On November 25, 1915, Einstein‘s paper on general relativity, Die Feldgleichungen der Gravitation (The Field Equations of Gravity), was published in the *Sitzungsberichte der* *Königlich Preussische Akademie der Wissenschaften.* Several scans of the original are available online, with this being a relatively clear and readable scan. Einstein’s equations definitely had major effects on physics, but they have also led to remarkable mathematics. Despite rumors to the contrary, Einstein was rather good at mathematics. Much of his work relies on mathematical derivations of physical results. An accessible example is his derivation of the equivalence of mass and energy, which is the content of his Gibbs Lecture. More challenging is his work on the existence of singularities in space-times that contain either mass or charge [first paper, second paper – with Pauli]. Note that the paper with Pauli was published in the *Annals of Mathematics*. Indeed, a MathSciNet search turns up 13 papers by Einstein that were published in the *Annals*. Not bad for someone reputed to be “bad at math”.

The influence I would like to concentrate on here is the concept of an *Einstein manifold*. The definition is quite simple, if you are already conversant in differential geometry: a Riemannian (or pseudo-Riemannian) manifold is Einstein if the Ricci tensor is proportional to the metric tensor. That is to say, the (Ricci) curvature is a scalar multiple of the metric. An important special family of Einstein manifolds is formed by the Kähler-Einstein manifolds, which include the Calabi-Yau manifolds. Wherever they occur, Einstein manifolds tend to have special properties. When starting with a manifold without a metric, the goal is often to determine whether it supports an Einstein metric.

There is a great introduction to the subject in the book *Einstein Manifolds*, by Arthur Besse. The review of the original edition of the book is appended below. The book emphasizes compact Einstein manifolds. Examples of such are few and far between. The author of the book used to have a standing offer of a meal at a Michelin-starred restaurant for anyone who came up with a genuinely new example. I don’t know the status of that offer today. Also, I am not sure how you would collect on it, as Arthur Besse is a pseudonym.

Calabi-Yau manifolds are complex manifolds that are Einstein manifolds, Kähler manifolds, and Ricci flat. They can also be described as Ricci-flat Kähler manifolds whose first Chern class vanishes. Calabi-Yau manifolds play important roles in mathematical physics, particularly in string theory and mirror symmetry. A supersymmetric picture of the universe is ten dimensional, having the usual four dimensions of space-time, plus compact Calabi-Yau manifolds “attached” at each point. (The ten-dimensional space is fibered by Calabi-Yau manifolds over the four-dimensional base space.) The Calabi-Yau manifolds have three complex dimensions, which means six real dimensions. In the late 1980s, physicists were trying to generate as many examples of compact Calabi-Yau manifolds as possible. I was quite impressed to see that some people in Philip Candelas‘s group at the University of Texas had written computer programs to generate possible examples. This was done without the aid of Maple, Mathematica, or any other computer algebra system. They wrote code, which generated pages of output (on the old side-punched, 132-column computer paper), with each line being a candidate. They then used methods of algebraic geometry and representation theory to determine if the candidates were valid examples.

In algebraic geometry, Calabi-Yau manifolds can be viewed as higher-dimensional generalizations of K3 surfaces. Their existence was conjectured by Eugenio Calabi, first in his talk at the ICM in Amsterdam, then in his paper in a volume honoring Lefschetz. Their existence was proved by S.-T. Yau in his famous paper in CPAM. The MathSciNet review of the paper is appended below. The result is essentially a statement in algebraic geometry, but the proof involves some very serious work on existence of solutions to PDEs, e.g., the complex Monge-Ampère equation. Applications of geometric analysis to algebraic geometry in this vein continue. A current example is the work of Xiuxiong Chen, Simon Donaldson, and Song Sun on Kähler-Einstein metrics on Fano manifolds: see MR3194014, with full details in the review of the three-part paper MR3264766, MR3264767, MR3264768. Several talks at the 2015 Summer Institute in Algebraic Geometry were on this work.

Einstein’s influence in mathematics is truly remarkable. Searching MathSciNet for items with “Einstein” in the title leads to over 8,000 matches. Doing the search with “Einstein” anywhere leads to nearly 29,000 matches. That is quite a legacy.

Happy 100^{th} birthday, General Relativity! And thank you for all the great mathematics!

**MR0867684** **(88f:53087)**

Besse, Arthur L.

Einstein manifolds.

Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. *Springer-Verlag, Berlin,*1987. xii+510 pp. ISBN: 3-540-15279-2

53C25 (53-02 53C21 53C30 53C55 58D17 58E11)

A Riemannian metric on a smooth manifold $M$ is defined by a positive definite symmetric bilinar form $g$ on each tangent space $T_pM$. Part of the curvature of $g$ is represented by another symmetric bilinear form on $T_pM$, namely the Ricci tensor $r$. The metric is Einstein when $r$ is a constant multiple of $g$; if the constant if zero, $M$ is “Ricci-flat” and a positive definite analogue of a solution to Einstein’s vacuum field equations.

