Preface These notes were the basis for a series of te n lectures given from Januar y 6-10 , 1984 a t Polytechni c Institut e o f Ne w Yor k unde r th e sponsorshi p o f th e Con - ference Boar d o f th e Mathematica l Science s an d th e Nationa l Scienc e Founda - tion. Th e lecture s were aime d a t mathematician s wh o knew eithe r som e differen - tial geometr y o r partia l differentia l equations , althoug h other s coul d hopefull y understand th e lectures. Ostensibly, th e primar y proble m addresse d her e i s t o understan d th e variou s curvatures o n a Riemannia n manifold . Often thi s questio n ca n b e reduce d t o solving some nonlinear partia l differentia l equations . But from th e viewpoin t o f a geometer, thes e question s ar e onl y portal s t o see k a deepe r understandin g o f Riemannian manifolds . On the other hand, an analyst may find th e geometry dul l and stil l be delighted with the nonlinear partial differential equation s one is led t o understand. Th e questions are rich enough to serve as motivation fo r man y varie d tastes. One goal of thes e lectures was to state what is currently known an d no t know n about a variety of problems that involv e the curvature of a Riemannian manifold . My ow n inclinatio n wa s especially t o emphasize areas of curren t ignorance . Wit h this in mind , a t th e end of thes e notes I have collected a partial lis t of som e open questions that are drawn from th e main body of the lectures. Generally, I trie d t o giv e th e essentia l idea s behin d variou s proofs , an d hav e given references but certainly not detailed arguments. The only places where there are detaile d discussion s ar e wher e th e materia l i s either difficul t t o extrac t fro m the literature, or where it has not bee n written down. This is especially true of th e last sectio n o f thes e notes, where I give Caffarelli's simplificatio n o f a n importan t recent estimat e of Krylo v concernin g th e boundary regularit y fo r som e nonlinea r elliptic equations. While this material is a bit technical , it appear s nowher e in th e literature an d ma y hel p spee d progres s i n resolvin g variou s boundar y valu e problems that arise in geometry. There i s a considerabl e overla p betwee n thes e note s an d a serie s o f lecture s I gave i n Japa n i n Jul y 1983 . Bu t thes e note s hav e a muc h greate r emphasi s o n geometric issues , whil e th e note s [K4 ] fro m th e lecture s i n Japa n gav e a mor e detailed an d elementar y expositio n o f th e idea s fro m partia l differentia l equa - tions. Because of thos e earlier notes—which wer e distributed a t th e NSF/CBM S v
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