503:
491:
479:
36:
28:
1353:
2528:
1109:
2699:
E.J. Lobaton, T.R. Salamon. Computation of constant mean curvature surfaces: Application to the gas–liquid interface of a pressurized fluid on a superhydrophobic surface. Journal of
Colloid and Interface Science. Volume 314, Issue 1, 1 October 2007, Pages
2788:
Helmut
Pottmann, Yang Liu, Johannes Wallner, Alexander Bobenko, Wenping Wang. Geometry of Multi-layer Freeform Structures for Architecture. ACM Transactions on Graphics – Proceedings of ACM SIGGRAPH 2007 Volume 26 Issue 3, July 2007 Article No. 65
587:"force" along the asymptotic axis of the unduloid (where n is the circumference of the necks), the sum of which must be balanced for the surface to exist. Current work involves classification of families of embedded CMC surfaces in terms of their
2231:
where the different components have a nonzero interfacial energy or tension. CMC analogs to the periodic minimal surfaces have been constructed, producing unequal partitions of space. CMC structures have been observed in ABC triblock copolymers.
466:
with each genus bigger than one. In particular gluing methods appear to allow combining CMC surfaces fairly arbitrarily. Delaunay surfaces can also be combined with immersed "bubbles", retaining their CMC properties.
2709:
D. M. Anderson, H. T. Davis, L. E. Scriven, J. C. C. Nitsche, Periodic
Surfaces of Prescribed Mean Curvature in Advances in Chemical Physics vol 77, eds. I. Prigogine and S. A. Rice, John Wiley & Sons, 2007, p.
2397:
1805:
can be described in purely algebro-geometric data. This can be extended to a subset of CMC immersions of the plane which are of finite type. More precisely there is an explicit bijection between CMC immersions of
1976:
1098:
2453:
Nikolaos
Kapouleas, Christine Breiner, Stephen Kleene. Conservation laws and gluing constructions for constant mean curvature (hyper)surfaces. Notices Amer. Math. Soc. 69 (2022), no.5, 762–773.
688:
1877:
1774:
930:
749:
1348:{\displaystyle X_{z}(z)={\frac {-1}{H(1+\phi (z){\bar {\phi }}(z))^{2}}}\left\{(1-\phi (z)^{2},i(1+\phi (z)^{2}),2\phi (z)){\frac {\bar {\partial \phi }}{\partial {\bar {z}}}}(z)\right\}}
2124:
1480:
970:
1906:
1833:
1803:
1718:
1689:
1660:
1554:
1002:
819:
547:
464:
435:
395:
358:
303:
244:
185:
152:
2090:
2063:
876:
585:
2016:
2184:
2036:
1996:
615:
270:
211:
1623:
1382:
2790:
2144:
1588:
1402:
1506:
329:
2164:
1422:
896:
854:
790:
123:
2317:
A. D. Alexandrov, Uniqueness theorem for surfaces in the large, V. Vestnik, Leningrad Univ. 13, 19 (1958), 5–8, Amer. Math. Soc. Trans. (Series 2) 21, 412–416.
2721:
2557:. Triunduloids: Embedded constant mean curvature surfaces with three ends and genus zero. J. Reine Angew. Math., 564, pp. 35–61 2001 arXiv:math/0102183v2
2838:
830:
2675:
Smith, J. 2003. Three
Applications of Optimization in Computer Graphics. PhD thesis, Robotics Institute, Carnegie Mellon University, Pittsburgh, PA
2511:
Shoichi
Fujimori, Shimpei Kobayashi and Wayne Rossman, Loop Group Methods for Constant Mean Curvature Surfaces. Rokko Lectures in Mathematics 2005
2288:
Carl Johan
Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2003:E11
2488:
Korevaar N., Kusner R., Solomon B., The structure of complete embedded surfaces with constant mean curvature, J. Diff. Geom. 30 (1989) 465–503.
2277:
Nick
Korevaar, Jesse Ratzkin, Nat Smale, Andrejs Treibergs, A survey of the classical theory of constant mean curvature surfaces in R3, 2002
1911:
2638:
2197:
can be used to produce approximations to CMC surfaces (or discrete counterparts), typically by minimizing a suitable energy functional.
2805:
1720:, which are spanned by a minimal patch that can be extended into a complete surface by reflection, and then turned into a CMC surface.
549:. Korevaar, Kusner and Solomon proved that a complete embedded CMC surface will have ends asymptotic to unduloids. Each end carries a
2554:
2497:
2437:
155:
2224:
2687:
1010:
2824:
2289:
2194:
2762:
2299:
C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl., 6 (1841), 309–320.
624:
1838:
1735:
2810:
2210:
901:
697:
79:
2690:. ACM Transactions on Graphics – SIGGRAPH 2012 Conference Proceedings. Volume 31 Issue 4, July 2012 Article No. 85
2095:
2440:. Coplanar constant mean curvature surfaces. Comm. Anal. Geom. 15:5 (2008) pp. 985–1023. ArXiv math.DG/0509210.