This book is designed to be an exhaustive survey of results concerning Einstein metrics on compact manifolds of dimension at least four, and has its origins in a 1979 symposium held at Espalion in southern France. At that time the recent solution by S. T. Yau and T. Aubin of conjectures of E. Calabi had established the existence of large classes of Kähler-Einstein manifolds whose scalar curvature $s=\textrm{trace}_gr$ is nonpositive. A Kähler manifold is one with strongly compatible complex and Riemannian structures, and for example an algebraic hypersurface of degree $m+1$ in the complex projective space $\mathbf{C}\textrm{P}^m$ was shown to admit a Ricci-flat Kähler metric.

Complementary and more explicit classes of Einstein metrics arise on homogeneous spaces $G/H$. If $G$ is a compact Lie group, any orbit of the adjoint representation admits a canonical Kähler-Einstein metric with $s>0$. In another direction, if $H$ has irreducible isotropy representation, an invariant Riemannian metric $g$ on $G/H$ is forced to be proportional to its Ricci tensor $r$. Further progress towards a classification of homogeneous Einstein manifolds has been made, particularly by M. Wang and W. Ziller . An early observation was that $\mathbf{C}\textrm{P}^3$ has a nonstandard homogeneous Einstein metric, a fact that may be explained in terms of a Riemannian submersion $\pi\colon\mathbf{C}\textrm{P}^3\to S^4$. Such techniques were exploited by L. Bérard-Bergery to generalize an example due to D. Page of an Einstein metric on the connected sum $\mathbf{C}\textrm{P}^2\,\#\,\overline{\mathbf{C}\textrm{P}^2}$.

The description of the above examples and methods forms the central part of the book (Chapters 7, 8, 9, 11 out of a total of 16). With the exception of a few difficult proofs, the exposition is self-contained, and contains a wealth of previously unpublished or inaccessible material. The remainder of the book is organized as follows.

A comprehensive exposition of “basic material” precedes a digression into Lorentzian geometry and relativity. Motivation from physics permeates into Chapter 4, which develops the fact that Einstein metrics are the solutions to the Euler-Lagrange equations for stationary points of the integral of the scalar curvature $s$. Some attractive analysis then follows from the question of which functions arise as scalar curvatures, and which symmetric bilinear tensors are Ricci tensors. Chapter 6 is an examination of topological consequences of the Einstein condition, the most successful being the inequality $\chi(M)\geq\frac 32|\tau(M)|$ relating the Euler characteristic and the signature of a compact oriented Einstein 4-manifold, and resulting from successive work of M. Berger, J. A. Thorpe, and N. J. Hitchin . Other techniques give conditions for the existence of Einstein metrics with $s>0$, but it is still unknown whether there is a manifold of dimension five or more admitting no Einstein metric.

Later chapters cover relevant, but more specialised topics. A sequel to the variational approach is provided by a study of deformations of Einstein metrics, and work of N. Koiso on the moduli space problem. Three chapters which could be read consecutively deal respectively with holonomy groups, self-duality in four dimensions, and quaternionic manifolds. The last of these is followed by a rather brief report on important constructions in the noncompact case. Finally there is a discussion of generalizations of the Einstein condition; interesting though these are, the reader may be left in agreement with the author’s thesis that Einstein metrics are the nicest sort. The book includes an Appendix on Sobolev spaces and elliptic operators, and an Addendum of results on five topics, too recent to make it into the main text. The fact that four of these topics are related to Kähler-Einstein manifolds is indicative of the ever-developing links between complex and Riemannian geometry.

The book under review serves several purposes. It is an efficient reference book for many fundamental techniques of Riemannian geometry. On the other hand, despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of the interaction of geometry with partial differential equations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title.

Reviewed by S. M. Salamon

**MR0480350** **(81d:53045)**

Yau, Shing Tung

On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I.

*Comm. Pure Appl. Math.* **31 **(1978), no. 3, 339–411.

53C55 (32C10 35J60)

This remarkable paper establishes several related results which are of fundamental importance in the study of complex manifolds. These results, the main points of which were conjectured more than twenty years ago by E. Calabi [Proceedings of the International Congress of Mathematicians (Amsterdam, 1954), Vol. 2, pp. 206–207, Noordhoff, Groningen, 1954], have to do with the existence of Kähler metrics with certain special properties on compact Kähler manifolds. The results are established by reducing them to questions in nonlinear partial differential equations of Monge-Ampère type which are then treated by a continuity method involving difficult a priori estimates. The methods and results on partial differential equations have great interest in themselves in addition to their importance as the mechanism of establishing the Calabi conjectures and some similar but more general geometric results.

