Remembrance of Things Past


In my first year, the second quarter 1981, at UC Berkeley I passed the preliminary exam, to my surprise. That quarter I also took the course 'Differentiable Manifolds' and I found myself more interested in geometry than in number theory. I consulted my high school senior, Hyeong In Choi, and he advised me to try to become Rick's student.

Next quarter I took a reading course with Rick. In this course, I started learning undergraduate differential geometry with Spivak's book volume II. At the end of the reading course, Rick told me to teach him henceforth rather than to learn from him. I also sat in Rick's topics course. The room was full of students, postdocs, visitors, and professors. The main topics of the course were the Schauder theory and many recent results in geometric analysis. I did not quite like the Schuder theory because it was too analytic for me. I still have 184 pages of Rick's lecture notes printed in ditto machine. Next year, fortunately, Rick taught a graduate differential geometry course and he covered the minimal surface theory. I loved the course because it was very geometric for me.

In the third year, I took the qualifying exam. The exam committee members were Rick, Shiing-Shen Chern, Hung-Hsi Wu, and Tosio Kato. Wu asked me to show the cut locus has measure zero. I could not answer. This made Wu dissatisfied, and disappointingly, he opened a book to read. At that moment, Kato unexpectedly took his turn. He asked me the same question that I had asked him in his office one hour earlier. Then Rick's question led me to discuss Jenkins and Serrin's theorem on a minimal surface going off to plus-minus infinity over the boundary of a quadrilateral. This satisfied Rick and Chern. When the exam was over, Chern asked me an easy question: how to write my name in Chinese characters.

For months I tried hard to find problems for my thesis. I discussed these with Rick when he was visiting Stanford. Not quite interested, Rick talked about the recently discovered Costa surface and showed its picture to me, and suggested the problem of constructing a minimal surface with arbitrary genus. This problem was very interesting for me and I drew pictures of the Costa surface over and over again. After a few months, I drove to Stanford to talk with Rick about my idea. He listened to me for about 30 minutes. When I was finished, he said, "I don't want to discourage you, but Meeks called me last week and said he and David Hoffman had constructed the complete minimal surfaces with finite total curvature and arbitrary genus." Afterwards, though, I was not so discouraged because I improved that idea to construct the generalized Scherk's second surfaces. I was delighted when Rick dropped by my office at Berkeley to ask me whether my idea was still correct.

In the second semester of the third year, Rick wrote to us from Princeton.
To my students: I am spending next year at UC San Diego in La Jolla. I can arrange for students to accompany me for all or part of the year (partially TA, partially RA). There should be interesting mathematics there next year. Please let me know as soon as possible if you are interested in going.
Seven students decided to accompany Rick: Guojun, Chepe, Nat, Rob, Nicos, Will, and me. Two quit mathematics and one changed his adviser.

I drove a U-Haul truck with my wife and son from Berkeley to La Jolla on Mid-Autumn full moon night. Indeed there were lots of activities at UCSD by Rick, Yau, Hamilton, Freedman, and numerous visitors including Leon Simon, Uhlenbeck, Huisken. Rick's students and Yau's students played beach volleyball after a seminar every week. Rick's spike was devastating, almost as powerful as his theorems.

In the first two quarters, Rick taught geometric measure theory with the textbook written by Leon. I seriously took the course. One day I came across Rick in a hallway of the department. He said to me with a little smile that Freedman proposed a problem to him which seemed to be a good thesis problem for me. It was to show the existence and regularity of a fundamental domain with least boundary area in a compact 3-manifold. At first, I was not so enthusiastic about this problem. When I said to Rick that this problem seemed to be trivial, he said, "If this is trivial, Plateau's problem is trivial." Then I took the problem seriously. After a few months, I solved the problem trivially by using the functions of bounded variation. But in two days Rick found a gap in my proof.

One day I went to Rick's office for a discussion. A student was already there talking with Rick, so I waited outside. Having waited long, thinking deeply about my question, I saw a button on the wall and inadvertently I pressed that button. Immediately, an emergency bell rang loudly everywhere in the AP&M building. All the people had to evacuate the building. Soon a fire engine arrived. Since then Rick had never kept me waiting.

Toward the end of the fourth year, Rick surprised us again: he was going to visit Mittag-Leffler Institute and IHES for a quarter. Chepe, Nat, and I followed Rick, and Nicos occasionally visited. We lived together in Mittag-Leffer's brother's yellow villa. One day we had dinner at Rick's place and Doris cooked crawfish for us. She said, "All are equal when eating crawfish because even a king has to eat it the right way." I guess we are not all equal when doing mathematics.

When Leon joined us in volleyball at UCSD, I told him about the difficulty of my thesis. He suggested using the filigree lemma of his paper with Almgren. It turned out to me in Sweden that the filigree lemma was the key to fill the gap in my original proof. So I proved the rectifiability of the boundary of the limiting fundamental domain at Mittag-Leffler Institute and finished the proof of my thesis at IHES. I still remember when I talked to Rick about the rectifiability in front of Weierstrass's portrait in his office at Mittag-Leffler. Chepe, Nat, and Nicos also made big progress in their problems while we were in Europe.

Rick advised me to submit my thesis to the Acta Mathematica. Surprisingly, after 11 years, I received letters between the editor and the referee. After Almgren passed away, his former students distributed his letters to those involved and invited them to respond with contributions, which would be collected and distributed in memory of Almgren. According to these letters, Hoermander asked if Almgren considered my thesis to be in the upper bracket of papers accepted in the Annals of Mathematics. Almgren replied, "I find the principal result to be of substantial intrinsic result. It is a nice addition to the literature of the calculus of variations and 3-manifold theory. I do not see immediate applications to understanding the topology of 3-manifolds. I feel that this paper is comparable to a number of papers which have been published in the Annals. On the other hand, it is not the equal of the upper half of such papers." My thesis was then accepted by JDG.

After I finished my thesis I became the notetaker of Rick and Yau's seminar. Later, my notes formed part of their book 'Lectures on Harmonic Maps'. That year, Nat, Rob, and I gave talks on our theses in a special session of the AMS meeting in San Antonio. Leon, who also gave a talk there, called us Rick's army.

Next year we all left UCSD. Rick moved to Stanford, Yau to Harvard, Chepe to NYU, Rob, Nicos, and Will to Stanford, Nat and I to MSRI. Nat and I shared an office with a bay view.

After thirty-one years, Rick came to the conference for my 65th birthday at Gyeongju in Korea. More than 40 mathematicians participated in the conference. Many of them came, encouraged by Rick's participation.

Rick is still doing mathematics actively. Why not me?

presented in the Zoom gathering for Rick SchoenĄ¯s 70th birthday