437:
with most topological types and at least two ends. Subsequently, Kapouleas constructed compact CMC surfaces in
2248:
2236:
2216:
Besides macroscopic bubble surfaces CMC surfaces are relevant for the shape of the gas–liquid interface on a
2308:
J. H. Jellet, Sur la
Surface dont la Courbure Moyenne est Constant, J. Math. Pures Appl., 18 (1853), 163–167
1732:, Sterling and Bobenko showed that all constant mean curvature immersions of a 2-torus into the space forms
1430:
935:
2422:
Rafe Mazzeo, Daniel
Pollack, Gluing and Moduli for Noncompact Geometric Problems. 1996 arXiv:dg-ga/9601008
2500:, A Complete Family of CMC Surfaces. In Integrable Systems, Geometry and Visualization, 2005, pp 237–245.
83:
2756:
2371:
405:
44:
1882:
1809:
1779:
1694:
1665:
1636:
1511:
975:
795:
523:
440:
411:
371:
334:
279:
220:
161:
128:
2736:
2686:
Hao Pan, Yi-King Choi, Yang Liu, Wenchao Hu, Qiang Du, Konrad Polthier, Caiming Zhang, Wenping Wang,
2464:
Existence and classification of constant mean curvature multibubbletons of finite and infinite type
2676:
2068:
2041:
859:
514:
Triunduloids with different neck sizes. As neck sizes are varied the asymptotic directions change.
2658:
2619:
2512:
2476:
404:
Up until this point it had seemed that CMC surfaces were rare. Using gluing techniques, in 1987
276:
conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in
63:
552:
772:
Like for minimal surfaces, there exist a close link to harmonic functions. An oriented surface
2001:
502:
2169:
2021:
1981:
594:
249:
190:
2744:
2650:
2611:
2582:
2352:
2228:
2217:
1593:
1361:
87:
56:
2602:
Pinkall, U.; Sterling, I. (1989). "On the classification of constant mean curvature tori".
2501:
2129:
1559:
1387:
2278:
2258:
2326:
H. Hopf, Differential geometry in the large. Springer-Verlag, Berlin, 1983. vii+184 pp.
1485:
308:
2740:
2239:
such as inflatable domes and enclosures, as well as a source of flowing organic shapes.
331:
sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in
2410:
2336:
2149:
1729:
1407:
881:
839:
775:
361:
108:
52:
17:
2832:
2662:
2654:
2541:
2451:
2777:
Observation of a non-constant mean curvature interface in an ABC triblock copolymer
2776:
2253:
2206:
826:
588:
490:
2340:
365:
2775:
Samuel P. Gido, Dwight W. Schwark, Edwin L. Thomas, Maria do Carmo Goncalves,
273:
214:
2587:
2570:
2357:
822:
154:
with constant mean curvature, then it is the standard sphere. Subsequently,
2722:"Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries"
2720:
Meinhard Wohlgemuth; Nataliya Yufa; James Hoffman; Edwin L. Thomas (2001).
520:
Meeks showed that there are no embedded CMC surfaces with just one end in
2815:
2477:
The topology and geometry of embedded surfaces of constant mean curvature
95:
91:
2463:
2441:
478:
408:
constructed a plethora of examples of complete immersed CMC surfaces in
86:
with constant mean curvature were the surfaces obtained by rotating the
2623:
2558:
2384:
2748:
2423:
2516:
99:
67:
2615:
1971:{\displaystyle (\Sigma ,\lambda ,\rho ,\lambda _{1},\lambda _{2},L)}
35:
2385:
Complete constant mean curvature surfaces in Euclidean three space
398:
34:
27:
26:
2411:
Constant mean curvature surfaces constructed by fusing Wente tori
2398:
Compact constant mean curvature surfaces in Euclidean three-space
2227:
there has been interest in periodic CMC surfaces as models for
1691:. This allows constructions starting from geodesic polygons in
62:
Note that these surfaces are generally different from constant
2639:"Surfaces of constant mean curvature and integrable equations"
829:. Kenmotsu’s representation formula is the counterpart to the
2529:
Weierstrass Formula for Surfaces of Prescribed Mean Curvature
59:
as a subset, but typically they are treated as special case.
2372:
Constant mean curvature surfaces in Euclidean three-space.
90:
of the conics. These are the plane, cylinder, sphere, the
2374:. Bull. Amer. Math. Soc. (N.S.) 17 (1987), no.2, 318–320.
2211:
curvature corresponding to a nonzero pressure difference
1093:{\displaystyle X(z)=\Re \int _{z_{0}}^{z}X_{z}(z')\,dz'}
2466:, Indiana Univ. Math. J. 42 (1993), no. 4, 1239–1266.