To discuss the contents of the paper more specifically: Let $M$ be a compact Kähler manifold with Kähler metric $g$. Then it is a well-known result of S. S. Chern that the Ricci form (= in local coordinate representation

$$ (-\surd(-1)/(2\pi))\sum_{i,j}\left(\partial^2[\log\text{}\det(g_{s\overline t})]/\partial z^i\partial z^{\overline j}\right)dz^i\wedge d\overline z^j=\\ (\surd(-1)/(2\pi))\sum_{i,j}R_{i\overline j}dz^i\wedge d\overline z^j, $$ where$\sum_{i,j}R^{i\overline j}dz^i\otimes d\overline z^j$ is the Ricci tensor) is a closed real $(1,1)$ form whose cohomology class is the first Chern class of $M$. In particular, this cohomology class depends only on the complex structure of $M$ and not on the choice of Kähler metric $g$. The conjecture [Calabi, op. cit.], which is proved in the paper under review, was that the converse held: Every closed real $(1,1)$ form whose cohomology class is the first Chern class of $M$ is the Ricci form of some Kähler metric on $M$. More precisely, Calabi conjectured that there is a $C^\infty$ function $\varphi\colon M\rightarrow{\bf R}$ such that $\tilde g={}_{\text{df}}\sum_{i,j}(g_{i\overline j}+\partial^2\varphi/\partial z^i\partial\overline z^j)dz_i\otimes d\overline z_j$ is a positive definite Kähler metric the Ricci form of which is the given form; he proved that at most one metric of this form with given Ricci form exists, and that in fact $\varphi$ is uniquely determined up to an additive constant by the Ricci form of the metric $\tilde g$ [Calabi, Algebraic geometry and topology, pp. 78–89, Princeton Univ. Press, Princeton, N.J., 1957; MR0085583 (19,62b)]. A second conjecture of Calabi [op. cit., 1954] concerned the existence of Einstein-Kähler metrics. A Kähler metric $g$ is by definition an Einstein-Kähler metric if its Ricci form is a multiple at each point of the Kähler form $\omega$ of the metric $g$. This condition is of course automatic by type considerations in complex dimension one. In complex dimension two or higher, it is a standard result that the multiple must be constant, so that by replacing $g$ by a positive constant multiple it is sufficient to consider the cases $R=\pm\omega$ or $R\equiv 0$. The case $R\equiv 0$ is a special case of the first conjecture: that there is a Kähler metric with Ricci form identically zero if (and only if) the first Chern class of the manifold is zero. If $R$ is to be $-\omega$, then the first Chern class of $M$ must be negative or, equivalently, the canonical bundle of $M$ must be ample. The present paper establishes the existence of an Einstein-Kähler metric with $R=-\omega$ on any compact complex manifold with ample canonical bundle. (A theorem of Nakano that negativity of the first Chern class implies nonexistence of nontrivial holomorphic vector fields relates this result to the original form of Calabi’s conjecture for this case.) This metric is canonical in the sense that it is uniquely determined by the complex structure of $M$. (Some results which are related to the Calabi conjectures and to which reference is made in the present article are given in the works of T. Aubin; these and their relationship to the present paper are discussed in the review of Aubin’s paper [53047 below].)

