456:(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture). The Schoen(-Yau) conjecture has since been disproved.
353:
781:
217:
451:
405:
704:
665:
228:
709:
so the question is whether or not they are "harmonically diffeomorphic". However, as the truth of Heinz's theorem and the falsity of the Schoen–Yau conjecture demonstrate,
112:
82:
735:
552:
504:
727:
615:
583:
127:
875:
786:
Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.
468:
diffeomorphism, and that this property is a "one-way" property. In more detail: suppose that we consider two
Riemannian manifolds
911:
868:
416:
370:
676:
51:
1012:
861:
1017:
627:
986:
809:
Collin, Pascal; Rosenberg, Harold (2010). "Construction of harmonic diffeomorphisms and minimal graphs".
348:{\displaystyle \mathrm {d} s^{2}=4{\frac {\mathrm {d} x^{2}+\mathrm {d} y^{2}}{(1-(x^{2}+y^{2}))^{2}}}.}
921:
594:
776:{\displaystyle \mathbb {C} \propto \mathbb {H} {\text{ but }}\mathbb {H} \not \propto \mathbb {C} .}
906:
590:
95:
65:
531:
981:
941:
936:
896:
482:
712:
991:
670:
It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:
976:
848:
600:
586:
568:
410:
In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism
8:
961:
951:
89:
25:
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618:
931:
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115:
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832:
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362:
47:
36:
32:
1006:
840:
85:
29:
212:{\displaystyle \mathbb {H} :=\{(x,y)\in \mathbb {R} ^{2}|x^{2}+y^{2}<1\}}
359:
43:
17:
853:
885:
796:
Heinz, Erhard (1952). "Über die Lösungen der
Minimalflächengleichung".
823:
119:
798:
Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt
57:
464:
The emphasis is on the existence or non-existence of an
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98:
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498:
445:
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211:
106:
76:
446:{\displaystyle g:\mathbb {C} \to \mathbb {H} .\,}
400:{\displaystyle f:\mathbb {H} \to \mathbb {C} .\,}
1004:
557:if there exists an harmonic diffeomorphism from
358:E. Heinz proved in 1952 that there can exist no
699:{\displaystyle \mathbb {H} \sim \mathbb {C} ,}
869:
92:with its usual (flat) Riemannian metric. Let
46:(1952). One method of disproof is the use of
206:
139:
476:(with their respective metrics), and write
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162:
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597:of Riemannian manifolds. In particular,
58:Setting and statement of the conjecture
1005:
509:if there exists a diffeomorphism from
857:
795:
660:{\displaystyle M\sim N\iff N\sim M.}
565:. It is not difficult to show that
222:endowed with the hyperbolic metric
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14:
1029:
42:It was inspired by a theorem of
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173:
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142:
1:
789:
729:is not a symmetric relation:
24:is a disproved conjecture in
833:10.4007/annals.2010.172.1879
585:(being diffeomorphic) is an
107:{\displaystyle \mathbb {H} }
77:{\displaystyle \mathbb {C} }
7:
517:(in the usual terminology,
459:
10:
1034:
547:{\displaystyle M\propto N}
525:are diffeomorphic). Write
54:and Pascal Collin (2006).
892:
499:{\displaystyle M\sim N\,}
722:{\displaystyle \propto }
777:
723:
700:
661:
611:
579:
548:
500:
447:
401:
349:
213:
108:
78:
1013:Disproved conjectures
912:Euler's sum of powers
778:
724:
701:
662:
612:
610:{\displaystyle \sim }
580:
578:{\displaystyle \sim }
549:
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448:
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214:
109:
79:
22:Schoen–Yau conjecture
736:
713:
677:
628:
601:
587:equivalence relation
569:
532:
483:
417:
371:
229:
128:
96:
66:
1018:Hyperbolic geometry
90:Riemannian manifold
26:hyperbolic geometry
902:Chinese hypothesis
773:
719:
696:
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619:symmetric relation
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575:
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397:
345:
209:
104:
74:
28:, named after the
1000:
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340:
1025:
952:Ono's inequality
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871:
864:
855:
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844:
826:
817:(3): 1879–1906.
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763:∝ ̸
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116:hyperbolic plane
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88:considered as a
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52:Harold Rosenberg
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1024:
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792:
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48:Scherk surfaces
12:
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5:
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984:
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927:Hauptvermutung
924:
919:
914:
909:
904:
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893:
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806:
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707:
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461:
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454:
453:
441:
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433:
429:
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408:
407:
395:
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379:
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363:diffeomorphism
356:
355:
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147:
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141:
138:
134:
102:
72:
59:
56:
37:Shing-Tung Yau
33:Richard Schoen
30:mathematicians
9:
6:
4:
3:
2:
1030:
1019:
1016:
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1011:
1010:
1008:
993:
990:
988:
985:
983:
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978:
975:
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958:
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948:
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938:
935:
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928:
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923:
920:
918:
915:
913:
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908:
905:
903:
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834:
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820:
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803:
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770:
762:
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732:
731:
730:
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693:
685:
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671:
654:
651:
648:
645:
637:
634:
631:
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623:
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620:
604:
596:
592:
588:
572:
564:
560:
541:
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528:
527:
526:
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467:
457:
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361:
342:
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167:
157:
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117:
91:
87:
86:complex plane
55:
53:
50:, as used by
49:
45:
40:
38:
34:
31:
27:
23:
19:
966:
922:Hedetniemi's
824:math/0701547
814:
811:Ann. of Math
810:
801:
797:
785:
708:
669:
562:
558:
556:
522:
518:
514:
510:
508:
473:
469:
465:
463:
455:
409:
357:
221:
61:
44:Erhard Heinz
41:
21:
15:
982:Von Neumann
886:conjectures
118:, i.e. the
114:denote the
18:mathematics
1007:Categories
992:Williamson
987:Weyl–Berry
967:Schoen–Yau
884:Disproved
790:References
841:0003-486X
745:∝
717:∝
686:∼
649:∼
642:⟺
635:∼
605:∼
573:∼
539:∝
490:∼
432:→
386:→
298:−
158:∈
120:unit disc
962:Ragsdale
942:Keller's
937:Kalman's
897:Borsuk's
804:: 51–56.
595:category
466:harmonic
460:Comments
360:harmonic
972:Seifert
947:Mertens
849:2726102
593:of the
591:objects
589:on the
84:be the
977:Tait's
932:Hirsch
907:Connes
839:
20:, the
957:Pólya
917:Ganea
819:arXiv
813:. 2.
617:is a
561:onto
513:onto
837:ISSN
802:1952
521:and
472:and
201:<
62:Let
35:and
829:doi
815:172
16:In
1009::
846:MR
835:.
827:.
800:.
621::
137::=
39:.
877:e
870:t
863:v
843:.
831::
821::
771:.
767:C
759:H
749:H
741:C
694:,
690:C
682:H
655:.
652:M
646:N
638:N
632:M
563:N
559:M
542:N
536:M
523:N
519:M
515:N
511:M
493:N
487:M
474:N
470:M
440:.
436:H
428:C
424::
421:g
394:.
390:C
382:H
378::
375:f
343:.
335:2
331:)
327:)
322:2
318:y
314:+
309:2
305:x
301:(
295:1
292:(
285:2
281:y
276:d
272:+
267:2
263:x
258:d
251:4
248:=
243:2
239:s
234:d
207:}
204:1
196:2
192:y
188:+
183:2
179:x
174:|
168:2
163:R
155:)
152:y
149:,
146:x
143:(
140:{
133:H
101:H
71:C
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