368:
In 1990, Claus
Gerhardt showed that this situation is characteristic of the more general case of mean-convex star-shaped smooth hypersurfaces of Euclidean space. In particular, for any such initial data, the inverse mean curvature flow exists for all positive time and consists only of mean-convex and
673:
As a consequence of
Huisken and Ilmanen's extension of Geroch's monotonicity, they were able to use the Hawking mass to interpolate between the surface area of an "outermost" minimal surface and the ADM mass of an asymptotically flat three-dimensional Riemannian manifold of nonnegative scalar
1015:
369:
star-shaped smooth hypersurfaces. Moreover the surface area grows exponentially, and after a rescaling that fixes the surface area, the surfaces converge smoothly to a round sphere. The geometric estimates in
Gerhardt's work follow from the
183:
529:
624:. However, the elliptic and weak setting gives a broader context, as such boundaries can have irregularities and can jump discontinuously, which is impossible in the usual inverse mean curvature flow.
908:
903:
808:
373:; the asymptotic roundness then becomes a consequence of the Krylov-Safonov theorem. In addition, Gerhardt's methods apply simultaneously to more general curvature-based hypersurface flows.
360:
864:
727:
1085:
895:
107:
767:
747:
1172:
666:
increases. In the simpler case of a smooth inverse mean curvature flow, this amounts to a local calculation and was shown in the 1970s by the physicist
433:
441:
670:. In Huisken and Ilmanen's setting, it is more nontrivial due to the possible irregularities and discontinuities of the surfaces involved.
1291:
1226:
1128:
620:} moves through the hypersurfaces arising in a inverse mean curvature flow, with the initial condition given by the boundary of
376:
As is typical of geometric flows, IMCF solutions in more general situations often have finite-time singularities, meaning that
1337:
867:
1010:{\displaystyle {\begin{aligned}{\frac {\text{d}}{{\text{d}}t}}r(t)=&{\frac {r(t)}{m}},\\r(0)=&r_{0}.\end{aligned}}}
675:
50:
248:
has mean curvature which is nowhere zero, then there exists a unique inverse mean curvature flow whose "initial data" is
1287:
772:
365:
So a family of concentric round hyperspheres forms an inverse mean curvature flow when the radii grow exponentially.
46:
820:
304:
1332:
1021:
813:
Due to the rotational symmetry of the sphere (or in general, due to the invariance of mean curvature under
697:
547:
22:
1030:
1181:
571:
1286:. Second Session of the Centro Internazionale Matematico Estivo (Cetraro, Italy, June 15–22, 1996).
397:
1278:; Polden, Alexander (1999). "Geometric evolution equations for hypersurfaces". In Hildebrandt, S.;
409:
298:. As such, such a family of concentric spheres forms an inverse mean curvature flow if and only if
42:
1197:
54:
26:
370:
562:
which is asymptotically flat or asymptotically conic, and for any precompact and open subset
178:{\displaystyle {\frac {\partial F}{\partial t}}={\frac {-\mathbf {H} }{|\mathbf {H} |^{2}}},}
1311:
1295:
1263:
1245:
1235:
1205:
1189:
1155:
1137:
636:
550:. Huisken and Ilmanen proved that for any complete and connected smooth Riemannian manifold
1307:
1259:
1151:
873:
1315:
1303:
1275:
1267:
1255:
1250:
1217:
1209:
1159:
1147:
413:
265:
221:
73:
690:
A simple example of inverse mean curvature flow is given by a family of concentric round
260:
A simple example of inverse mean curvature flow is given by a family of concentric round
1185:
1279:
1193:
752:
732:
401:
195:
34:
1326:
1201:
1167:
667:
543:
640:
691:
417:
405:
261:
38:
1240:
1221:
1142:
1123:
639:, Huisken and Ilmanen showed that a certain geometric quantity known as the
814:
1299:
1222:"The inverse mean curvature flow and the Riemannian Penrose inequality"
590:; moreover such a function is uniquely determined on the complement of
524:{\displaystyle \operatorname {div} _{g}{\frac {du}{|du|_{g}}}=|du|_{g}}
420:
replaced the IMCF equation, for hypersurfaces in a
Riemannian manifold
79:, an inverse mean curvature flow consists of an open interval
1284:
Calculus of
Variations and Geometric Evolution Problems
582:
which is a positive weak solution on the complement of
1033:
906:
876:
823:
775:
755:
735:
700:
444:
307:
110:
574:, there is a proper and locally Lipschitz function
1079:
1009:
889:
858:
802:
761:
741:
721:
523:
354:
177:
49:. It has been used to prove a certain case of the
803:{\displaystyle H={\frac {m}{r}}\in \mathbb {R} }
1124:"Flow of nonconvex hypersurfaces into spheres"
674:curvature. This settled a certain case of the
396:Following the seminal works of Yun Gang Chen,
1020:The solution of this ODE (obtained, e.g., by
60:Formally, given a pseudo-Riemannian manifold
8:
817:) the inverse mean curvature flow equation
1173:Annals of the New York Academy of Sciences
682:Example: inverse mean curvature flow of a
662:}, and is monotonically non-decreasing as
1249:
1239:
1141:
1067:
1063:
1053:
1032:
994:
945:
917:
911:
907:
905:
881:
875:
859:{\displaystyle \partial _{t}F=H^{-1}\nu }
844:
828:
822:
796:
795:
782:
774:
754:
734:
707:
703:
702:
699:
515:
510:
498:
486:
481:
469:
458:
449:
443:
328:
306:
163:
158:
152:
147:
140:
134:
111:
109:
380:often cannot be taken to be of the form
355:{\displaystyle r'(t)={\frac {r(t)}{n}}.}
1097:
546:of this equation can be specified by a
268:. If the dimension of such a sphere is
434:elliptic partial differential equation
7:
392:Huisken and Ilmanen's weak solutions
643:can be defined for the boundary of
255:
1194:10.1111/j.1749-6632.1973.tb41445.x
870:, for an initial sphere of radius
825:
722:{\displaystyle \mathbb {R} ^{m+1}}
236:, and if a given smooth immersion
122:
114:
31:inverse mean curvature flow (IMCF)
14:
1080:{\displaystyle r(t)=r_{0}e^{t/m}}
1227:Journal of Differential Geometry
1129:Journal of Differential Geometry
153:
141:
1251:11858/00-001M-0000-0013-5581-4
1043:
1037:
982:
976:
957:
951:
937:
931:
868:ordinary differential equation
749:-dimensional sphere of radius
511:
499:
482:
470:
391:
340:
334:
322:
316:
256:Gerhardt's convergence theorem
159:
148:
21:In the mathematical fields of
1:
1170:(1973). "Energy extraction".
676:Riemannian Penrose inequality
400:, and Shun'ichi Goto, and of
276:, then its mean curvature is
51:Riemannian Penrose inequality
1288:Lecture Notes in Mathematics
586:and which is nonpositive on
1113:Huisken and Polden, page 59
729:. The mean curvature of an
601:increases, the boundary of
570:whose boundary is a smooth
534:for a real-valued function
1354:
1290:. Vol. 1713. Berlin:
53:, which is of interest in
47:pseudo-Riemannian manifold
631:is three-dimensional and
627:In the special case that
1122:Gerhardt, Claus (1990).
1220:; Ilmanen, Tom (2001).
1022:separation of variables
1241:10.4310/jdg/1090349447
1143:10.4310/jdg/1214445048
1081:
1011:
891:
860:
804:
763:
743:
723:
525:
356:
179:
1338:Differential geometry
1082:
1012:
892:
890:{\displaystyle r_{0}}
861:
805:
764:
744:
724:
597:The idea is that, as
548:variational principle
526:
357:
196:mean curvature vector
180:
23:differential geometry
1031:
904:
874:
821:
773:
753:
733:
698:
686:-dimensional spheres
572:embedded submanifold
442:
305:
108:
1186:1973NYASA.224..108G
410:mean curvature flow
1300:10.1007/BFb0092667
1294:. pp. 45–84.
