The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)
Chapter 5. PROVING CALABI
A mathematical proof is a bit like climbing a mountain. The first stage, of course, is discovering a mountain worth climbing. Imagine a remote wilderness area yet to be explored. It takes some wit just to find such an area, let alone to know whether something worthwhile might be found there. The mountaineer then devises a strategy for getting to the top—a plan that appears flawless, at least on paper. After acquiring the necessary tools and equipment, as well as mastering the necessary skills, the adventurer mounts an ascent, only to be stopped by unexpected difficulties. But others follow in their predecessor’s footsteps, using the successful strategies, while also pursuing different avenues—thereby reaching new heights in the process. Finally someone comes along who not only has a good plan of attack that avoids the pitfalls of the past but also has the fortitude and determination to reach the summit, perhaps planting a flag there to mark his or her presence. The risks to life and limb are not so great in math, and the adventure may not be so apparent to the outsider. And at the end of a long proof, the scholar does not plant a flag. He or she types in a period. Or a footnote. Or a technical appendix. Nevertheless, in our field there are thrills as well as perils to be had in the pursuit, and success still rewards those of us who’ve gained new views into nature’s hidden recesses.
Eugenio Calabi found his mountain some decades earlier, but in the early 1970s, I (among many others) still needed some convincing that it was more than a molehill. I didn’t buy the provocative statement he’d put before us. As I saw it, there were a number of reasons to be skeptical. For starters, people were doubtful that a nontrivial Ricci-flat metric—one that excludes the flat torus—could exist on a compact manifold without a boundary. We didn’t know of a single example, yet here was this guy Calabi saying it was true for a large, and possibly infinite, class of manifolds.
Then there was the fact that Calabi was, as Robert Greene puts it, taking a very general topological condition and using it to find a very specific geometric corollary that was uniform over the whole space. That doesn’t happen in real manifolds that are lacking in complex structure, but the conjecture suggested that with complex manifolds, it does.1 To elaborate on Greene’s point, the Calabi conjecture basically says, starting with the case of one complex dimension (and two real dimensions), that if you have a general topology or shape where the average curvature is zero, then you can find a metric, or geometry, where the curvature is zero everywhere. In higher dimensions, the conjecture specifically refers to Ricci curvature (which is the same as Gauss curvature in real dimension two but is different in dimensions above two), and the condition of the average Ricci curvature being zero is replaced by the condition of the first Chern class being zero. Calabi asserts that if this topological condition of a zero first Chern class is met, then there exists a Kähler metric with zero Ricci curvature. You’ve thus replaced a rather broad, nonspecific statement with a much stronger, more restrictive statement, which is why Greene (and practically everyone else) regarded the proposition as surprising.
I was also wary for some additional technical reasons. It was widely held that no one could ever write down a precise solution to the Calabi conjecture, except perhaps in a small number of special cases. If that supposition were correct—and it was eventually proven to be so—the situation thus seemed hopeless, which is another reason the whole proposition was deemed too good to be true.
There’s a comparison to number theory we can make. While there are many numbers we can write down in a straightforward way, there is a much larger class of numbers that we can never write down explicitly. These numbers, called transcendental, include e (2.718 . . . ) and π (3.1415 . . . ), which we can write out to a trillion digits or more but can never write out in full. In technical terms, this is because the numbers cannot be constructed by algebraic manipulation, nor can they be the solution to a polynomial equation whose coefficients are rational numbers. They can only be defined by certain rules, which means they can be narrowed down to a large degree without ever being spelled out verbatim.
Nonlinear equations, such as those pertaining to the Calabi conjecture, are similar. The solution to a nonlinear equation is itself a function. We don’t expect to be able to solve them in a clean, explicit manner—as in writing down an exact formula for the solution—because that’s simply not possible in most cases. We try to approximate them with the functions we know very well: some polynomial functions, trigonometric functions (such as sine, cosine, and tangent curves), and a few others. If we cannot approximate them with functions we know how to handle, we have trouble.