2172:
2152:
2132:
2098:
2071:
2044:
2024:
2004:
1984:
1914:
1885:
1841:
1812:
1782:
1738:
1697:
1668:
1639:
1596:
1562:
1514:
1488:
1433:
1410:
1390:
1364:
1112:
1013:
978:
938:
904:
884:
862:
842:
798:
778:
700:
627:
597:
555:
526:
443:
414:
374:
337:
311:
282:
252:
223:
193:
164:
131:
111:
1998:
is a hyperelliptic curve called the spectral curve,
2688:
Robust modeling of constant mean curvature surfaces
932:is a harmonic function into the Riemann sphere. If
2178:
2158:
2138:
2118:
2084:
2057:
2030:
2010:
1990:
1970:
1900:
1871:
1827:
1797:
1768:
1712:
1683:
1654:
1617:
1582:
1548:
1500:
1474:
1416:
1396:
1376:
1347:
1092:
996:
964:
924:
890:
870:
848:
813:
784:
743:
683:{\displaystyle \sum _{i=1}^{k}n_{i}\leq (k-1)\pi }
682:
609:
579:
541:
458:
429:
389:
352:
323:
297:
264:
238:
205:
179:
146:
117:
1872:{\displaystyle \mathbb {R} ^{3},\mathbb {S} ^{3}}
1769:{\displaystyle \mathbb {R} ^{3},\mathbb {S} ^{3}}
2804:CMC surfaces at the Scientific Graphics Project
2400:J. Differential Geom. 33 (1991), no. 3, 683-715.
2205:CMC surfaces are natural for representations of
898:be an arbitrary non-zero real constant. Suppose
39:Unduloid, a surface with constant mean curvature
1633:Lawson showed in 1970 that each CMC surface in
925:{\displaystyle \phi :V\rightarrow \mathbb {C} }
821:has constant mean curvature if and only if its
2571:"Harmonic maps from a 2-torus to the 3-sphere"
2235:In architecture CMC surfaces are relevant for
759: − 2 ends can be cylindrical.
744:{\displaystyle \sum _{i=1}^{k}n_{i}\leq k\pi }
66:surfaces, with the important exception of the
31:Nodoid, a surface with constant mean curvature
2779:, Macromolecules, 1993, 26 (10), pp 2636–2640
2436:Karsten Grosse-Brauckmann, Robert B. Kusner,
2387:Ann. of Math. (2) 131 (1990), no. 2, 239-330.
1662:has an isometric "cousin" minimal surface in
8:
2553:Karsten Grosse-Brauckmann, Robert B Kusner,
2341:"Counterexample to a conjecture of H. Hopf."
2113:
2107:
2751:. Archived from the original on 2015-06-23.
2544:”, Annals of Mathematics 92 (1970) 335–374.
2432:
2430:
2119:{\displaystyle \mathbb {C} \setminus \{0\}}
2413:Invent. Math. 119 (1995), no. 3, 443-518.
158:proved that a compact embedded surface in
2586:
2356:
2171:
2151:
2131:
2100:
2099:
2097:
2076:
2070:
2049:
2043:
2023:
2003:
1983:
1953:
1940:
1913:
1892:
1888:
1887:
1884:
1863:
1859:
1858:
1848:
1844:
1843:
1840:
1819:
1815:
1814:
1811:
1789:
1785:
1784:
1781:
1760:
1756:
1755:
1745:
1741:
1740:
1737:
1704:
1700:
1699:
1696:
1675:
1671:
1670:
1667:
1646:
1642:
1641:
1638:
1595:
1572:
1561:
1537:
1513:
1487:
1461:
1460:
1455:
1432:
1409:
1389:
1363:
1317:
1316:
1297:
1295:
1265:
1231:
1195:
1171:
1170:
1135:
1117:
1111:
1078:
1058:
1048:
1041:
1036:
1012:
977:
944:
943:
937:
918:
917:
903:
883:
864:
863:
861:
841:
805:
801:
800:
797:
777:
726:
716:
705:
699:
653:
643:
632:
626:
596:
554:
533:
529:
528:
525:
450:
446:
445:
442:
421:
417:
416:
413:
381:
377:
376:
373:
344:
340:
339:
336:
310:
289:
285:
284:
281:
272:must be a standard sphere. Based on this
251:
230:
226:
225:
222:
192:
171:
167:
166:
163:
138:
134:
133:
130:
110:
2270:
2104:
2754:
1475:{\displaystyle \phi (z)=-1/{\bar {z}}}
965:{\displaystyle \phi _{\bar {z}}\neq 0}
856:be an open simply connected subset of
49:constant-mean-curvature (CMC) surfaces
2146:is an antiholomorphic involution and
7:
831:Weierstrass–Enneper parameterization
125:is a compact star-shaped surface in
105:In 1853 J. H. Jellet showed that if
2809:GeometrieWerkstatt surface gallery
2479:, J. Diff. Geom. 27 (1988) 539–552.
