This is the complex analogue due to Abe of the classical Hartman-Nirenberg theorem. By previous work of Dajczer-Gromoll, Bourgain, Wu and Vitter, non-cylinder examples exist for r≥2.
   The first main result of the paper is that when r=2 and Mn is not a cylinder, then Mn is the total space of a holomorphic fiber bundle over a Riemann surface, whose fibers are (mapped by the immersion of Mn in CN into) linear subvarieties of dimension (n−1) in CN, each of which is the union of parallel (n−2)-dimensional level sets of its Gauss map. Moreover, if Mn is, in addition, an embedded hypersurface in Cn+1, they show that it can be explicitly described in terms of a complex plane curve S in C2={(u1,u2)} whose projection Ω=u1(S) into the first coordinate axis is a non-empty open subset of C, and a holomorphic map f: Ω→Cn∖{0}.
   The second main result concerns the general case when the rank restriction is removed. Namely, when r≤4 or r=n−1 they are able to prove that Mn is a twisted cylinder, that is, it is foliated by cylinders (which reduce to linear subvarieties of dimension (n−1) when r=2) whose generators are the level sets of the Gauss map. This was conjectured by Vitter for any value of r (and first proved by him for r=2), but the authors also give counterexamples showing that it fails to be true for r=5.
   In the last part of this interesting paper, the authors raise some questions related to topological aspects of developable submanifolds and the holomorphic deformability of a developable submanifold into a cylinder.

As a natural generalization of the classical developable surfaces in R3, an n-dimensional immersed submanifold (M,i) of RN, with n≥2 and i:M→RN an immersion of an n-manifold M into RN, is a developable submanifold in RN if for a positive integer r≤n, there is a foliation L on M whose leaves {Ls} are r-dimensional, each i(Ls) is an open subset of an r-dimensional linear subvariety of RN, and the tangent space di(TxM) for x∈M, when identified as a subset of RN, is constant along each i(Lx). If G(n,N) denotes the Grassmannian of n-planes in RN, then such an M has a degenerate Gauss map Γ:M→G(n,N) in the sense that the differential dΓ is singular at each point; in fact rank dΓ≤n−r. In this paper, the author studies complete submanifolds of RN consisting of those with a degenerate Gauss map. Clearly, developable submanifolds all are such submanifolds. For convection, Γ is said to have rank s if dΓ has rank at most s and has rank s at some point.
   The main results of this paper are as follows. (1) Let (M,i) be a complete n-dimensional immersed submanifold of RN with nonnegative Ricci curvature in RN. If i(M) contains an r-dimensional linear subvariety (1≤r≤n), then (M,i) is a cylinder with an r-dimensional generator. In particular, (M,i) is a developable submanifold. Clearly, if the Gauss map Γ of a complete (M,i) has rank n−r, where 1≤r≤n, then i(M) must contain an r-dimensional linear subvariety. Hence, the above result is a generalization of the Hartman-Nirenberg theorem [cf. P. Hartman and L. Nirenberg, Amer. J. Math. 81 (1959), 901–920; MR0126812]. (2) Let (M,i) be a complete n-dimensional submanifold of RN (n≥2) and let the rank of the Gauss map Γ:M→G(n,N) be n−r, where 1≤r≤n. Then a component of M∨ is a cylinder with r-dimensional generators iff the tangent subbundle F of TM∨ in that component is integrable. Here M∨ is the open subset of M on which rank dΓx=n−r for all x∈M∨. (3) Let (M,i) be an n-dimensional properly immersed complex submanifold of CN such that its Gauss map Γ:M→GC(n,N), where GC(n,N) is the complex Grassmannian, has rank ≤1. Then (M,i) is a complex cylinder in the sense that there is a closed complex curve C⊂M such that M is biholomorphic to C×Cn−1 and, after a unitary transformation of CN, i(C)⊂CN−n+1 and I:M→CN−n+1×Cn−1≡CN is given by (p,z)↦(i(p),z) for all p∈C and for all z∈Cn−1.
   This paper is a natural outgrowth of a preceding paper by G. Fischer and Wu [Internat. J. Math. 6 (1995), no. 2, 229–272; MR1316302].