With that as a backdrop, I tried to find counterexamples to the Calabi conjecture in my spare time. There were moments of excitement: I’d find a line of attack that seemed likely to disprove the conjecture, only to later discover a flaw in my ostensibly impeccable constructions. This happened repeatedly. In 1973, inspiration seized me. This time, I felt I really had something. The approach I took, which we call proof by contradiction, is similar to the approach Richard Schoen and I used for the positive mass conjecture proof. As far as I could tell, my argument was airtight.
Coincidentally, the idea came to me during an international geometry conference held at Stanford in 1973—the same conference at which Geroch spoke about the positive mass conjecture. As a general matter, conferences are a great way of keeping abreast of developments in your field and in fields outside your own specialty, and this one was no exception. They’re a terrific venue for exchanging ideas with colleagues you don’t see every day. Nevertheless, it’s rare for a single conference to change the course of your career. Twice.
During the meeting, I casually mentioned to some colleagues that I might have found a way to disprove Calabi, once and for all. After some prompting, I agreed to discuss my idea informally one night after dinner, even though I was already scheduled to give several other formal talks. Twenty or so people showed up for my presentation, and the atmosphere was charged. After I made my case, everyone seemed to agree that the reasoning was sound. Calabi was there, and he raised no objections, either. I was told that by virtue of this work, I’d made a big contribution to the conference, and afterward, I felt quite proud of myself.
Calabi contacted me a few months later, asking me to write down the argument, as he was puzzled over certain aspects of it. I then set out to prove, in a more rigorous way, that the conjecture was false. Upon receiving Calabi’s note, I felt that the pressure was on me to back up my bold assertion. I worked very hard and barely slept for two weeks, pushing myself to the brink of exhaustion. Each time I thought I’d nailed the proof, my argument broke down at the last second, always in an exceedingly frustrating manner. After those two weeks of agony, I decided there must be something wrong with my reasoning. My only recourse was to give up and try working in the opposite direction. I had concluded, in other words, that the Calabi conjecture must be right, which put me in a curious position: After trying so hard to prove that the conjecture was false, I then had to prove that it was true. And if the conjecture were true, all the stuff that went with it—all the stuff that was supposedly too good to be true—must also be true.
Proving the Calabi conjecture meant proving the existence of a Ricci-flat metric, which meant solving partial differential equations. Not just any partial differential equations, but highly nonlinear ones of a certain type: complex Monge-Ampère equations.
Monge-Ampère equations are named after the French mathematician Gaspard Monge, who started studying equations of this sort around the time of the French Revolution, and the French physicist and mathematician André-Marie Ampère, who continued this work a few decades later. These are not easy equations to work with.
Perhaps the simplest example of one that we can draw from the real world, Calabi suggests, concerns a flat plastic sheet affixed to an immovable rim. Suppose the surface either stretches or shrinks. The question is, when that happens, how does the surface bend or otherwise change from being totally flat? If the middle expands, it will create a bulge extending upward with positive curvature, and the solution to the Monge-Ampère equation will be of an elliptic form. Conversely, if the interior shrinks instead, and the surface becomes a saddle with negative curvature everywhere, the solution will be of a hyperbolic form. Finally, if the curvature is zero everywhere, the solution will be of a parabolic form. In each case, the original Monge-Ampère equation you want to solve is the same, but, as Calabi explains, “they must be solved by completely different techniques.”2
Of the three forms of differential equations, we have the best techniques for analyzing the elliptic type. Elliptic equations pertain to simpler, stationary situations where things do not move around in time or space. These equations model physical systems that are no longer changing with respect to time, such as a drum that has stopped vibrating and returned to equilibrium. The solutions to elliptic equations, moreover, are the easiest of the three to understand because when you graph them as functions, they look smooth, and you rarely encounter troublesome singularities, though singularities can crop up in the solutions to some nonlinear elliptic equations.
