1569:
308:
577:
986:
174:
1319:
450:
817:
662:
1211:
719:
153:
427:
303:{\displaystyle \Delta _{0}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}=4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\bar {z}}}}.}
791:
1223:
759:
1529:, Wiley Classics Library, vol. 3 (Reprint ed.), New York - Chichester - Brisbane - Toronto - Singapore: John Wiley & Sons, pp. X+637,
572:{\displaystyle \Delta _{0}u=-2Ke^{u}\quad \Longleftrightarrow \quad {\frac {\partial ^{2}u}{{\partial z}{\partial {\bar {z}}}}}=-{\frac {K}{2}}e^{u}.}
581:
Note that it is exactly in the first one of the preceding two forms that
Liouville's equation was cited by David Hilbert in the formulation of his
1372:
981:{\displaystyle u(z,{\bar {z}})=\ln \left(4{\frac {\left|{\mathrm {d} f(z)}/{\mathrm {d} z}\right|^{2}}{(1+K\left|f(z)\right|^{2})^{2}}}\right)}
1597:
605:
40:
61:
1214:
1564:
22:
1186:, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation
673:
318:
1534:
1494:
1646:
1486:
582:
346:
84:
100:
1651:
1570:
Nachrichten von der Kƶniglichen
Gesellschaft der Wissenschaften zu Gƶttingen, Mathematisch-Physikalische Klasse
1474:
805:
811:, the general solution of Liouville's equation can be found by using Wirtinger calculus. Its form is given by
1520:
1483:
Modern
GeometryāMethods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields
597:
432:
Other two forms of the equation, commonly found in the literature, are obtained by using the slight variant
377:
1583:
735:
1592:
328:
can be described as the conformal factor with respect to the flat metric. Occasionally it is the square
1314:{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}=e^{f}}
764:
1059:
1183:
802:
739:
314:
1489:, vol. 93 (2nd ed.), BerlināHeidelbergāNew York: Springer Verlag, pp. xv+468,
1075:
Liouville's equation can be used to prove the following classification results for surfaces:
444:
49:
1628:
1544:
1504:
1346:
1003:
31:
1620:
1578:
1552:
1512:
8:
358:
744:
88:
1530:
1490:
1163:
1611:
1616:
1606:
1574:
1548:
1508:
1371:
Ehrendorfer, Martin. "The
Liouville Equation: Background - Historical Background".
166:
57:
1624:
1540:
1524:
1500:
1478:
1149:
1640:
1588:
1560:
342:
593:
In a more invariant fashion, the equation can be written in terms of the
738:
for minimal immersions into the 3-space, when the metric is written in
657:{\displaystyle \Delta _{\mathrm {LB} }={\frac {1}{f^{2}}}\Delta _{0}}
588:
371:, another commonly found form of Liouville's equation is obtained:
1397:, p. 288): Hilbert does not cite explicitly Joseph Liouville.