The proofs of these results are far too lengthy and intricate to treat in any detail in a review. But it is possible to indicate how the questions are transformed to nonlinear partial differential equations problems: As before, let $M$ be a compact Kähler manifold with Kähler metric $g$ and let$(\surd(-1)/(2\pi))\sum_{i,j}R_{i\overline j}dz_i\wedge d\overline z_j$ be the Ricci form of $g$. Suppose that $(\surd(-1)/(2\pi))\sum_{i,j}\tilde R_{i\overline j}dz_i\wedge d\overline z_j$ is a closed $(1,1)$ form which also represents the first Chern class of $M$. By standard results, there is then a (smooth) function $F$ on$M$ such that $\tilde R_{i\overline j}-R_{i\overline j}=-\partial^2F/\partial z^id\overline z^j$. If $\sum\tilde R_{i\overline j}dz_i\otimes d\overline z_j$ is in fact the Ricci tensor of a Kähler metric $\tilde g$ then $\tilde R_{i\overline j}=-(\partial^2/\partial z^i\partial\overline z^j)\log(\det\tilde g_{s\overline t})$. From the maximum principle one sees that $\det(\tilde g_{s\overline t})=C(\exp F)\det(g_{s\overline t})$, where $C$ is a constant. To make a unique choice of $\tilde g$, one seeks $\tilde g$ in the form $\tilde g_{i\overline j}=g_{i\overline j}+(\partial^2\varphi/\partial z^i\partial\overline z^j)$. Then the problem is reduced to solving the equation for $\varphi\colon\det(g_{i\overline j}+\partial^2\varphi/\partial z^i\partial\overline z^j)=C(\exp F)\det(g_{st})$, where the constant $C=\text{Vol}(M)/\int_M\exp F$. The solution of this is unique [Calabi, op. cit., 1957]. It is natural in this situation to attempt the solution of the equation by the continuity method, where the family of equations to which the method is to be applied is obtained by varying $F$. Specifically, fix $k\geq 3$ and $\alpha\in(0,1]$. Let $S=$ {$t\in[0,1]$: the equation $\det(g_{i\overline j}+(\partial^2\varphi)/(\partial z^i\partial\overline z^j))\det(g_{i\overline j})^{-1}=\text{Vol}(M)[\int\exp\{tF\}]^{-1}\exp\{tF\}$ has a solution in $C^{k+1,\alpha}(M)$}. Then clearly $0\in S$ and if $1\in S$ then one has solved the equation originally considered. So now it suffices to show that $S$ is open and closed in $[0,1]$. That $S$ is open can be established by applying the Banach space version of the implicit function theorem (because of ellipticity, there are no technical difficulties involving derivative loss). This step was carried out by Calabi. To see that $S$ is closed, one tries to find a solution for the value $t_0\in\overline S$ by taking the limit of a (sub)sequence of the solutions $\varphi_{t_i}$ of the equations for $t_i\in S$ with $t_i\rightarrow t_0$, where for normalization one assumes $\int_M\varphi_{t_i}=0$. (This normalization actually makes each $\varphi_{t_i}$ uniquely determined.) To be sure that such a limit exists, one must obtain a priori estimates on the $\varphi_{t_i}$ and their derivatives. To obtain estimates that are sufficient to allow the Schauder theory to apply (i.e. estimates up to third order derivatives) is difficult. The third derivative estimates can be obtained (in the presence of the lower order estimates) by a process similar to that used by Calabi in a slightly different context (the real Monge-Ampère equation: [Calabi, Michigan Math. J. 5 (1958), 105–126; MR0106487 (21 #5219)]). But the obtaining of the estimates up to second order involves an intricate, new procedure. Once these estimates are obtained, the Schauder theory allows the proof to be completed by the continuity method.

Similar considerations imply the existence of Einstein-Kähler metrics on any compact complex manifold $M$ with ample canonical bundle (such a manifold is automatically projective algebraic and hence a Kähler manifold). Indeed by hypothesis there is a closed positive $(1,1)$ form $\surd(-1)\sum_{i,j}g_{i\overline j}dz^i\wedge d\overline z^j$, which represents the negative of the first Chern class of $M$. This form is the Kähler form of a Kähler metric $g$. By a theorem of Chern as before, the first Chern class of $M$ is also represented by $-\partial\overline\partial\log\text{}\det(g_{i\overline j})$ (= the Ricci form of $g$). So there is a smooth function $f$ with $\partial\overline\partial\log\text{}\det(g_{i\overline j})=\surd(-1)\sum g_{i\overline j}dz^i\wedge d\overline z^j+\partial\overline\partial f$. One seeks a function $\varphi$ such that $\tilde g=\sum(g_{i\overline j}+(\partial^2\varphi/\partial z^i\partial\overline z^j))dz_i\otimes dz_j$ is a Kähler metric and such that $\det(g_{i\overline j}+\partial^2\varphi/\partial z^i\partial\overline z^j)=\exp(\varphi-f)\det(g_{i\overline j})$. A brief calculation then shows that the Ricci tensor of $\tilde g$ is $-\tilde g$. To find $\varphi$, one again uses the continuity method to solve the partial differential equation given. (Actually, a somewhat more general equation is solved.) The metric $\tilde g$ is shown to be uniquely determined; i.e., it depends only on the complex structure of $M$.

Numerous technical refinements of the results stated, some of them of immediate importance in applications to geometry, are given. Detailed discussion of these refinements must be omitted here. Although the author does not discuss applications in this paper, it would be inappropriate to conclude this review without noting that not only are the results given in this article of great interest in themselves but also they have numerous important applications in geometry. These are discussed in part in a previous article of the author [Proc. Nat. Acad. Sci. 74 (1977), no. 5, 1798–1799; MR0451180 (56 #9467)]. They include: the existence of Ricci flat but nonflat compact manifolds, the uniqueness of complex structure on complex projective 2-space, and the proof that any complex surface that is oriented homotopic to a surface covered by the ball is biholomorphic to the latter surface. The results of the presently reviewed article represent a major advance in differential and transcendental algebraic geometry. These results and the methods by which they are obtained will no doubt become a permanent part of the investigative techniques of these areas.

{This review was received in April 1979.}

Reviewed by Robert E. Greene