Hyperbolic differential equations model things like waves or vibrations, which may never reach equilibrium. The solutions to these equations, unlike the elliptic ones, generally have singularities and are therefore much harder to work with. While we can handle linear versions of hyperbolic equations reasonably well—in cases where changing one variable leads to a proportional change in another—we don’t really have effective tools for handling nonlinear hyperbolic equation’s in a way that controls the singularities.
Parabolic equations lie somewhere in between. They model a stable physical system—such as a vibrating drum—that will eventually reach equilibrium but may not have gotten there yet, which brings time evolution or change into the picture. These equations are less prone to singularities than hyperbolic ones, and any singularities there are more readily smoothed out, which again places them somewhere between elliptic and hyperbolic in terms of difficulty.
And there are even tougher mathematical challenges to be had. Although the simplest Monge-Ampère equations have only two variables, many have more. These equations are beyond hyperbolic (and are sometimes called ultrahyperbolic), the solutions of which we know even less about. As Calabi puts it, “we have no clue about these other solutions that go beyond the familiar three because we have absolutely no physical picture to draw on.”3 As a result of the varying difficulty of the three types of equations, most of the contributions from geometric analysis to date have involved either elliptic or parabolic equations. We’re interested in all these equations, of course, and there are plenty of exciting problems that involve hyperbolic equations (such as the full Einstein equations) that we’d like to address, if only we had the wherewithal.
The equations of the Calabi conjecture, fortunately, were of the nonlinear, elliptic variety. That’s because, even though it is related to the Einstein equation, which is itself hyperbolic, the Calabi conjecture is based on a somewhat different geometric framework. In this case, we assume that time is frozen, like the scene in Sleeping Beauty where nobody moves for a hundred years. That placed the conjecture in the somewhat simpler elliptic category, with time taken out of the equation. For this reason, I felt hopeful that the tools of geometric analysis—including some of those we’ve touched on so far—might be successfully brought to bear on the problem.
Even with those tools at my disposal, there was still a good deal of preparatory work to be done. Part of the challenge was that no one had ever solved a complex Monge-Ampère equation before except in one dimension. Just as a mountain climber constantly strives to reach higher elevations, I was going to need to push to higher dimensions. To gird myself for higher-dimensional Monge-Ampère equations (whose nonlinearity goes without saying), I set out with my friend S. Y. Cheng to tackle some higher-dimensional cases, starting with problems expressed in real numbers before getting to the more difficult, complex equations.
First we took on a famous problem posed by Hermann Minkowski around the turn of the twentieth century. The Minkowski problem involves taking a prescribed set of information and then determining whether a structure that meets those criteria actually exists. Take a simple polyhedron. Upon examining such a structure, you could characterize it by counting the numbers of faces and edges and measuring their dimensions. The Minkowski problem is like the flip side of that: If you are told the shape, area, number, and orientation of the faces, can you determine whether a polyhedron meeting those criteria actually exists and, if so, whether such an object is unique?
The problem is actually more general than that, because it could apply to an arbitrary convex surface rather than just a polyhedron. Instead of talking about which way the faces are pointing, you could specify the curvature in terms of perpendicular (or “normal”) vectors at each spot on the surface, which is equivalent to describing which way the surface is pointing. You could then ask whether an object with the specified curvature could exist. The useful thing is that this problem needn’t only be presented in purely geometric terms. It can also be written as a partial differential equation. “If you can solve the geometric problem,” says Erwin Lutwak of the Polytechnic Institute of New York University (NYU), “there is this huge bonus: You’re also solving this horrific partial differential equation. That interplay between geometry and partial differential equations is one of the things that makes this problem so important.”4
5.1—The mathematician S. Y. Cheng (Photo by George M. Bergman)
Cheng and I found a way to solve the problem, and our paper on the subject was published in 1976. (As it turned out, another independent solution had been presented several years earlier in a 1971 paper by the Russian mathematician Aleksei Pogorelov. Cheng and I had not seen that paper, because it was published in Pogorelov’s native tongue.) In the end, it came down to solving a complex nonlinear partial differential equation of the sort that had never been solved before.