1135:
352:
1347:"Sur la Theorie de la Variation des constantes arbitraires"
334:
that is referred to as the conformal factor, instead of
1472:
1455:
1407:
729:
1226:
820:
767:
747:
676:
608:
453:
380:
341:
Liouville's equation was also taken as an example by
177:
103:
1374:
The
Liouville Equation in Atmospheric Predictability
796:
714:{\displaystyle \Delta _{\mathrm {LB} }\log \;f=-K.}
30:For Liouville's equation in quantum mechanics, see
21:For Liouville's equation in dynamical systems, see
1313:
1210:, therefore the equation appears as the following
980:
785:
753:
713:
656:
571:
421:
302:
147:
1081:. A surface in the Euclidean 3-space with metric
589:A formulation using the LaplaceāBeltrami operator
39:For Liouville's equation in Euclidean space, see
1638:
1598:Bulletin of the American Mathematical Society
1389:
1387:
313:Liouville's equation appears in the study of
1354:Journal de mathƩmatiques pures et appliquƩes
1526:Applied and Computational Complex Analysis
1384:
734:Liouville's equation is equivalent to the
695:
353:Other common forms of Liouville's equation
148:{\displaystyle \Delta _{0}\log f=-Kf^{2},}
1610:
1344:
443:of the previous change of variables and
1587:
1559:
1519:
1442:
1437:
1435:
1433:
1424:
1411:
1394:
1370:
62:nonlinear partial differential equation
1639:
422:{\displaystyle \Delta _{0}u=-Ke^{2u}.}
1126:, and with constant scalar curvature
1456:Dubrovin, Novikov & Fomenko 1992
1430:
1408:Dubrovin, Novikov & Fomenko 1992
761:such that the Hopf differential is
730:Relation to GaussāCodazzi equations
13:
1282:
1268:
1245:
1231:
899:
875:
769:
686:
683:
678:
645:
618:
615:
610:
521:
513:
498:
455:
382:
279:
275:
264:
260:
236:
226:
204:
194:
179:
105:
64:satisfied by the conformal factor
14:
1663:
786:{\displaystyle \mathrm {d} z^{2}}
23:Liouville's theorem (Hamiltonian)
16:Equation in differential geometry
797:General solution of the equation
41:LiouvilleāBratuāGelfand equation
1612:10.1090/S0002-9904-1902-00923-3
1487:Graduate Studies in Mathematics
493:
489:
1465:
1448:
1417:
1400:
1364:
1338:
1197:
1070:
961:
946:
940:
919:
888:
882:
845:
839:
824:
530:
490:
317:in differential geometry: the
288:
1:
1582:, translated into English by
724:
1331:
7:
1584:Mary Frances Winston Newson
1177:
1078:
324:are the coordinates, while
10:
1668:
1573:(in German) (3): 253ā297,
1345:Liouville, Joseph (1838).
345:in the formulation of his
1130:is locally isometric to:
598:LaplaceāBeltrami operator
1565:"Mathematische Probleme"
1190:
1593:"Mathematical Problems"
736:GaussāCodazzi equations
1647:Differential equations
1315:
1184:Liouville field theory
982:
787:
755:
740:isothermal coordinates
715:
658:
573:
423:
315:isothermal coordinates
304:
149:
1652:Differential geometry
1316:
983:
788:
756:
716:
659:
574:
424:
319:independent variables
305:
150:
50:differential geometry
1458:, pp. 118ā120).
1427:, pp. 287ā294).
1410:, p. 118) and (
1224:
1004:meromorphic function
818:
765:
745:
674:
606:
451:
378:
175:
101:
54:Liouville's equation
32:Von Neumann equation
1164:Lobachevskian plane
1043: ā Ω
359:change of variables
1311:
978:
783:
751:
711:
654:
583:nineteenth problem
569:
445:Wirtinger calculus
419:
347:nineteenth problem
300:
145:
89:Gaussian curvature
1473:Dubrovin, B. A.;
1380:. pp. 48ā49.