2173:
2025:
1985:
1918:
1313:
1300:
1029:
25:
2839:Differential geometry of surfaces
217:proved that a sphere immersed in
2575:Journal of Differential Geometry
2225:triply periodic minimal surfaces
1908:, and spectral data of the form
1901:{\displaystyle \mathbb {H} ^{3}}
1828:{\displaystyle \mathbb {R} ^{2}}
1798:{\displaystyle \mathbb {H} ^{3}}
1713:{\displaystyle \mathbb {S} ^{3}}
1684:{\displaystyle \mathbb {S} ^{3}}
1655:{\displaystyle \mathbb {R} ^{3}}
1549:{\displaystyle \phi (z)=-e^{ix}}
1404:as Gauss map and mean curvature
997:{\displaystyle X:V\rightarrow R}
814:{\displaystyle \mathbb {R} ^{3}}
542:{\displaystyle \mathbb {R} ^{3}}
501:
489:
477:
459:{\displaystyle \mathbb {R} ^{3}}
430:{\displaystyle \mathbb {R} ^{3}}
390:{\displaystyle \mathbb {R} ^{3}}
353:{\displaystyle \mathbb {R} ^{4}}
298:{\displaystyle \mathbb {R} ^{n}}
239:{\displaystyle \mathbb {R} ^{3}}
180:{\displaystyle \mathbb {R} ^{3}}
147:{\displaystyle \mathbb {R} ^{3}}
2655:10.1070/RM1991v046n04ABEH002826
2542:Complete minimal surfaces in S3
2531:, Math. Ann., 245 (1979), 89–99
2345:Pacific Journal of Mathematics
2195:Discrete differential geometry
1965:
1915:
1524:
1518:
1466:
1443:
1437:
1337:
1331:
1322:
1307:
1292:
1289:
1283:
1271:
1262:
1255:
1243:
1228:
1221:
1209:
1192:
1188:
1182:
1176:
1167:
1161:
1149:
1129:
1123:
1075:
1064:
1023:
1017:
988:
949:
914:
674:
662:
621:-unduloids of genus 0 satisfy
574:
559:
401:with constant mean curvature.
1:
2819:Noid, software for computing
2814:GANG gallery of CMC surfaces
2462:I. Sterling and H. C. Wente,
2018:is a meromorphic function on
246:with constant mean curvature
187:with constant mean curvature
2186:obeying certain conditions.
2085:{\displaystyle \lambda _{2}}
2058:{\displaystyle \lambda _{1}}
1384:is a regular surface having
871:{\displaystyle \mathbb {C} }
305:must be a standard embedded
51:are surfaces with constant
2855:
2190:Discrete numerical methods
1508:this produces the sphere.
580:{\displaystyle n(2\pi -n)}
2761:: CS1 maint: unfit URL (
2358:10.2140/pjm.1986.121.193
2249:Double bubble conjecture
2237:air-supported structures
2011:{\displaystyle \lambda }
2637:Bobenko, A. I. (1991).
2569:Hitchin, Nigel (1990).
2179:{\displaystyle \Sigma }
2031:{\displaystyle \Sigma }
1991:{\displaystyle \Sigma }
1629:Conjugate cousin method
1590:gives a cylinder where
610:{\displaystyle k\geq 3}
265:{\displaystyle H\neq 0}
206:{\displaystyle H\neq 0}
18:Constant mean curvature
2588:10.4310/jdg/1214444631
2209:, since they have the
2180:
2160:
2140:
2120:
2086:
2059:
2032:
2012:
1992:
1972:
1902:
1873:
1829:
1799:
1770:
1714:
1685:
1656:
1619:
1618:{\displaystyle z=x+iv}
1584:
1550:
1502:
1476:
1418:
1398:
1378:
1377:{\displaystyle z\in V}
1349:
1094:
998:
966:
926:
892:
872:
850:
815:
786:
768:Representation formula
745:
721:
684:
648:
611:
581:
543:
460:
431:
391:
354:
325:
299:
266:
240:
213:must be a sphere, and
207:
181:
148:
119:
84:surfaces of revolution
40:
32:
2643:Russian Math. Surveys
2604:Annals of Mathematics
2370:Nikolaos Kapouleas.
2181:
2161:
2141:
2139:{\displaystyle \rho }
2121:
2087:
2060:
2033:
2013:
1993:
1973:
1903:
1874:
1830:
1800:
1771:
1715:
1686:
1657:
1620:
1585:
1583:{\displaystyle H=1/2}
1551:
1503:
1477:
1419:
1399:
1397:{\displaystyle \phi }
1379:
1350:
1095:
999:
967:
927:
893:
873:
851:
833:of minimal surfaces:
816:
787:
746:
701:
685:
628:
612:
591:. In particular, for
582:
544:
461:
432:
392:
355:
326:
300:
267:
241:
208:
182:
149:
120:
82:proved that the only
45:differential geometry
38:
30:
2409:Nikolaos Kapouleas.
2396:Nikolaos Kapouleas.
2383:Nikolaos Kapouleas.