Even though no one had previously managed to solve a problem of this exact type (except for Pogorelov, whose work we weren’t aware of), there was a well-prescribed procedure for attacking nonlinear partial differential equations. The approach, called the continuity method, is based on making a series of estimates. Although the general approach was by no means new, the trick, as always, lies in coming up with a strategy specifically tailored to the problem at hand. The basic idea is to try to successively approximate a solution through a process that keeps yielding better and better results. The essence of the proof lies in showing that this process will, with a sufficient number of iterations, eventually converge on a good solution. If everything works out, what you’re left with in the end is not the solution to an equation that you can write down as an exact formula but merely a proof that a solution to this equation exists. In the case of the Calabi conjecture and other problems of this sort, showing that a solution can be found to a partial differential equation is equivalent to an existence proof in geometry—showing that for a given “topological” condition, a specified geometry is indeed viable. This is not to say that you’d know nothing about the solution whose existence you’ve just proved. That’s because the scheme you used to prove the existence of a solution can often be turned into a numerical technique for approximating the solution on a computer. (We’ll talk about numerical techniques in Chapter 9.)
The continuity method is so named because it involves taking the solution to an equation you know how to solve and continuously deforming it until you arrive at a solution to the equation you want to solve. This procedure, which was used in the Calabi conjecture proof, is typically broken down into two parts, one of which only works in the immediate vicinity of the known solution.
One of these parts we’ll call Newton’s method, because it’s roughly based on a technique Isaac Newton invented more than three hundred years ago. To see how Newton’s method works, let’s take a function, y = x3 - 3x + 1, which describes a curve that intersects the x-axis at three points (or roots). Newton’s approach enables us to find where those three roots lie, which is not obvious just by looking at the equation. Let’s assume we can’t solve the equation outright, so we guess that one root can be found at a point called x1. If we take a tangent to the curve at that point, the tangent line will intersect the x-axis at another point, which we’ll call x2, that is even closer to the root. If we then take a tangent to the curve at x2, it will intersect the x-axis closer still to the root at x3. The process will converge quickly on the root itself if the initial guess is not too far off the mark.
5.2—An introduction to Newton’s method. To find out where a particular curve, or function, crosses the X-axis, we start off with our best guess—a point we’ll call x0. We then take the tangent of the curve at x0 and see where the tangent line crosses the X-axis (at a point we’ll call x1). We continue this process at x, and so forth. Assuming our initial guess was not too far off, we’ll come closer and closer to the true answer, point x.
As another example, let’s suppose we have a whole series of equations, Et, only one of which, E0 (pertaining to the case when t = 0), we know how to solve. But the equation we really want to solve is E1, which concerns the case when t = 1. We could use Newton’s method when we’re very close to 0, where we know the solution, but this approach may not carry us all the way to 1. In that event, we’ll need to bring in another, more broadly applicable estimation technique.
So how do we do that? Imagine somebody shot a missile over the Pacific Ocean that landed within a hundred-mile radius of the Bikini Atoll. That gives us some idea of where the missile could be—its general position, in other words—but we’d like to know more, such as its velocity or its acceleration over the course of its flight or how its acceleration changed. We’d do this with calculus, taking the first, second, and third derivatives of an equation describing the rocket’s location. (We could take even more derivatives, but for second-order elliptic equations of the type I was interested in, the third derivative is usually high enough.)
Just getting these derivatives is not enough, even though this alone may be extremely challenging. We also have to be able to “control” them. And by that I mean putting bounds on them—making sure they cannot get too big or too small. Making sure, in other words, that the solutions we get are “stable” and do not blow up on us, in which case they’d disqualify themselves as solutions, dashing our dreams in the process. So we start with the zeroth derivative, the location of the missile, and see if we can set some upper and lower bounds—make estimates, in other words, that suggest that an answer is at least possible. We’d then do the same for each higher-order derivative, making sure that they are not too big or too small and that the functions describing them do not fluctuate too wildly. This involves making a priori estimates for the velocity, acceleration, change in acceleration, and so forth. If we can establish control in this way, from the zeroth through the third derivative, normally we have good control over the equation as a whole and have a decent chance of solving it. This process of estimation, and proving that the estimates themselves can be controlled, is typically the hardest part of the whole process.