1296:
1259:
1215:elliptic equation
1171: < 0
1143: > 0
971:
842:
754:{\displaystyle z}
642:
554:
538:
533:
295:
291:
271:
250:
218:
1659:
1631:
1614:
1581:
1555:
1515:
1459:
1452:
1446:
1439:
1428:
1421:
1415:
1404:
1398:
1391:
1382:
1381:
1379:
1368:
1362:
1361:
1351:
1342:
1325:
1320:
1318:
1317:
1312:
1310:
1309:
1297:
1295:
1294:
1293:
1280:
1276:
1275:
1265:
1260:
1258:
1257:
1256:
1243:
1239:
1238:
1228:
1209:
1203:Hilbert assumes
1201:
1172:
1158:
1144:
1129:
1125:
1124:
1123:
1122:
1117:
1107:
1106:
1105:
1100:
1080:
1065:
1057:
1044:
1037:
1031:
1029:
1028:
1022:
1019:
1001:
987:
985:
984:
979:
977:
973:
972:
970:
969:
968:
959:
958:
953:
949:
917:
916:
911:
907:
906:
902:
896:
891:
878:
866:
844:
843:
835:
810:
803:simply connected
792:
790:
789:
784:
782:
781:
772:
760:
758:
757:
752:
720:
718:
717:
712:
691:
690:
689:
663:
661:
660:
655:
653:
652:
643:
641:
640:
628:
623:
622:
621:
578:
576:
575:
570:
565:
564:
555:
547:
539:
537:
536:
535:
534:
526:
519:
510:
506:
505:
495:
488:
487:
463:
462:
442:
434:2 log
428:
426:
425:
420:
415:
414:
390:
389:
370:
337:
333:
327:
323:
309:
307:
306:
301:
296:
294:
293:
292:
284:
274:
272:
270:
259:
251:
249:
248:
247:
234:
233:
224:
219:
217:
216:
215:
202:
201:
192:
187:
186:
167:Laplace operator
164:
154:
152:
151:
146:
141:
140:
113:
112:
93:
82:
67:
58:Joseph Liouville
1667:
1666:
1662:
1661:
1660:
1658:
1657:
1656:
1637:
1636:
1635:
1605:(10): 437ā479,
1537:
1497:
1468:
1463:
1462:
1453:
1449:
1445:, p. 294).
1440:
1431:
1422:
1418:
1414:, p. 294).
1405:
1401:
1392:
1385:
1377:
1369:
1365:
1349:
1343:
1339:
1334:
1329:
1328:
1305:
1301:
1289:
1285:
1281:
1271:
1267:
1266:
1264:
1252:
1248:
1244:
1234:
1230:
1229:
1227:
1225:
1222:
1221:
1204:
1202:
1198:
1193:
1180:
1167:
1153:
1150:Euclidean plane
1139:
1127:
1118:
1115:
1114:
1113:
1101:
1098:
1097:
1096:
1082:
1073:
1063:
1048:
1039:
1036:) ā 0
1023:
1020:
1014:
1013:
1011:
1010:
992:
964:
960:
954:
936:
932:
931:
918:
912:
898:
897:
892:
874:
873:
872:
868:
867:
865:
861:
857:
834:
833:
819:
816:
815:
808:
799:
777:
773:
768:
766:
763:
762:
746:
743:
742:
732:
727:
682:
681:
677:
675:
672:
671:
648:
644:
636:
632:
627:
614:
613:
609:
607:
604:
603:
591:
560:
556:
546:
525:
524:
520:
512:
511:
501:
497:
496:
494:
483:
479:
458:
454:
452:
449:
448:
433:
407:
403:
385:
381:
379:
376:
375:
361:
355:
335:
329:
325:
321:
283:
282:
278:
273:
263:
258:
243:
239:
235:
229:
225:
223:
211:
207:
203:
197:
193:
191:
182:
178:
176:
173:
172:
163:
159:
136:
132:
108:
104:
102:
99:
98:
91:
69:
65:
17:
12:
11:
5:
1665:
1655:
1654:
1649:
1634:
1633:
1589:Hilbert, David
1561:Hilbert, David
1557:
1535:
1521:Henrici, Peter
1517:
1495:
1479:Fomenko, A. T.
1475:Novikov, S. P.
