Some Personal Recollections about Yvonne

I met Yvonne on my first visit to Paris after finishing my PhD, probably in the summer of 1980. As she describes herself in her “Mémoires”, I came to her office to ask her opinion whether my recent global existence result for quasilinear wave equations in space dimensions ≥ 6 could be of physical interest. This was mostly a pretext to meet her; her name loomed large in my imagination in view of her foundational results in General Relativity (GR), especially her famous local existence result for the Einstein vacuum equations (EVE). That result was in fact based on the observation that EVE reduces, in wave (or harmonic) coordinates, to a system of quasilinear hyperbolic equations, similar to the type I have treated in my work. Though ignorant of GR when I first wrote my PhD thesis, I became interested in the subject during my stay in Berkeley as a postdoc, largely due to the influence of S. T. Yau who made me aware that my work could be relevant to the problem of the nonlinear stability of the Minkowski space.

My result had however a serious defect, it was restricted to high space dimensions, and as such it was inapplicable, in principle, to the physical case of dimension n = 3. Yvonne, though, liked my result to the point that she could not resist the competitive temptation to point out that in such high dimensions she could have also proved a similar result. This was true, of course, but for me that remark was actually flattering. More importantly she mentioned to me that Einstein conjectured¹, at the time of her visit to the IAS in 1951-1952, that the Minkowski space is non-linearly unstable in the physical case of space dimension 3. I was not too bothered by this crucial, negative, piece of information, since, at that time, I would have been happy to improve my result to all dimensions ≥ 4. Interestingly, as I learned later, in one of her less known papers² Yvonne seemed to confirm Einstein’s conjecture by identifying a possible obstruction in the case of a simple nonlinear wave equation³ in dimension 3. This short paper played a very important psychological role in my later work with D. Christodolou on the nonlinear stability of the Minkowski space, as it meant that we had to avoid wave coordinates and rely instead on a more covariant approach.

During the same visit Yvonne told me that her brilliant young collaborator Demetrios Christodoulou was also interested in the stability of Minkowski space and mentioned to me that he planned to visit the Courant Institute on a Humboldt grant.  This was great news to me as I was going to go to Courant too, in the fall of 1980, as an assistant professor.  In fact I did actually meet Demetrios earlier, during a short visit by him at Berkeley, where I spent two years (1978-1980) as a postdoc. Demetrios told me at the time that he was trying to prove the stability of Minkowski space using the conformal compactification method. This very clever method, first introduce by Penrose as a heuristic tool to derive the decay rates of the gravitational field at null infinity, had been rigorously used by him, in collaboration with Yvonne, to prove the first global existence result for Yang-Mills-Higgs type equations. It turned out that the method was not quite applicable⁴ to the Einstein equations, because of the non-triviality of the ADM mass, yet the method turned out to be very influential in many other situations.

Demetrios’ visit at the Courant Institute (1981-1983) started indeed a very fruitful collaboration that led ultimately to a proof of the global nonlinear stability of the Minkowski space in space dimension 3 disproving, among other things, Einstein’s conjecture.

It is hard to overstate the importance of Yvonne’s work. She, together with a very select few others such as Leray and Lichnerowicz, was among the first mathematicians in the world to concentrate their attention to the great mathematical challenges of the general theory of relativity. At the time when she wrote her famous Acta Matematica paper ”Théorème d’existence pour certains systèmes d’équations aux derivées partielles non linéaires,” it was still debatable in the physics community if the Einstein field equations were hyperbolic. Her, work which treats the initial value problem for the Einstein field equations in full generality, can be rightly considered as the starting point of the systematic mathematical study of the problem of evolution in GR. A later extension of her local (in time) existence result, obtained in collaboration with R. Geroch, associates to any initial data set a unique maximal, future, global hyperbolic development, a crucial concept which provides the natural framework for the global study of the problem of evolution in GR.

Here are some other important mathematical contributions of Yvonne:

1. On the constraint equations. Together with collaborators such as D. Christodoulou (Elliptic systems in H_s,δ spaces on manifolds which are Euclidean at infinity. Acta Math. 146 (1981)), J. York (The Cauchy problem. General relativity and gravitation, Vol. 1, pp. 99–172, Plenum, New York-London), J. Isenberg and J. York (Einstein constraints on asymptotically Euclidean manifolds. Phys. Rev. D (3) 61 (2000)).

2. Global solutions to the Einstein vacuum equations with a U (1) symmetry. With V. Moncrief. (Future global in time Einsteinian spacetimes with U(1) isometry group. Ann. Henri Poincaré 2 (2001)).

3. Global solutions for the Einstein vacuum equations in higher dimensions. With P. Chrusciel and L. Loizelet (Global solutions of the Einstein-Maxwell equations in higher dimensions. Classical Quantum Gravity 23 (2006), no. 24, 7383–7394).

4. Characteristic Cauchy problem. With P. Chrusciel and M-G, José (The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions. Ann. Henri Poincaré 12 (2011)).

I have met Yvonne many more times through the years, and it was always a great pleasure to talk to her, not only about mathematics and physics, but about any subject concerning our “strange universe”. Her wisdom, warmth, broad interests, generosity and intellectual honesty are something I will always remember about her.

Sergiu Klainerman

March 7, 2025