So in the end, it all boils down to estimates. There was something ironic in my recognizing their relevance to the problem I faced. When I first entered graduate school, I remember encountering two Italian postdocs in the halls of the Berkeley math department. They were jumping up and down, screaming jubilantly. I asked them what had happened and was told they’d just obtained an estimate. When I asked them what an estimate was, they acted as if I were an ignoramus who had no business setting foot in the building. That’s when I decided I might want to learn about a priori estimates. Calabi received a similar dressing-down some decades earlier from his friend and collaborator Louis Nirenberg: “Repeat after me,” Nirenberg said, “you cannot solve partial differential equations without a priori estimates!”5 And in the early 1950s, Calabi had written to André Weil about his conjecture. Weil, who thought the “technology” of that era was simply not mature enough to yield a solution, asked him, “How are you going to get the estimates?”6
Two decades later, which is when I entered the game, the problem itself had not changed. It was still incredibly difficult, yet the tools had advanced to the point that a solution was within the realm of possibility. It was a question of figuring out a line of attack or at least establishing some sort of foothold. So I picked an easier equation and then sought to show that the solution to it could eventually be “deformed” to the solution of the harder equation.
Suppose you want to solve the equation f(x) = x2 - x when f(x) = 0. For starters, we could try x = 2, which doesn’t quite work; f(2) = 2, not 0. Nevertheless, I now have a solution, if not to the original equation then to something similar. So next I rewrite the equation as f(x) = 2t. I know how to solve it when t = 1, but I’d really like to solve it for t = 0, which is the equation we started with in the first place. So what do I do? I look at the parameter t. I know how to solve the equation for t = 1, but what happens when I move t a bit so that it’s not exactly 1 but is still close? I then guess that when t is close to 1, there exists a solution to f(t) that is close to 2. That assumption turns out to be true in most cases, which means that when t is close to 1, I can solve the equation.
Now I want to make t smaller and smaller and approach 0 so that we can get to our original equation. I keep on making t smaller and smaller, and for each value I pick, I can find a solution. So I have a sequence of points where I can solve the equation, and each of these points corresponds to a different value of x, which I’ll collectively call xi. The whole point of this exercise is to prove that the sequence xi converges on a particular value. To do that, we must show that xi is bounded and cannot move to infinity, because for any bounded sequence, at least some part of it must converge. Showing that xi converges is equivalent to showing that we can reduce the value of t all the way down to 0 without running into any insurmountable difficulties. And if we can do that, we’ll have solved the equation by showing that t = 0 can be solved. In other words, we’ve shown that a solution to our original function, x2 - x = 0, must exist.
This is exactly the kind of argument I used in the Calabi proof. A key part of that proof, in fact, was showing that xi is a convergent sequence. Of course, in tackling the Calabi conjecture, the starting equation is obviously more complicated than x2 - x = 0. In the Calabi case, x is a function rather than a number, which increases the complexity immensely because the convergence of a function is normally a very tough thing to prove.
Again, we break the big problem into smaller pieces. The equation of the Calabi conjecture is a second-order elliptic equation, and to solve equations like this, we need to work out the zeroth-, first-, second-, and third-order estimates. Once those estimates are completed, and you can prove they converge to the desired solution, you’ve proved the entire conjecture. That’s easier said than done, of course, because solving those four estimates is not necessarily easy at all. I guess that’s why they call it work.
