Celebratio Mathematica

I first met Sho in 1962 at one of the AMS Sum­mer In­sti­tutes in Santa Bar­bara. I had just fin­ished my first year as a gradu­ate stu­dent at MIT, and he told me he was on his way to Berke­ley. We ended up be­ing col­leagues for forty-sev­en years when I my­self got to Berke­ley in 1965. Al­though as col­leagues we could not help but run in­to each oth­er of­ten, I think it was in the ten or so years from 1980 to 1990 that I had ex­ten­ded con­tact with him every week when he drove me home after each dif­fer­en­tial geo­metry sem­in­ar late in the day on Fri­day. We had to walk a bit be­fore we could get to his car and that gave us even more of a chance to chat and gos­sip. I am afraid the in­tel­lec­tu­al qual­ity of the con­ver­sa­tions was not par­tic­u­larly high, but the en­ter­tain­ment value was off the charts. It was most en­joy­able. However, the one thing that has stuck in my mind about Sho all these years is prob­ably the for­tu­it­ous con­flu­ence of events sur­round­ing the dis­cov­ery of the Kobay­ashi met­ric in 1966.

In the sum­mer of 1966, Pro­fess­or Chern and I at­ten­ded the AMS Sum­mer In­sti­tute on en­tire func­tions in La Jolla, and Pro­fess­or Chern gave a lec­ture on his new res­ult on the volume de­creas­ing prop­erty for holo­morph­ic map­pings from (ba­sic­ally) the unit ball in ${\mathbb{C}}^n$ in­to an Ein­stein man­i­fold of the same di­men­sion with neg­at­ive curvature in a suit­able sense. This is a gen­er­al­iz­a­tion of the fam­ous Ahlfors–Schwarz lemma on the unit disc, and it is to Pro­fess­or Chern’s cred­it that he re­cog­nized it as the spe­cial case of a gen­er­al the­or­em about holo­morph­ic map­pings on com­plex man­i­folds. At the time, the idea of put­ting the sub­ject of holo­morph­ic func­tions in a geo­met­ric set­ting was very much on his mind. In the fol­low­ing fall, he gave a sim­il­ar talk in one of the first lec­tures of the Fri­day geo­metry sem­in­ar, but this time he had a manuscript ready. The main in­gredi­ent of his proof is ba­sic­ally a Weitzenböck for­mula for the volume form; his ob­ser­va­tion was that neg­at­ive curvature (in one form or an­oth­er) of the tar­get man­i­fold lim­its the be­ha­vi­or of holo­morph­ic map­pings. This is the be­gin­ning of what Phil­lip Grif­fiths later called hy­per­bol­ic com­plex ana­lys­is. Of course both Sho and I were in the audi­ence, and with­in two or three weeks Sho came up with a gen­er­al­iz­a­tion us­ing more ele­ment­ary meth­ods.

At the time I was fas­cin­ated with Bloch’s the­or­em (in one com­plex vari­able) and was try­ing to un­der­stand why there would be a uni­valent disc for holo­morph­ic func­tions in­to the unit disc. Pro­fess­or Chern’s pa­per con­tains a ref­er­ence to a pa­per of Grauert–Reck­ziegel which can also be said to be an ap­plic­a­tion of the Ahlfors–Schwarz lemma. Upon read­ing it, I got the idea that Bloch’s the­or­em was a con­sequence of the phe­nomen­on of nor­mal fam­il­ies and, as a res­ult, I could prove a qual­it­at­ive gen­er­al­iz­a­tion for Bloch’s the­or­em for holo­morph­ic map­pings between com­plex man­i­folds. I wrote up my find­ings and, as it was the tra­di­tion then among the geo­met­ers at Berke­ley, put cop­ies of my manuscript in my geo­metry col­leagues’ mail­boxes.

In a few days, Sho came up with the manuscript of his an­nounce­ment that all com­plex man­i­folds carry an in­trins­ic met­ric, the met­ric that now bears his name. Sho re­cog­nized that with the avail­ab­il­ity of the Ahlfors–Schwarz lemma, the con­struc­tion of the Carathéodory met­ric could be “du­al­ized” to define the Kobay­ashi met­ric. His pa­per took the fo­cus com­pletely out of the holo­morph­ic map­pings them­selves and put it right­fully on the Kobay­ashi met­ric of the tar­get com­plex man­i­fold in ques­tion. This in­sight sheds light on nu­mer­ous clas­sic­al res­ults (such as the little and big Pi­card the­or­ems) and in­aug­ur­ates a new era in com­plex man­i­folds.