920:
478:
29:
915:{\displaystyle {\begin{aligned}\Gamma (g)_{ij}^{k}&={\frac {1}{2}}\sum _{\ell =1}^{m}g^{k\ell }{\Big (}{\frac {\partial g_{j\ell }}{\partial x^{i}}}+{\frac {\partial g_{i\ell }}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{\ell }}}{\Big )}\\\Gamma (h)_{\alpha \beta }^{\gamma }&={\frac {1}{2}}\sum _{\delta =1}^{n}h^{\gamma \delta }{\Big (}{\frac {\partial h_{\beta \delta }}{\partial y^{\alpha }}}+{\frac {\partial h_{\alpha \delta }}{\partial y^{\beta }}}-{\frac {\partial h_{\alpha \beta }}{\partial y^{\delta }}}{\Big )}\end{aligned}}}
1313:
980:
4203:
2194:
1308:{\displaystyle \nabla (df)_{ij}^{\alpha }={\frac {\partial ^{2}f^{\alpha }}{\partial x^{i}\partial x^{j}}}-\sum _{k=1}^{m}\Gamma (g)_{ij}^{k}{\frac {\partial f^{\alpha }}{\partial x^{k}}}+\sum _{\beta =1}^{n}\sum _{\gamma =1}^{n}{\frac {\partial f^{\beta }}{\partial x^{i}}}{\frac {\partial f^{\gamma }}{\partial x^{j}}}\Gamma (h)_{\beta \gamma }^{\alpha }\circ f.}
4408:
3992:
2733:
3754:
to a harmonic map. The very existence of a harmonic map in each homotopy class, which is implicitly being asserted, is part of the result. This is proven by constructing a heat equation, and showing that for any map as initial condition, solution that exists for all time, and the solution uniformly
152:
and a smooth stone can both be naturally viewed as
Riemannian manifolds. Any way of stretching the rubber band over the stone can be viewed as a mapping between these manifolds, and the total tension involved is represented by the Dirichlet energy. Harmonicity of such a mapping means that, given any
4665:
must be biholomorphic or anti-biholomorphic if they are homotopic to each other; the biholomorphism (or anti-biholomorphism) is precisely the harmonic map produced as the limit of the harmonic map heat flow with initial data given by the homotopy. By an alternative formulation of the same approach,
3777:
extended their methods to study uniqueness of harmonic maps within homotopy classes, additionally showing that the convergence in the EellsâSampson theorem is strong, without the need to select a subsequence. That is, if two maps are initially close, the distance between the corresponding solutions
3875:
was necessary. Following the work of Kung-Ching Chang, Wei-Yue Ding, and Rugang Ye in 1992, it is widely accepted that the maximal time of existence of a harmonic map heat flow cannot "usually" be expected to be infinite. Their results strongly suggest that there are harmonic map heat flows with
1979:
5084:
3900:
are taken to be the two-dimensional sphere with its standard metric. Since elliptic and parabolic partial differential equations are particularly smooth when the domain is two dimensions, the ChangâDingâYe result is considered to be indicative of the general character of the flow.
4232:
4198:{\displaystyle {\Big (}{\frac {\partial }{\partial t}}-\Delta ^{g}{\Big )}e(f)=-{\big |}\nabla (df){\big |}^{2}-{\big \langle }\operatorname {Ric} ^{g},f^{\ast }h{\big \rangle }_{g}+\operatorname {scal} ^{g}{\big (}f^{\ast }\operatorname {Rm} ^{h}{\big )}.}
2546:
1466:
3928:
of the harmonic map heat flow. Moreover, he found that his weak solutions are smooth away from finitely many spacetime points at which the energy density concentrates. On microscopic levels, the flow near these points is modeled by a
1905:
2189:{\displaystyle {\frac {1}{2}}\sum _{i=1}^{m}\sum _{j=1}^{m}\sum _{\alpha =1}^{n}\sum _{\beta =1}^{n}g^{ij}{\frac {\partial f^{\alpha }}{\partial x^{i}}}{\frac {\partial f^{\beta }}{\partial x^{j}}}(h_{\alpha \beta }\circ f).}
3941:
at singular times, meaning that the
Dirichlet energy of Struwe's weak solution, at a singular time, drops by exactly the sum of the total Dirichlet energies of the bubbles corresponding to singularities at that time.