But there were still more preliminaries to be done before Cheng and I could attack complex Monge-Ampère equations. We started working on the Dirichlet problem, named after the German mathematician Lejeune Dirichlet. It’s what we call a boundary value problem, and tackling such problems is usually the first thing we do when trying to solve an elliptic differential equation. We discussed an example of a boundary value problem in Chapter 3, the Plateau problem, which is often visualized in terms of soap films and which stated that for an arbitrary closed curve, one could find a minimal surface stretching across that same boundary. Every point on that surface is also a solution to a particular differential equation. The question boils down to this: If you know the boundary of the solution to such an equation, can you find the interior surface that not only connects to the boundary but solves the equation as well? While the Calabi conjecture is not a boundary value problem, Cheng and I needed to test the methods, which we could then bring to bear on complex Monge-Ampère equations like Calabi’s. For practice, we were trying to solve the Dirichlet problem in certain domains in complex Euclidean spaces.
To solve the Dirichlet problem, one has to go through the same steps we’ve just outlined, finding the zeroth-, first-, second-, and third-order estimates for the boundary. But we also need to make the same set of estimates for the interior of the curve, as there could be discontinuities, singularities, or other departures from smoothness in the “soap bubble” under consideration. That means eight estimates in all.
By early 1974, Calabi and Nirenberg, who were also working on this Dirichlet problem, had gotten the second-order estimate, as had Cheng and I. The zeroth-order estimate turned out to be relatively easy. And the first-order estimate could be derived from the zeroth and second. That left the third order, wherein lay the key to the whole Dirichlet problem.
The tools required for finding the solution first emerged in the late 1950s, while I was still in grade school, when Calabi solved a major problem in geometry that later proved critical in understanding how to obtain the third-order interior estimate for the real Monge-Ampère equation. Calabi’s contributions to this estimate stemmed, in part, from a coincidence. He had been working on a seemingly unrelated problem in affine geometry (a generalization of Euclidean geometry that, being somewhat far afield, I won’t take the time to explain here), while Nirenberg and Charles Loewner of Stanford were working on the Dirichlet problem of a Monge-Ampère equation with a singular boundary (like the cresting edge of a wave) rather than a smooth boundary. After seeing the equation that Nirenberg and Loewner had studied, Calabi realized it was related to his work on affine geometry. Calabi and Nirenberg were able to figure out how Calabi’s efforts from the 1950s could be applied to the third-order interior estimate problem we were facing in the 1970s. “A lot of mathematical discoveries occur through lucky accidents like that,” Calabi notes. “It’s often a matter of connecting up ideas that might seem unrelated and then exploiting this newfound connection.”7
Later in 1974, Calabi and Nirenberg announced the solution to the boundary value problem for the complex Monge-Ampère equation. But it turns out they had made a mistake: The third-order boundary estimate was still missing.
5.3—The mathematician Louis Nirenberg
Cheng and I soon came up with our own solution for the third-order boundary estimate. Chern had invited us to join him and Nirenberg for dinner, during which we’d present our proof. Nirenberg was a big shot at the time, while we were barely out of graduate school, so we checked our proof carefully the night before the dinner and, to our dismay, found mistakes. We stayed up the entire night fixing those mistakes and thereby fixing the proof as well. We showed it to Nirenberg the next night before dinner. He thought everything looked fine; we thought it looked fine, too, and everyone enjoyed the meal. But after dinner, Cheng and I went through the proof again and found additional mistakes. It wasn’t until late in 1974, about six months later, that Cheng and I solved the boundary value problem. We did it by studying an equation similar to what Loewner and Nirenberg had looked at, only in higher dimensions. The method we used bypassed the third-order boundary estimate, showing why it was unnecessary.
That work done, I was ready to take on the complex case of Calabi itself—a problem that was set on a complex manifold, in contrast to the Dirichlet problem, which was set in complex Euclidean space. So eager was I, in fact, that we didn’t get around to publishing the paper on the Dirichlet problem until 1979—about five years later.
With the Dirichlet problem put to rest, much of the work that lay ahead involved generalizing or translating the estimates for the real Monge-Ampère equation to the complex case. But I was to travel alone from this point on, as Cheng’s interests lay in other directions.