2170:
2166:is a line bundle on
2150:
2130:
2096:
2069:
2042:
2022:
2002:
1982:
1912:
1883:
1839:
1810:
1780:
1736:
1695:
1666:
1637:
1594:
1560:
1512:
1486:
1431:
1408:
1388:
1362:
1110:
1011:
976:
936:
902:
882:
860:
840:
796:
776:
698:
625:
595:
553:
524:
441:
412:
372:
368:, an immersion into
335:
309:
280:
250:
221:
191:
162:
129:
109:
2823:-noid CMC surfaces
2741:2001MaMol..34.6083W
1501:{\displaystyle H=1}
1053:
324:{\displaystyle n-1}
2176:
2156:
2136:
2116:
2082:
2055:
2028:
2008:
1988:
1968:
1898:
1869:
1825:
1795:
1766:
1710:
1681:
1652:
1615:
1580:
1546:
1498:
1472:
1414:
1394:
1374:
1345:
1090:
1032:
994:
962:
922:
888:
868:
846:
811:
782:
763:Generation methods
741:
680:
607:
577:
539:
496:Unequal neck sizes
456:
427:
406:Nikolaos Kapouleas
387:
350:
321:
295:
262:
236:
203:
177:
144:
115:
64:Gaussian curvature
41:
33:
2749:10.1021/ma0019499
2735:(17): 6083–6089.
2159:{\displaystyle L}
1469:
1417:{\displaystyle H}
1329:
1325:
1310:
1202:
1179:
952:
891:{\displaystyle H}
849:{\displaystyle V}
785:{\displaystyle S}
118:{\displaystyle S}
16:(Redirected from
2846:
2792:
2786:
2780:
2773:
2767:
2766:
2760:
2752:
2726:
2717:
2711:
2707:
2701:
2697:
2691:
2684:
2678:
2673:
2667:
2666:
2634:
2628:
2627:
2599:
2593:
2592:
2590:
2566:
2560:
2551:
2545:
2538:
2532:
2525:
2519:
2509:
2503:
2498:John M. Sullivan
2495:
2489:
2486:
2480:
2473:
2467:
2460:
2454:
2449:
2443:
2438:John M. Sullivan
2434:
2425:
2420:
2414:
2407:
2401:
2394:
2388:
2381:
2375:
2368:
2362:
2361:
2360:
2333:
2327:
2324:
2318:
2315:
2309:
2306:
2300:
2297:
2291:
2286:
2280:
2275:
2229:block copolymers
2218:superhydrophobic
2185:
2183:
2182:
2177:
2165:
2163:
2162:
2157:
2145:
2143:
2142:
2137:
2125:
2123:
2122:
2117:
2103:
2091:
2089:
2088:
2083:
2081:
2080:
2064:
2062:
2061:
2056:
2054:
2053:
2037:
2035:
2034:
2029:
2017:
2015:
2014:
2009:
1997:
1995:
1994:
1989:
1977:
1975:
1974:
1969:
1958:
1957:
1945:
1944:
1907:
1905:
1904:
1899:
1897:
1896:
1891:
1878:
1876:
1875:
1870:
1868:
1867:
1862:
1853:
1852:
1847:
1834:
1832:
1831:
1826:
1824:
1823:
1818:
1804:
1802:
1801:
1796:
1794:
1793:
1788:
1775:
1773:
1772:
1767:
1765:
1764:
1759:
1750:
1749:
1744:
1719:
1717:
1716:
1711:
1709:
1708:
1703:
1690:
1688:
1687:
1682:
1680:
1679:
1674:
1661:
1659:
1658:
1653:
1651:
1650:
1645:
1624:
1622:
1621:
1616:
1589:
1587:
1586:
1581:
1576:
1555:
1553:
1552:
1547:
1545:
1544:
1507:
1505:
1504:
1499:
1481:
1479:
1478:
1473:
1471:
1470:
1462:
1459:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1354:
1352:
1351:
1346:
1344:
1340:
1330:
1328:
1327:
1326:
1318:
1311:
1306:
1298:
1296:
1270:
1269:
1236:
1235:
1203:
1201:
1200:
1199:
1181:
1180:
1172:
1144:
1136:
1122:
1121:
1099:
1097:
1096:
1091:
1089:
1074:
1063:
1062:
1052:
1047:
1046:
1045:
1003:
1001:
1000:
995:
971:
969:
968:
963:
955:
954:
953:
945:
931:
929:
928:
923:
921:
897:
895:
894:
889:
877:
875:
874:
869:
867:
855:
853:
852:
847:
820:
818:
817:
812:
810:
809:
804:
791:
789:
788:
783:
750:
748:
747:
742:
731:
730:
720:
715:
689:
687:
686:
681:
658:
657:
647:
642:
616:
614:
613:
608:
586:
584:
583:
578:
548:
546:
545:
540:
538:
537:
532:
505:
493:
484:Equal neck sizes
481:
465:
463:
462:
457:
455:
454:
449:
436:
434:
433:
428:
426:
425:
420:
396:
394:
393:
388:
386:
385:
380:
364:constructed the
359:
357:
356:
351:
349:
348:
343:
330:
328:
327:
322:
304:
302:
301:
296:
294:
293:
288:
271:
269:
268:
263:
245:
243:
242:
237:
235:
234:
229:
212:
210:
209:
204:
186:
184:
183:
178:
176:
175:
170:
156:A. D. Alexandrov
153:
151:
150:
145:
143:
142:
137:
124:
122:
121:
116:
57:minimal surfaces
55:. This includes
21:
2854:
2853:
2849:
2848:
2847:
2845:
2844:
2843:
2829:
2828:
2801:
2796:
2795:
2787:
2783:
2774:
2770:
2753:
2724:
2719:
2718:
2714:
2708:
2704:
2698:
2694:
2685:
2681:
2674:
2670:
2636:
2635:
2631:
2616:10.2307/1971425
2601:
2600:
2596:
2568:
2567:
2563:
2555:John M Sullivan
2552:
2548:
2539:
2535:
2526:
2522:
2510:
2506:
2496:
2492:
2487:
2483:
2474:
2470:
2461:
2457:
2450:
2446:
2435:
2428:
2421:
2417:
2408:
2404:
2395:
2391:
2382:
2378:
2369:
2365:
2337:Wente, Henry C.