4889:
4620:
must be holomorphic, provided that the target manifold has appropriately negative curvature. As an application, by making use of the EellsâSampson existence theorem for harmonic maps, he was able to show that if
4403:{\displaystyle \Delta ^{g}e(f)={\big |}\nabla (df){\big |}^{2}+{\big \langle }\operatorname {Ric} ^{g},f^{\ast }h{\big \rangle }_{g}-\operatorname {scal} ^{g}{\big (}f^{\ast }\operatorname {Rm} ^{h}{\big )}.}
4611:
The general idea of deforming a general map to a harmonic map, and then showing that any such harmonic map must automatically be of a highly restricted class, has found many applications. For instance,
3332:
191:, has been particularly influential, as their analysis has been adapted to many other geometric contexts. Notably, Uhlenbeck's parallel discovery of bubbling of YangâMills fields is important in
483:
2358:
5419:
4751:
One of the original ideas in grid generation methods for computational fluid dynamics and computational physics was to use either conformal or harmonic mapping to generate regular grids.
4829:
2728:{\displaystyle \int _{M}{\frac {\partial }{\partial s}}{\Big |}_{s=0}e(f_{s})\,d\mu _{g}=-\int _{M}h\left({\frac {\partial }{\partial s}}{\Big |}_{s=0}f_{s},\Delta f\right)\,d\mu _{g}}
153:
hypothetical way of physically deforming the given stretch, the tension (when considered as a function of time) has first derivative equal to zero when the deformation begins.
4785:
4855:
4701:. By an extension of the EellsâSampson theorem together with an extension of the SiuâCorlette Bochner formula, they were able to prove new rigidity theorems for lattices.
1344:
3813:
215:
and
Uhlenbeck to study the regularity theory of harmonic maps have likewise been the inspiration for the development of many analytic methods in geometric analysis.
5387:
5772:
5331:
3853:
3833:
1785:
5227:
5545:
This means that, relative to any local coordinate charts, one has uniform convergence on compact sets of the functions and their first partial derivatives.
5744:
5459:
3953:. Their results achieved less than in low dimensions, only being able to prove existence of weak solutions which are smooth on open dense subsets.
2759:
of the
Dirichlet energy; correspondingly, a harmonic map is a critical point of the Dirichlet energy. This can be done formally in the language of
6900:. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). Vol. 11 (Second edition of 2005 original ed.). Pisa: Edizioni della Normale.
5079:{\displaystyle e_{\epsilon }(u)(x)={\frac {\int _{M}d^{2}(u(x),u(y))\,d\mu _{x}^{\epsilon }(y)}{\int _{M}d^{2}(x,y)\,d\mu _{x}^{\epsilon }(y)}}}
6128:
6854:
6812:
3288:
7165:
7115:
6967:
6913:
5629:
3024:
being geodesic, although this does not imply that it is a constant-speed parametrization, and hence does not imply that
7202:. Conference Proceedings and Lecture Notes in Geometry and Topology. Vol. 2. Cambridge, MA: International Press.
7311:
7257:
7207:
7065:
7023:
6780:
6730:
6680:
6554:
6454:
72:
50:
43:
7354:
6608:
6406:
6342:
5877:
3863:
For many years after Eells and
Sampson's work, it was unclear to what extent the sectional curvature assumption on
2295:
6452:(1976). "Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature".
6182:
6070:
4790:
6115:
6714:
5724:
98:
5499:
7369:
6947:
5203:
3763:
4616:
found an important complex-analytic version of the
Bochner formula, asserting that a harmonic map between
7379:
7374:
3945:
Struwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold is
3337:
Eells and
Sampson introduced the harmonic map heat flow and proved the following fundamental properties:
6502:(1980). "The complex-analyticity of harmonic maps and the strong rigidity of compact KĂ€hler manifolds".