Sometime in 1974, Calabi and Nirenberg, along with J. J. Kohn of Princeton, started to work on the complex version of the Dirichlet problem in Euclidean space. They made some headway on the third-order estimate, and I was able to extend their result for curved space. Later that year, I got some ideas for the second-order estimate of the Calabi conjecture as well, drawing on some work I’d done in 1972 on something called the Schwarz lemma. The lemma, or mini-theorem, dates back to the nineteenth century and originally had nothing to do with geometry until it was reinterpreted by Lars Ahlfors of Harvard in the first half of the twentieth century. Ahlfors’s theorem pertained just to Riemann surfaces (of one complex dimension, by definition), but I generalized his theorem to any complex dimension.
I finished the preliminaries on the second-order estimate for the Calabi conjecture during the summer of 1975. (A year later, I learned that the French mathematician Thierry Aubin had independently arrived at an estimate for the second-order problem.) In addition to determining the second-order estimate, I showed how it depended on the zeroth-order estimate and how one could go from the zeroth to the second. After I’d finished the work on the second-order estimate, I knew that the entire proof now rested on a single question, the zeroth-order problem. Once that was knocked off, I could not only get the second, but the first as well, which was a kind of freebie. For when we had the zeroth- and second-order estimates, we’d know exactly how to do the first-order estimate. It was one of those lucky breaks, a matter of how the cookie crumbles, and this time, it crumbled the right way. Not only that, but the third-order estimate also depended on the zeroth and second, so in the end, it all came down to the zeroth-order estimate. With that in hand, everything else would fall into place. Without it, I would have nothing.
I had done this latest work at NYU’s Courant Institute, securing a visiting position through the help of Nirenberg. Before long, my fiancé Yu-Yun, who had been working at Princeton, had been offered a job in Los Angeles. I took another visiting faculty position at UCLA to be with her. In 1976, we drove across country together, planning to get married when we got to California. (And we did, in fact, get married in California.) It was a memorable trip: We were in love. The scenery was beautiful. And we shared our visions of a future life together. But I must confess to being more than a bit distracted: I still had the Calabi conjecture on my mind and the zeroth-order estimate in particular, which was proving to be most obstinate. I spent the entire year working on it. Finally, in September 1976, right after our wedding, I found the zeroth-order estimate, and the rest of the proof fell into place with it. Married life appears to have been just what I needed.
The problem of getting the zeroth-order estimate was similar to getting the other estimates: You have an equation or a function that you’d like to put some constraints on—both from above and from below. In other words, you want to put the function in a metaphorical box and show that the box doesn’t have to be too big for the function to “fit in.” If you can do that, you’ve bounded the function from above. At the other end, you need to show that the function cannot be so small that it will somehow “leak out” of the box, thereby bounding it from below.
One way to approach a problem of this sort is to take the absolute value of the function, which tells you the function’s overall magnitude—how big it gets in either the negative or the positive direction. In order to control a function, u, we just have to show that its absolute value at any point in space is less than or equal to a constant, c. Since c is a well-defined number, we’ve shown that u cannot be arbitrarily big or small. What we want to prove, in other words, is a simple inequality, that the absolute value of u was less than or equal to c: |u| ≤ c. While that may not look so hard, it can be challenging when u happens to be a particularly messy object.
I won’t go through the details of the proof, but I will say that it drew on the second-order Monge-Ampère problem, which I’d already solved. I also used a famous inequality by Poincaré as well as an inequality by the Russian mathematician Sergei Sobolev. Both inequalities involved integrals and derivatives (of various orders) of the absolute value of u taken to a certain power or exponent. The last part, involving exponents, is crucial for making estimates, because if you can show that u, in various forms, taken to the pth power is neither too big nor too small—even when p is a very large number—you’ve done your work. You’ve stabilized the function. In the end—by using these inequalities and various theorems, as well as some lemmas I’d developed along the way—that’s what I was able to do. With the zeroth-order estimate in hand, the case was closed, or so it appeared.
