## Remembrance of Things Past

JAIGYOUNG CHOE

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*