2335:
2334:
2330:
2325:
2321:
2316:
2312:
2307:
2303:
2298:
2294:
2287:
2283:
2276:
2272:
2267:
2259:Minimal surface
2245:
2203:
2192:
2168:
2167:
2148:
2147:
2128:
2127:
2094:
2093:
2072:
2067:
2066:
2045:
2040:
2039:
2020:
2019:
2000:
1999:
1980:
1979:
1949:
1936:
1910:
1909:
1886:
1881:
1880:
1857:
1842:
1837:
1836:
1813:
1808:
1807:
1783:
1778:
1777:
1754:
1739:
1734:
1733:
1726:
1698:
1693:
1692:
1669:
1664:
1663:
1640:
1635:
1634:
1631:
1592:
1591:
1558:
1557:
1533:
1510:
1509:
1484:
1483:
1429:
1428:
1406:
1405:
1386:
1385:
1360:
1359:
1312:
1299:
1261:
1227:
1208:
1204:
1191:
1145:
1137:
1113:
1108:
1107:
1082:
1067:
1054:
1037:
1009:
1008:
974:
973:
939:
934:
933:
900:
899:
880:
879:
858:
857:
838:
837:
799:
794:
793:
774:
773:
770:
765:
722:
696:
695:
649:
623:
622:
593:
592:
551:
550:
527:
522:
521:
518:
517:
516:
515:
511:
510:
509:
508:With nodoid end
506:
498:
497:
494:
486:
485:
482:
470:
444:
439:
438:
415:
410:
409:
375:
370:
369:
338:
333:
332:
307:
306:
283:
278:
277:
248:
247:
224:
219:
218:
189:
188:
165:
160:
159:
132:
127:
126:
107:
106:
76:
23:
22:
15:
12:
11:
5:
2852:
2850:
2842:
2841:
2831:
2830:
2827:
2826:
2817:
2812:
2807:
2800:
2799:External links
2797:
2794:
2793:
2781:
2768:
2729:Macromolecules
2712:
2702:
2692:
2679:
2668:
2629:
2610:(2): 407–451.
2594:
2581:(3): 627–710.
2561:
2546:
2540:Lawson H.B., “
2533:
2520:
2504:
2490:
2481:
2468:
2455:
2444:
2426:
2415:
2402:
2389:
2376:
2363:
2328:
2319:
2310:
2301:
2292:
2281:
2269:
2268:
2266:
2263:
2262:
2261:
2256:
2251:
2244:
2241:
2202:
2199:
2191:
2188:
2175:
2155:
2135:
2115:
2112:
2109:
2106:
2102:
2092:are points on
2079:
2075:
2052:
2048:
2027:
2007:
1987:
1967:
1964:
1961:
1956:
1952:
1948:
1943:
1939:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1895:
1890:
1866:
1861:
1856:
1851:
1846:
1822:
1817:
1792:
1787:
1763:
1758:
1753:
1748:
1743:
1725:
1722:
1707:
1702:
1678:
1673:
1649:
1644:
1630:
1627:
1614:
1611:
1608:
1605:
1602:
1599:
1579:
1575:
1571:
1568:
1565:
1543:
1540:
1536:
1532:
1529:
1526:
1523:
1520:
1517:
1497:
1494:
1491:
1468:
1465:
1458:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1413:
1393:
1373:
1370:
1367:
1356:
1355:
1343:
1339:
1336:
1333:
1324:
1321:
1315:
1309:
1305:
1302:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1268:
1264:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1234:
1230:
1226:
1223:
1220:
1217:
1214:
1211:
1207:
1198:
1194:
1190:
1187:
1184:
1178:
1175:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1143:
1140:
1134:
1131:
1128:
1125:
1120:
1116:
1101:
1100:
1088:
1085:
1081:
1077:
1073:
1070:
1066:
1061:
1057:
1051:
1044:
1040:
1035:
1031:
1028:
1025:
1022:
1019:
1016:
993:
990:
987:
984:
981:
961:
958:
951:
948:
942:
920:
916:
913:
910:
907:
887:
866:
845:
808:
803:
781:
769:
766:
764:
761:
751:for even
740:
737:
734:
729:
725:
719:
714:
711:
708:
704:
679:
676:
673:
670:
667:
664:
661:
656:
652:
646:
641:
638:
635:
631:
606:
603:
600:
576:
573:
570:
567:
564:
561:
558:
536:
531:
513:
512:
507:
500:
499:
495:
488:
487:
483:
476:
475:
474:
473:
472:
453:
448:
424:
419:
384:
379:
362:Henry C. Wente
347:
342:
320:
317:
314:
292:
287:
261:
258:
255:
233:
228:
202:
199:
196:
174:
169:
141:
136:
114:
75:
72:
53:mean curvature
24:
14:
13:
10:
9:
6:
4:
3:
2:
2851:
2840:
2837:
2836:
2834:
2825:
2822:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2802:
2798:
2791:
2785:
2782:
2778:
2772:
2769:
2764:
2758:
2750:
2746:
2742:
2738:
2734:
2730:
2723:
2716:
2713:
2706:
2703:
2696:
2693:
2689:
2683:
2680:
2677:
2672:
2669:
2664:
2660:
2656:
2652:
2648:
2644:
2640:
2633:
2630:
2625:
2621:
2617:
2613:
2609:
2605:
2598:
2595:
2589:
2584:
2580:
2576:
2572:
2565:
2562:
2559:
2556:
2550:
2547:
2543:
2537:
2534:
2530:
2527:K. Kenmotsu,
2524:
2521:
2518:
2514:
2508:
2505:
2502:
2499:
2494:
2491:
2485:
2482:
2478:
2475:Meeks W. H.,
2472:
2469:
2465:
2459:
2456:
2452:
2448:
2445:
2442:
2439:
2433:
2431:
2427:
2424:
2419:
2416:
2412:
2406:
2403:
2399:
2393:
2390:
2386:
2380:
2377:
2373:
2367:
2364:
2359:
2354:
2350:
2346:
2342:
2338:
2332:
2329:
2323:
2320:
2314:
2311:
2305:
2302:
2296:
2293:
2290:
2285:
2282:
2279:
2274:
2271:
2264:
2260:
2257:
2255:
2252:
2250:
2247:
2246:
2242:
2240:
2238:
2233:
2230:
2226:
2221:
2219:
2214:
2212:
2208:
2200:
2198:
2196:
2189:
2187:
2153:
2133:
2110:
2077:
2073:
2050:
2046:
2005:
1962:
1959:
1954:
1950:
1946:
1941:
1937:
1933:
1930:
1927:
1924:
1921:
1893:
1864:
1854:
1849:
1820:
1790:
1761:
1751:
1746:
1731:
1723:
1721:
1705:
1676:
1647:
1628:
1626:
1612:
1609:
1606:
1603:
1600:
1597:
1577:
1573:
1569:
1566:
1563:
1541:
1538:
1534:
1530:
1527:
1521:
1515:
1495:
1492:
1489:
1463:
1456:
1452:
1449:
1446:
1440:
1434:
1425:
1411:
1391:
1371:
1368:
1365:
1341:
1334:
1319:
1303:
1286:
1280:
1277:
1274:
1266:
1258:
1252:
1249:
1246:
1240:
1237:
1232:
1224:
1218:
1215:
1212:
1205:
1196:
1185:
1173:
1164:
1158:
1155:
1152:
1146:
1141:
1138:
1132:
1126:
1118:
1114:
1106:
1105:
1104:
1086:
1083:
1079:
1071:
1068:
1059:
1055:
1049:
1042:
1038:
1033:
1026:
1020:
1014:
1007:
1006:
1005:
991:
985:
982:
979:
959:
956:
946:
940:
911:
908:
905:
885:
843:
834:
832:
828:
824:
806:
779:
767:
762:
760:
758:
754:
738:
735:
732:
727:
723:
717:
712:
709:
706:
702:
693:
677:
671:
668:
665:
659:
654:
650:
644:
639:
636:
633:
629:
620:
604:
601:
598:
590:
589:moduli spaces
571:
568:
565:
562:
556:
534:
504:
492:
480:
471:
468:
451:
422:
407:
402:
400:
382:
367:
363:
345:
318:
315:
312:
290:
275:
259:
256:
253:
231:
216:
200:
197:
194:
172:
157:
139:
112:
103:
101:
97:
93:
89:
85:
81:
73:
71:
69:
65:
60:
58:
54:
50:
46:
37:
29:
19:
2820:
2784:
2771:
2757:cite journal
2732:
2728:
2715:
2705:
2695:
2682:
2671:
2646:
2642:
2632:
2607:
2603:
2597:
2578:
2574:
2564:
2549:
2536:
2523:
2517:math/0602570
2507:
2493:
2484:
2471:
2458:
2447:
2418:
2405:
2392:
2379:
2366:
2348:
2344:
2331:
2322:
2313:
2304:
2295:
2284:
2273:
2254:Free surface
2234:
2222:
2215:
2207:soap bubbles
2204:
2201:Applications
2193:
1727:
1632:
1426:
1357:
1102:
835:
827:harmonic map
771:
756:
752:
691:
618:
519:
469:
403:
104:
77:
61:
48:
42:
2649:(4): 1–45.