Of course, people often say the proof of the pudding is in the eating, which presumably means that even when something looks good, you don’t know for sure until you put it to a test. And this time, with regard to this proof, I wasn’t taking any chances. I had already embarrassed myself once in 1973, when I’d claimed at Stanford, in a highly public forum, that I knew how to disprove the Calabi conjecture. My alleged proof of the negative case had been flawed, and if my proof of the affirmative case were similarly flawed, I would not be doing much to bolster my reputation. In fact, I was convinced that at this stage of my career, still in my twenties, I couldn’t afford to be wrong again—or at least not wrong on something so important and high-profile.
So I checked it and rechecked it, going through the proof four times in four different ways. I checked it so many times that I vowed that if I was wrong about this, I’d give up mathematics altogether. But try as I might, I couldn’t find anything wrong with my argument. As far as I could tell, it all added up. Since in those days, there was no World Wide Web where I could just post a draft of a paper and solicit comments, I did it the old-fashioned way: I mailed a copy of my proof to Calabi and then went to Philadelphia to discuss it with him and other geometers at the University of Pennsylvania math department, including Jerry Kazdan.
Calabi thought the proof appeared solid, but we decided to meet with Nirenberg and go through it together, step by step. As it was hard to find a time when all three of us were free, we met on Christmas Day 1976, when none of us had any other pressing business engagements. No flaws were uncovered during that meeting, but it would take more time to be sure. “Tentatively, it looked good,” Calabi recalls. “Still it was a difficult proof that would take about a month to check thoroughly.”8
By the end of that review period, both Calabi and Nirenberg had signed off. It appeared that the Calabi conjecture had been proved, and in the thirty-plus years that have transpired since, nothing has come up to alter that judgment. By now, the proof has been cross-examined so many times by so many people it’s virtually impossible to imagine that a significant defect will ever be exposed.
So what exactly had I accomplished? In proving the conjecture, I had validated my conviction that important mathematical problems could be solved by combining nonlinear partial differential equations with geometry. More specifically, I had proved that a Ricci-flat metric can be found for compact Kähler spaces with a vanishing first Chern class, even though I could not produce a precise formula for the metric itself. All I could say was that the metric was there without being able to say exactly what it was.
Although that might not sound like much, the metric I proved to be “there” turned out to be pretty magical. For as a consequence of the proof, I had confirmed the existence of many fantastic, multidimensional shapes (now called Calabi-Yau spaces) that satisfy the Einstein equation in the case where matter is absent. I had produced not just a solution to the Einstein equation, but also the largest class of solutions to that equation that we know of.
I also showed that by changing the topology continuously, one could produce an infinite class of solutions to the key equation of the Calabi conjecture (now called the Calabi-Yau equation), which is itself a special case of the Einstein equation. The solutions to this equation were themselves topological spaces, and the power of the proof lay in the fact that it was completely general. In other words, I had proved the existence of not just one example of such spaces, or a special case, but a broad class of examples. I showed, moreover, that when you fix certain topological data—such as some complex submanifolds within the bigger manifold—there is only one possible solution.
Before my proof, the only known compact spaces that satisfied the requirements set down by Einstein were locally homogeneous, meaning that any two points near each other will look the same. But the spaces I identified were both inhomogeneous and asymmetrical—or at least lacking in a sweeping global symmetry, though they were endowed with the less visible internal symmetry we discussed in the last chapter. To me, that was like overcoming a great barrier, because once you go beyond the strictures of global symmetry, you open up all kinds of possibilities, making the world more interesting and more confusing at the same time.
At first, I just reveled in the beauty of these intricate spaces and the beauty of the curvature itself, without thinking about possible applications. But before long, there would be many applications, both inside and outside mathematics. While we once believed that Calabi’s proposition was “too good to be true,” we’d just found out that it was even better.