1469:
1467:
1464:
1461:
1460:
1447:
1429:
1416:
1399:
1383:
1363:
1336:
1335:
1333:
1330:
1327:
1326:
1324:
1323:
1322:
1321:
1308:
1304:
1300:
1292:
1288:
1284:
1279:
1274:
1270:
1263:
1255:
1251:
1247:
1242:
1237:
1233:
1195:
1194:
1192:
1189:
1188:
1187:
1179:
1176:
1175:
1174:
1160:
1157: = 0
1146:
1072:
1069:
1068:
1067:
1046:
989:
988:
976:
967:
963:
957:
952:
948:
945:
942:
939:
935:
930:
927:
924:
921:
915:
910:
905:
901:
895:
890:
887:
884:
881:
877:
871:
864:
860:
856:
853:
850:
847:
841:
838:
832:
829:
826:
823:
798:
795:
780:
776:
771:
750:
731:
728:
726:
723:
722:
721:
710:
707:
704:
701:
698:
694:
688:
685:
680:
665:
664:
651:
647:
639:
635:
631:
626:
620:
617:
612:
590:
587:
568:
563:
559:
553:
550:
545:
542:
532:
529:
523:
518:
515:
509:
504:
500:
492:
486:
482:
478:
475:
472:
469:
466:
461:
457:
430:
429:
418:
413:
410:
406:
402:
399:
396:
393:
388:
384:
354:
351:
311:
310:
299:
290:
287:
281:
277:
269:
266:
262:
257:
254:
246:
242:
238:
232:
228:
222:
214:
210:
206:
200:
196:
190:
185:
181:
161:
156:
155:
144:
139:
135:
131:
128:
125:
122:
119:
116:
111:
107:
77: + d
56:, named after
46:
45:
36:
27:
15:
9:
6:
4:
3:
2:
1664:
1653:
1650:
1648:
1645:
1644:
1642:
1630:
1626:
1622:
1618:
1613:
1608:
1604:
1600:
1599:
1594:
1590:
1585:
1580:
1576:
1572:
1571:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1536:0-471-58986-1
1532:
1528:
1527:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1496:3-540-97663-9
1492:
1488:
1484:
1480:
1476:
1471:
1470:
1457:
1451:
1444:
1438:
1436:
1434:
1426:
1420:
1413:
1409:
1403:
1396:
1390:
1388:
1376:
1375:
1367:
1359:
1355:
1348:
1341:
1337:
1306:
1302:
1298:
1290:
1286:
1277:
1272:
1261:
1253:
1249:
1240:
1235:
1220:
1219:
1218:
1217:
1216:
1213:
1207:
1200:
1196:
1185:
1182:
1181:
1170:
1165:
1161:
1156:
1151:
1147:
1142:
1137:
1133:
1132:
1131:
1121:
1111:
1104:
1094:
1090:
1087: =
1086:
1076:
1061:
1055:
1051:
1047:
1042:
1035:
1027:
1018:
1009:
1008:
1007:
1005:
999:
995:
974:
965:
955:
950:
943:
937:
933:
928:
925:
922:
913:
908:
903:
893:
885:
879:
869:
862:
858:
854:
851:
848:
836:
830:
827:
821:
814:
813:
812:
807:
804:
794:
778:
774:
748:
741:
737:
708:
705:
702:
699:
696:
692:
670:
669:
668:
649:
637:
633:
629:
624:
602:
601:
600:
599:
596:
586:
584:
579:
566:
561:
557:
551:
548:
543:
540:
527:
516:
507:
502:
484:
480:
476:
473:
470:
467:
464:
459:
446:
441:
438: ā¦
437:
416:
411:
408:
404:
400:
397:
394:
391:
386:
374:
373:
372:
369:
366: ā¦
365:
360:
357:By using the
350:
348:
344:
343:David Hilbert
339:
332:
320:
316:
297:
285:
267:
255:
252:
244:
240:
230:
220:
212:
208:
198:
188:
183:
171:
170:
169:
168:
142:
137:
133:
129:
126:
123:
120:
117:
114:
109:
97:
96:
95:
90:
86:
80:
76:
72:
63:
59:
55:
51:
44:
42:
37:
35:
33:
28:
26:
24:
19:
18:
1602:
1596:
1568:
1525:
1482:
1450:
1443:Henrici 1993
1425:Henrici 1993
1419:
1412:Henrici 1993
1402:
1395:Hilbert 1900
1373:
1366:
1357:
1353:
1340:
1205:
1199:
1168:
1154:
1140:
1119:
1109:
1102:
1092:
1088:
1084:
1077:
1074:
1060:simple poles
1058:has at most
1053:
1049:
1040:
1033:
1025:
1016:
997:
993:
990:
800:
733:
667:as follows:
666:
594:
592:
580:
439:
435:
431:
367:
363:
356:
340:
330:
312:
165:is the flat
157:
87:of constant
78:
74:
70:
68:of a metric
53:
47:
38:
29:
20:
1466:Works cited
1071:Application
1641:Categories
1621:33.0976.07
1579:31.0068.03
1553:1107.30300
1513:0751.53001
1360:: 342ā349.