2351:: 193–243,
1004:defined by
366:Wente torus
2606:. Second.
2265:References
755:. At most
360:. In 1984
2663:250883973
2220:surface.
2174:Σ
2134:ρ
2105:∖
2074:λ
2047:λ
2026:Σ
2006:λ
1986:Σ
1951:λ
1938:λ
1931:ρ
1925:λ
1919:Σ
1728:Hitchin,
1531:−
1516:ϕ
1467:¯
1450:−
1435:ϕ
1392:ϕ
1369:∈
1323:¯
1314:∂
1308:¯
1304:ϕ
1301:∂
1281:ϕ
1253:ϕ
1219:ϕ
1216:−
1177:¯
1174:ϕ
1159:ϕ
1139:−
1034:∫
1030:ℜ
989:→
957:≠
950:¯
941:ϕ
915:→
906:ϕ
823:Gauss map
739:π
733:≤
703:∑
678:π
669:−
660:≤
630:∑
617:coplanar
602:≥
569:−
566:π
316:−
257:≠
198:≠
88:roulettes
2833:Category
2339:(1986),
2243:See also
1724:CMC Tori
1087:′
1072:′
690:for odd
96:unduloid
92:catenoid
80:Delaunay
78:In 1841
2737:Bibcode
2710:337–396
2700:184–198
2624:1971425
1730:Pinkall
274:H. Hopf
215:H. Hopf
74:History
2661:
2622:
1978:where
694:, and
100:nodoid
94:, the
68:sphere
2725:(PDF)
2659:S2CID
2620:JSTOR
2513:arXiv
2223:Like
1835:into
1103:with
972:then
825:is a
399:torus
397:of a
2763:link
2065:and
1879:and
1776:and
1556:and
1482:and
1427:For
1358:for
878:and
836:Let
98:and
2745:doi
2651:doi
2612:doi
2608:130
2583:doi
2353:doi
2349:121
792:in
43:In
2835::
2759:}}
2755:{{
2743:.
2733:34
2731:.
2727:.
2657:.
2647:46
2645:.
2641:.
2618:.
2579:31
2577:.
2573:.
2429:^
2347:,
2343:,
2213:.
2126:,
2038:,
1625:.
1424:.
102:.
70:.
47:,
2821:n
2765:)
2747::
2739::
2665:.
2653::
2626:.
2614::
2591:.
2585::
2515::
2355::
2154:L
2114:}
2111:0
2108:{
2101:C
2078:2
2051:1
1966:)
1963:L
1960:,
1955:2
1947:,
1942:1
1934:,
1928:,
1922:,
1916:(
1894:3
1889:H
1865:3
1860:S
1855:,
1850:3
1845:R
1821:2
1816:R
1791:3
1786:H
1762:3
1757:S
1752:,
1747:3
1742:R
1706:3
1701:S
1677:3
1672:S
1648:3
1643:R
1613:v
1610:i
1607:+
1604:x
1601:=
1598:z
1578:2
1574:/
1570:1
1567:=
1564:H
1542:x
1539:i
1535:e
1528:=
1525:)
1522:z
1519:(
1496:1
1493:=
1490:H
1464:z
1457:/
1453:1
1447:=
1444:)
1441:z
1438:(
1412:H
1372:V
1366:z
1342:}
1338:)
1335:z
1332:(
1320:z
1293:)
1290:)
1287:z
1284:(
1278:2
1275:,
1272:)
1267:2
1263:)
1259:z
1256:(
1250:+
1247:1
1244:(
1241:i
1238:,
1233:2
1229:)
1225:z
1222:(
1213:1
1210:(
1206:{
1197:2
1193:)
1189:)
1186:z
1183:(
1168:)
1165:z
1162:(
1156:+
1153:1
1150:(
1147:H
1142:1
1133:=
1130:)
1127:z
1124:(
1119:z
1115:X
1084:z
1080:d
1076:)
1069:z
1065:(
1060:z
1056:X
1050:z
1043:0
1039:z
1027:=
1024:)
1021:z
1018:(
1015:X
992:R
986:V
983::
980:X
960:0
947:z
919:C
912:V
909::
886:H
865:C
844:V
807:3
802:R
780:S
757:k
753:k
736:k
728:i
724:n
718:k
713:1
710:=
707:i
692:k
675:)
672:1
666:k
663:(
655:i
651:n
645:k
640:1
637:=
634:i
619:k
605:3
599:k
575:)
572:n
563:2
560:(
557:n
535:3
530:R
452:3
447:R
423:3
418:R
383:3
378:R
346:4
341:R
319:1
313:n
291:n
286:R
260:0
254:H
232:3
227:R
201:0
195:H
173:3
168:R
140:3
135:R
113:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