In this exciting paper, the authors study the classical problem of complex geometry concerning developable submanifolds in complex Euclidean space: a complex submanifold Mn in CN is called developable if there exists a holomorphic foliation on M whose leaves are open subsets of linear subvarieties in CN, and such that along each leaf the tangent space TM is constant. The first main result (Theorems 1 and 2) is the following: Suppose the Gauss map Γ:M→G(n,N) of a complex n-submanifold M in CN has rank n−r<n. Let S⊆M be the set where Γ has rank <n−r, and let F be the foliation on M∖S determined by Γ. Then M∖S is developable (under F) and all the leaves of F are closed in M. They also prove (Theorem 3) that S=∅, when r≥n/2.
   The second main result (Theorem 4) says that for a smooth hypersurface Mn⊆Cn+1, rankdΓ=rankH−2, where H is the extended Hesse matrix of the defining function of M. For a real surface in R3, this already gives a nice determinant formula for the Gaussian curvature. The authors also prove some other results (e.g., Theorems 5 and 6) which are very interesting but longer to state.
   It is worth noting that the technical aspects of this paper are equally exciting as the results. The most interesting new concept introduced and analyzed in the paper is the so-called holmet, which has good potential and just might become a fundamental object in complex geometry one day.

An interesting question concerning open manifolds M of nonnegative sectional curvature is to determine to what extent, if any, the geometry far away from the soul affects the global structure of M. One of the earlier results in this direction, due to the authors, states that if M is simply connected at infinity and flat outside a compact set, then M is, in fact, (isometric to) Euclidean space. In this paper, the authors observe that the key ingredient in their earlier proof actually implies the same conclusion if M merely has nonnegative Ricci curvature. They also generalize the above method to deal with the non-simply connected case: if M has nonnegative curvature and is flat outside a compact set, then any soul S of M is flat, and in fact, if S has codimension ≠2, then M itself is flat.

In this paper the author gives an interesting survey of invariant metrics on complex manifolds and their use in the study of complex analysis and geometry. In the first part he introduces in chronological order the classical invariant metrics on complex manifolds: Bergman, Carathéodory, Kobayashi and Kähler-Einstein, recalling the basic properties, the relationships among the different metrics and their boundary behavior.
   He also focuses attention on the importance of Carathéodory and Kobayashi metrics for the study of holomorphic mappings and the importance of Bergman and Kähler-Einstein metrics for a geometric-topological study of complex manifolds. The author then introduces a whole class of new invariant metrics (§6, §8, §9), with the main purpose of investigating the relationship between hyperbolicity and the existence of Hermitian metrics of strongly negative holomorphic curvature, or strongly negative Ricci curvature. In this direction the following theorem is proved: Theorem 2. Every compact Carathéodory-hyperbolic complex manifold admits a C∞ Hermitian metric of negative Ricci curvature, and is hence algebraic.

The main concern of this paper is the following natural question of complex differential geometry: when does the sum of Hermitian metrics of negative curvature also have negative curvature? It was well known that such a result holds for the holomorphic curvature [H. H. Wu, Indiana Univ. Math. J. 22 (1972/73), 1103–1108; MR0315642] and, more generally, for the bisectional curvature [N. Mok, Ann. of Math. (2) 125 (1987), no. 1, 105–152; MR0873379] of the sum of two Hermitian metrics with corresponding conditions. A simple example (p. 40 of this paper) shows that it is false in terms of all sectional curvatures, but the authors prove this statement for the sum of two admissible metrics in two complex dimensions. A Kähler metric G on a domain E⊂Cn is said to be admissible (it automatically has constant holomorphic curvature −4) if (a) G=−∂∂¯¯¯logϕ for some positive C∞ function ϕ:E→(0,∞); (b) −ϕ is plurisubharmonic; (c) ϕ is a linear function of each zi and each z¯¯¯j for the holomorphic coordinate system (z1,⋯,zn) of Cn. For instance, any ellipsoidal domain and any fractional linear transform of the unit ball admit such metrics. The main result of the paper is the following (Theorem 2): Let E,F be domains in C2 with admissible metrics H and S respectively. If M=E∩F is nonempty, then G=H+S is a Kähler metric of strongly negative curvature on M. In spite of "the obvious truth'' of this result it is nontrivial in view of the example on p. 40 and the difficulties involved in suitable H+S curvature estimation. In addition, an inspection of the authors' technique allows one to construct a complete Kähler metric of negatively pinched bisectional curvature on the intersection of two complex-ellipsoidal domains E and F in C2. According to a theorem of P. C. Yang [Duke Math. J. 43 (1976), no. 4, 871–874; MR0419819], this is impossible for the polydisk Δn for n>1. Thus this confirms one's intuition about the small role played by nonsmoothness of the boundary of the polydisk.