1212:semilinear
1038:for every
1006:such that
725:Properties
1523:(1993) ,
1481:(1992) ,
1332:Citations
1283:∂
1269:∂
1246:∂
1232:∂
855:
840:¯
703:−
679:Δ
646:Δ
611:Δ
595:intrinsic
544:−
531:¯
522:∂
514:∂
499:∂
491:⟺
471:−
456:Δ
398:−
383:Δ
362:log
289:¯
280:∂
276:∂
265:∂
261:∂
237:∂
227:∂
205:∂
195:∂
180:Δ
127:−
118:
106:Δ
60:, is the
1591:(1902),
1563:(1900),
1178:See also
338:itself.
1629:1557926
1545:0822470
1505:0736837
1079:Theorem
1052: (
1030:
1012:
1002:is any
996: (
85:surface
1627:
1619:
1577:
1551:
1543:
1533:
1511:
1503:
1493:
1208:= -1/2
1136:sphere
1064:Ω
991:where
809:Ω
806:domain
158:where
1454:See (
1441:See (
1423:See (
1406:See (
1393:See (
1378:(PDF)
1350:(PDF)
1191:Notes
801:In a
83:on a
1531:ISBN
1491:ISBN
1162:the
1148:the
1134:the
1617:JFM
1607:doi
1586:as
1575:JFM
1549:Zbl
1509:Zbl
1166:if
1152:if
1138:if
1062:in
693:log
322:x,y
115:log
48:In
1643::
1625:MR
1623:,
1615:,
1601:,
1595:,
1567:,
1547:,
1541:MR
1539:,
1507:,
1501:MR
1499:,
1485:,
1477:;
1432:^
1386:^
1356:.
1352:.
1108:)d
852:ln
793:.
585:.
447::
349:.
94::
73:(d
52:,
1632:.
1609::
1603:8
1556:.
1516:.
1358:3
1307:f
1303:e
1299:=
1291:2
1287:y
1278:f
1273:2
1262:+
1254:2
1250:x
1241:f
1236:2
1206:K
1173:.
1169:K
1159:;
1155:K
1145:;
1141:K
1128:K
1120:z
1116:_
1112:d
1110:z
1103:z
1099:_
1095:,
1093:z
1091:(
1089:g
1085:l
1083:d
1066:.
1056:)
1054:z
1050:f
1045:.
1041:z
1034:z
1032:(
1026:z
1024:d
1021:/
1017:f
1015:d
1000:)
998:z
994:f
975:)
966:2
962:)
956:2
951:|
947:)
944:z
941:(
938:f
934:|
929:K
926:+
923:1
920:(
914:2
909:|
904:z
900:d
894:/
889:)
886:z
883:(
880:f
876:d
870:|
863:4
859:(
849:=
846:)
837:z
831:,
828:z
825:(
822:u
779:2
775:z
770:d
749:z
709:.
706:K
700:=
697:f
687:B
684:L
650:0
638:2
634:f
630:1
625:=
619:B
616:L
567:.
562:u
558:e
552:2
549:K
541:=
528:z
517:z
508:u
503:2
485:u
481:e
477:K
474:2
468:=
465:u
460:0
440:u
436:f
417:.
412:u
409:2
405:e
401:K
395:=
392:u
387:0
368:u
364:f
336:f
331:f
326:f
298:.
286:z
268:z
256:4
253:=
245:2
241:y
231:2
221:+
213:2
209:x
199:2
189:=
184:0
162:0
160:ā
143:,
138:2
134:f
130:K
124:=
121:f
110:0
92:K
81:)
79:y
75:x
71:f
66:f
43:.
34:.
25:.
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