1104:Huisken and Polden
1077:
1007:
1005:
887:
856:
800:
759:
739:
719:
521:
352:
272:and its radius is
216:is Riemannian, if
175:
55:general relativity
27:geometric analysis
964:
926:
920:
915:
790:
762:{\displaystyle r}
742:{\displaystyle m}
493:
371:maximum principle
347:
198:of the immersion
170:
129:
83:and a smooth map
1345:
1319:
1276:Huisken, Gerhard
1271:
1253:
1243:
1218:Huisken, Gerhard
1213:
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1111:
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766:
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748:
746:
745:
740:
728:
726:
725:
720:
718:
717:
706:
685:
665:
661:
637:scalar curvature
635:has nonnegative
634:
630:
623:
619:
600:
593:
589:
585:
581:
577:
569:
565:
561:
541:
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315:
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145:
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82:
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71:
1353:
1352:
1348:
1347:
1346:
1344:
1343:
1342:
1323:
1322:
1274:
1216:
1166:
1121:
1118:
1117:
1112:
1108:
1103:
1099:
1094:
1059:
1049:
1029:
1028:
1004:
1003:
990:
988:
970:
969:
947:
943:
916:
902:
901:
877:
872:
871:
866:reduces to the
840:
824:
819:
818:
771:
770:
751:
750:
731:
730:
701:
696:
695:
688:
683:
663:
644:
632:
628:
621:
602:
598:
591:
587:
583:
579:
575:
567:
563:
551:
539:
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509:
480:
468:
460:
445:
440:
439:
421:
414:Gerhard Huisken
394:
381:
377:
330:
308:
303:
302:
289:
286:
281:
280:
278:
277:
273:
269:
266:Euclidean space
258:
249:
245:
241:
237:
225:
217:
213:
199:
189:
157:
146:
136:
121:
113:
106:
105:
98:
88:
84:
80:
76:
74:smooth manifold
61:
19:
12:
11:
5:
1351:
1349:
1341:
1340:
1335:
1333:Geometric flow
1325:
1324:
1321:
1320:
1272:
1234:(3): 353–437.
1214:
1180:(1): 108–117.
1168:Geroch, Robert
1164:
1136:(1): 299–314.
1116:
1115:
1106:
1096:
1095:
1093:
1090:
1089:
1088:
1074:
1070:
1066:
1062:
1056:
1052:
1048:
1045:
1042:
1039:
1036:
1018:
1017:
1002:
997:
993:
989:
987:
984:
981:
978:
975:
972:
971:
968:
963:
959:
956:
953:
950:
944:
942:
939:
936:
933:
930:
924:
910:
909:
884:
880:
855:
850:
847:
843:
839:
836:
831:
827:
798:
794:
789:
786:
781:
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758:
738:
716:
713:
710:
705:
687:
680:
544:Weak solutions
532:
531:
518:
513:
508:
505:
501:
497:
489:
484:
479:
476:
472:
466:
463:
457:
452:
448:
402:Lawrence Evans
398:Yoshikazu Giga
393:
390:
363:
362:
351:
346:
342:
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324:
321:
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314:
311:
257:
254:
186:
185:
174:
166:
161:
155:
150:
143:
139:
133:
127:
124:
119:
116:
35:geometric flow
18:Geometric flow
17:
13:
10:
9:
6:
4:
3:
2:
1350:
1339:
1336:
1334:
1331:
1330:
1328:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1252:
1247:
1242:
1237:
1233:
1229:
1228:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1174:
1169:
1165:
1161:
1157:
1153:
1149:
1144:
1139:
1135:
1131:
1130:
1125:
1120:
1119:
1110:
1107:
1101:
1098:
1091:
1072:
1068:
1064:
1060:
1054:
1050:
1046:
1040:
1034:
1027:
1026:
1025:
1023:
1000:
995:
991:
985:
979:
973:
966:
961:
954:
948:
940:
934:
928:
922:
900:
899:
898:
882:
878:
869:
853:
848:
845:
841:
837:
834:
829:
816:
811:
792:
787:
784:
779:
776:
756:
736:
714:
711:
708:
693:
681:
679:
677:
671:
669:
668:Robert Geroch
660:
656:
652:
648:
642:
638:
625:
618:
614:
610:
606:
595:
573:
559:
555:
549:
545:
516:
506:
503:
495:
487:
477:
474:
464:
461:
455:
450:
446:
438:
437:
436:
435:
429:
425:
419:
415:
411:
407:
403:
399:
389:
385:
374:
372:
366:
349:
344:
337:
331:
325:
319:
312:
309:
301:
300:
299:
292:
284:
267:
263:
253:
233:
229:
223:
210:
206:
202:
197:
192:
172:
164:
137:
131:
125:
117:
104:
103:
102:
95:
91:
75:
69:
65:
58:
56:
52:
48:
44:
40:
36:
32:
28:
24:
16:
1283:
1231:
1225:
1177:
1171:
1133:
1127:
1109:
1100:
1019:
812:
692:hyperspheres
689:
672:
658:
654:
650:
646:
641:Hawking mass
626:
616:
612:
608:
604:
596:
557:
553:
533:
427:
423:
395:
383:
375:
367:
364:
290:
282:
262:hyperspheres
259:
231:
227:
211:
204:
200:
190:
187:
93:
89:
67:
63:
59:
39:submanifolds
30:
20:
15:
418:Tom Ilmanen
406:Joel Spruck
1327:Categories
1316:0942.35047
1280:Struwe, M.
1268:1055.53052
1210:0942.53509
1160:0708.53045
1092:References
815:isometries
101:such that
43:Riemannian
1202:222086296
854:ν
846:−
826:∂
793:∈
456:
432:, by the
138:−
123:∂
115:∂
1292:Springer
1282:(eds.).
649: :
607: :
313:′
230:) = dim(
1308:1731639
1260:1916951
1182:Bibcode
1152:1064876
657:) <
615:) <
408:on the
295:
279:
194:is the
1314:
1306:
1266:
1258:
1208:
1200:
1158:
1150:
1024:) is
222:closed
188:where
72:and a
1198:S2CID
244:into
234:) + 1
224:with
97:into
87:from
41:of a
33:is a
416:and
404:and
386:, ∞)
226:dim(
207:, ⋅)
25:and
1312:Zbl
1296:doi
1264:Zbl
1246:hdl
1236:doi
1206:Zbl
1190:doi
1178:224
1156:Zbl
1138:doi
810:.
769:is
694:in
578:on
566:of
538:on
447:div
264:in
240:of
220:is
212:If
45:or
37:of
1329::
1310:.
1304:MR
1302:.
1262:.
1256:MR
1254:.
1244:.
1232:59
1230:.
1224:.
1204:.
1196:.
1188:.
1176:.
1154:.
1148:MR
1146:.
1134:32
1132:.
1126:.
897:,
678:.
594:.
556:,
542:.
426:,
412:,
388:.
252:.
209:.
92:×
66:,
57:.
29:,
1318:.
1298::
1270:.
1248::
1238::
1212:.
1192::
1184::
1162:.
1140::
1087:.
1073:m
1069:/
1065:t
1061:e
1055:0
1051:r
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1044:)
1041:t
1038:(
1035:r
1001:.
996:0
992:r
986:=
983:)
980:0
977:(
974:r
967:,
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955:t
952:(
949:r
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938:)
935:t
932:(
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914:d
883:0
879:r
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842:H
838:=
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830:t
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788:r
785:m
780:=
777:H
757:r
737:m
715:1
712:+
709:m
704:R
684:m
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659:t
655:x
653:(
651:u
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68:g
64:M
62:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.