In this paper, the author discusses in detail the interaction between the growth rate of the volume of a geodesic ball in a complete Riemannian manifold M and the growth rate of the maximum modulus of a nonconstant subharmonic function f and its differential df on M. One of the theorems of the paper asserts that if M admits a nonconstant continuous subharmonic function f satisfying f≤Aρν+B for some positive constants A and B and for a constant ν∈(0,2), then limr→∞V(r)/r2−δ=∞ for any δ∈(ν,2), where ρ stands for the distance function to a fixed point o of M and V(r) denotes the volume of the metric ball around o of radius r. As noted in this paper, an earlier result due to S. Y. Cheng and S.-T. Yau shows that if there is a nonconstant continuous subharmonic function bounded from above, then limr→∞V(r)/r2=∞. The conditions of these results are in a sense optimum.
   In fact, we have a family of complete metrics gν on R4 such that, (i) in the case ν<0, V(r)∼r2−ν and the space of bounded harmonic functions on Mν=(R4,gν) is of infinite dimension; (ii) in the case ν=0, V(r)∼r2, Mν possesses no nonconstant harmonic functions, but harmonic functions of order logρ form an infinite-dimensional space; (iii) in the case ν∈(0,2), V(r)∼r2−ν and the space Hν of harmonic functions f satisfying f≤Aρν+B for some positive constants A and B is infinite-dimensional; (iv) in the case ν=2, V(r)∼logr and dimH2=∞; (v) in the case ν>2, V(r)∼1, namely Mν has finite volume, but dimHν=∞.
   Moreover, the author also proves that if M is a complete noncompact Kähler manifold on which is defined a nonconstant holomorphic function h of polynomial growth, then limr→∞V(r)/rν=∞ for every ν<2.

The first part of this collection is a biography of Zhong. It is of interest not only because it presents details about the life of a mathematician who made important contributions to his field, but also because this particular life was so very much influenced by events which occurred during the Cultural Revolution. Based on information received from Zhong's wife, Wu gives us a vivid account of what life must have been like during a period of such great political and social conflict.
   The second part consists of a list of publications of Zhong. Many of these have been reviewed individually in MR (see the following paragraph).
   The third part contains fourteen of the papers of Zhong, several of which are appearing in English here for the first time. This work concerns differential geometry of symmetric spaces, questions about other bounded homogeneous domains, automorphic functions, etc. Eleven of the papers have been reviewed individually in MR: MR0549214;
   MR0619317;
   MR0619583;
   MR0616146;
   MR0615780;
   MR0635174;
   MR0635175;
   MR0658367;
   MR0897589;
   MR0794292; MR0840400. The reader should consult those reviews for more details.

Contents:

Hung Hsi Wu, "Zhong Jia Qing (1937–1987)”, 1–13.

"Publications of Zhong Jia Qing”, 15–17.

Peter Li and Andrejs Treibergs, "Applications of eigenvalue techniques to geometry”, 21–52.

Qi Keng Lu, "The theory of functions of several complex variables in China from 1949 to 1989”, 53–93.

Yum Tong Siu, "Uniformization in several complex variables”, 95–130.

Jia Qing Zhong, "Selected papers of Zhong Jia Qing”, 131–479.

   {The papers are being reviewed individually.}

The author defines a function f on a Kähler manifold M to be of polynomial growth if, for a fixed k, |f|≤Ark+B on any ball with center at a fixed point P0 and radius r. He proves that if M admits a nonconstant polynomial function then there are corresponding restrictions on the volumes of geodesic balls.

The authors list four additional references related to the paper cited in the heading.

The Bochner technique is the brainchild of Salomon Bochner and dates back some forty years or so. The underlying principle is that certain vector fields, e.g. of Killing or spinor type, or harmonic forms, are constrained to satisfy certain PDEs when curvature conditions are imposed. Careful applications of the technique have yielded a number of remarkable vanishing theorems; for instance, the famous result of Kodaira in the 1950s. The method is also particularly powerful in the theory of harmonic maps of Riemannian manifolds, where, in the presence of curvature bounds, special analytic properties of the maps may be deduced.
   An overall point is that, in its more elementary form, the method is quite direct and very workable. Although more recent and more sophisticated methods may emulate the Bochner technique (in some cases), the very ingenuity of the idea will no doubt survive the history of modern geometry.
   The author has performed a fine service to the community of modern geometric analysts by providing a self-contained and quite readable account of the background to the method and how it may be effectively applied to a number of problems. Space here certainly cannot accommodate very many details. For now, we will simply list the topics covered by the six chapters: coordinates and frames normal at a point, the Weitzenböck formulas, some results in the compact case, some results in the noncompact case, harmonic spinor fields, and harmonic mappings.

Let M=MK,D,.k,.,v(n) be the set of all connected, compact C∞ n-dimensional Riemannian manifolds, M, whose (sectional) curvature, KM, diameter, diamM, and volume, VolM, satisfy k≤KM≤K, diamM≤D, VolM≥v. By a theorem of M. Gromov[Structures métriques pour les variétés riemanniennes, CEDIC, Paris, 1981; MR0720933], M is precompact relative to the Hausdorff metric and convergence in this metric coincides with convergence relative to the Lipschitz metric. Moreover any limit space X∈M¯¯¯¯¯¯ is a connected, compact C∞ n-dimensional manifold with a nonsmooth "Riemannian'' metric gX. Several people have studied the regularity properties of gX. The best result is that gX is of class C1,α, for any 0<α<1. This is proved in the present paper and in a paper by S. Peters[Compositio Math. 62 (1987), no. 1, 3–16; MR0892147]. The essential tool is the use of "linear'' harmonic coordinates due to J. Jostand H. Karcher[Manuscripta Math. 40 (1982), no. 1, 27–77; MR0679120].

One of the fundamental concerns of the geometric theory of several complex variables is to understand the relationship between the existence of complete metrics with prescribed curvature properties and the function-theoretic properties of the manifold. In this partly expository paper, the author is mainly concerned with how the sign of the curvature affects the existence of holomorphic functions, and in particular, bounded holomorphic functions. After briefly surveying the known results and those questions which still remain open, he establishes the following new results: (a) If M is a complete Kähler manifold with nonnegative scalar curvature, then M does not admit any smooth negative plurisubharmonic function which is strictly plurisubharmonic at a point p; (b) If M is a complete Kähler manifold with nonnegative Ricci curvature which admits a smooth plurisubharmonic function which is strictly plurisubharmonic at a point p, then the holomorphic functions provide local coordinates at p.
   Related results were obtained by the author and R. E. Greeneand published elsewhere [Function theory on manifolds which possess a pole, Lecture Notes in Math., 699, Springer, Berlin, 1979; MR0521983].

A Riemannian manifold M is said to have q-positive [resp. q-nonnegative] curvature if for each x∈M and for all sets of q+1 orthonormal tangent vectors e0,e1,⋯,eq at x, ∑qi=1K(e0,ei)>0 [resp. ≥0], where K denotes the sectional curvature. Examples of manifolds of q-positive curvature (q>1) are compact locally symmetric spaces of rank >1. The author studies the topology of such manifolds through the existence of partially convex functions f in the sense that the Hessian D2f satisfies ∑qi=1D2f(ei,ei)>0. Theorem 1 says roughly that if M is compact, of q-nonnegative curvature and has convex boundary ∂M (the actual assumption is weaker than this) then (a) M has the homotopy type of a finite CW-complex with cells of dimension ≤q−1; (b) M also has the homotopy type of a CW-complex obtained from ∂M by attaching a finite number of cells of dimension ≥n−q+1. In particular Hi(M;Z)=πi(M)=0 for i≥q;Hi(M,∂M;Z)=πi(M,∂M)=0 for i≤n−q. In the proof a partially convex function is constructed using the distance function to ∂M, but in general it is not smooth; thus a smoothing theorem must be proved. Then the result follows from the approximation by a Morse function. In the special case where q=1 it is proved that (M,∂M) is diffeomorphic to the standard n-disk.
   In the case where M is a complete noncompact Riemannian manifold whose curvature is q-nonnegative and in addition q-positive outside a compact set, it is proved, using the Busemann function, that M carries a smooth partially convex function. As corollaries: (a) M contains no compact minimal submanifold of dimension ≥q; (b) if in addition the sectional curvature is nonnegative outside a compact set, M has the homotopy type of a finite CW-complex with cells of dimension ≤q−1.