5519:
3597:
as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that
7295:
7007:
4682:
1609:
1517:
256:
196:
4758:
5922:
4717:
Once existence is known, how can a harmonic map be constructed explicitly? (One fruitful method uses
4600:
is a closed
Riemannian manifold with nonpositive sectional curvature, then every continuous map from
3079:
4834:
1461:{\displaystyle (\Delta f)^{\alpha }=\sum _{i=1}^{m}\sum _{j=1}^{m}g^{ij}\nabla (df)_{ij}^{\alpha }.}
5291:
4711:
251:) is defined via the second fundamental form, and its vanishing is the condition for the map to be
37:
5367:
3961:
The main computational point in the proof of Eells and
Sampson's theorem is an adaptation of the
2914:
200:
102:
6993:
6124:"Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one"
4673:
Kevin
Corlette found a significant extension of Siu's Bochner formula, and used it to prove new
6898:
An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs
3792:
1724:
1613:
1526:
54:
5630:"Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3"
2258:-trace of the first fundamental form. Regardless of the perspective taken, the energy density
16:
This article is about harmonic maps between Riemannian manifolds. For harmonic functions, see
7240:. Lectures in Mathematics ETH ZĂŒrich. Based on lecture notes by Norbert HungerbĂŒhler. Basel:
6504:
6283:
5969:
5580:
5259:
5101:
86:
7321:
7267:
7217:
7175:
7125:
7075:
6997:
6977:
6923:
6875:
6833:
6790:
6740:
6690:
6631:
6575:
6533:
6475:
6429:
6365:
6312:
6253:
6205:
6149:
6099:
6045:
5998:
5943:
5900:
5179:
4729:
4484:
3962:
2493:; one supposes that the parametrized family is smooth in the sense that the associated map
1625:
334:
7329:
7275:
7241:
7225:
7183:
7133:
7083:
7049:
7033:
6985:
6931:
6883:
6841:
6798:
6748:
6698:
6639:
6591:
6541:
6491:
6437:
6373:
6320:
6269:
6221:
6165:
6107:
6053:
6006:
5959:
5908:
1900:{\displaystyle (\Delta f)_{p}=\sum _{i=1}^{m}{\big (}\nabla (df){\big )}_{p}(e_{i},e_{i})}
8:
6939:
4725:
4422:
3759:
3638:
3101:
2948:
2381:
1501:
224:
204:
187:
The discovery of the "bubbling" of sequences of harmonic maps, due to Jonathan Sacks and
169:
114:
90:
5920:(1989). "Existence and partial regularity results for the heat flow for harmonic maps".
6579:
6521:
6479:
6300:
6257:
6209:
6153:
6087:
5986:
5947:
5592:
5311:
3838:
3818:
434:
208:
181:
4617:
3933:, i.e. a smooth harmonic map from the round 2-sphere into the target. Weiyue Ding and
3090:
7307:
7253:
7203:
7161:
7111:
7061:
7019:
6963:
6909:
6893:
6776:
6726:
6676:
6583:
6483:
6261:
6157:
6123:
6065:
5649:
5610:
3835:
is homotopic to a map, such that the map is totally geodesic when restricted to each
3097:
2385:
2254:. It is also possible to consider the energy density as being given by (half of) the
391:
228:
161:
118:
17:
6213:
5951:
4226:
solution of the harmonic map heat flow, and so one gets from the above formula that
7325:
7299:
7271:
7245:
7221:
7179:
7153:
7149:
7129:
7103:
7079:
7053:
7029:
7011:
6981:
6955:
6927:
6901:
6879:
6863:
6837:
6821:
6794:
6768:
6764:
6744:
6718:
6694:
6668:
6635:
6617:
6587:
6563:
6537:
6513:
6487:
6463:
6433:
6415:
6380:
6369:
6351:
6316:
6292:
6265:
6241:
6217:
6191:
6161:
6137:
6103:
6079:
6049:
6031:
6002:
5978:
5955:
5931:
5904:
5886:
5641:
5602:
4737:
4733:
4667:
2813:
338:
106:
3924:
is two-dimensional, he established the unconditional existence and uniqueness for
7317:
7263:
7213:
7171:
7121:
7071:
6973:
6951:
6919:
6871:
6829:
6786:
6736:
6686:
6664:
6627:
6571:
6529:
6471:
6425:
6397:
6361:
6333:
6308:
6278:
6249:
6201:
6145:
6095:
6041:
5994:
5939:
5896:
4576:
In combination with the EellsâSampson theorem, this shows (for instance) that if
4414:
3950:
3946:
3086:
2836:
is harmonic; this follows directly from the above definitions. As special cases:
2764:
2760:
342:
276:
192:
188:
180:, in and of themselves, are among the most widely studied topics in the field of
7349:
7344:
7286:(1982). "Survey on partial differential equations in differential geometry". In
6713:. CBMS Regional Conference Series in Mathematics. Vol. 50. Providence, RI:
5645:
4710:
Existence results on harmonic maps between manifolds has consequences for their
7287:
7283:
7195:
7191:
7099:
6599:
6549:
6449:
6445:
6393:
6329:
6173:
6119:
5917:
5117:
4718:
4476:
4472:
3905:
3774:
232:
212:
7303:
7249:
7107:
7091:
7057:
7041:
6905:
6672:
6229:
6036:
6019:
7363:
7015:
6656:
6622:
6603:
6420:
6401:
6384:
6356:
6337:
5967:
Corlette, Kevin (1992). "Archimedean superrigidity and hyperbolic geometry".
5891:
5872:
5660:
5653:
5614:
5163:
4674:
3925:
2208:
1572:
1530:
6867:
6499:
6196:
6177:
4880:
4698:
4613:
1779:. By the definition of the trace operator, the laplacian may be written as
1671:
7157:
6825:
6772:
3782:
the set of totally geodesic maps in each homotopy class is path-connected;
7141:
7003:
6849:
6807:
6756:
6706:
6061:
4741:
4685:
and Richard Schoen extended much of the theory of harmonic maps to allow
2399:
for the Dirichlet energy compute the derivatives of the Dirichlet energy
164:, who showed that in certain geometric contexts, arbitrary maps could be
157:
149:
6552:(1985). "On the evolution of harmonic mappings of Riemannian surfaces".
6129:
Publications MathĂ©matiques de l'Institut des Hautes Ătudes Scientifiques
6020:"Energy identity for a class of approximate harmonic maps from surfaces"
4865:
The energy integral can be formulated in a weaker setting for functions
7233:
6959:
6722:
6567:
6525:
6467:
6304:
6245:
6141:
6091:
5990:
5935:
5606:
4745:
3609:
The primary result of Eells and Sampson's 1964 paper is the following:
2808:
is used to refer to the standard Riemannian metric on Euclidean space.
173:
101:. This partial differential equation for a mapping also arises as the
7002:. Cambridge Tracts in Mathematics. Vol. 150. With a foreword by
6015:
5873:"Finite-time blow-up of the heat flow of harmonic maps from surfaces"
5141:
5139:
5137:
4678:
3934:
6517:
6296:
6083:
5982:
5597:
4496:
must be constant. In summary, according to these results, one has:
4488:
3751:
3000:
2756:
272:
165:
110:
109:. As such, the theory of harmonic maps contains both the theory of
5134:
4588:
is a closed Riemannian manifold with positive Ricci curvature and
3522:
are two harmonic map heat flows as in the existence theorem, then
2414:
is deformed. To this end, consider a one-parameter family of maps
2392:
is compact; however, then the Dirichlet energy could be infinite.
2227:, and so one may define the energy density as the smooth function
2199:
Alternatively, in the bundle formalism, the Riemannian metrics on
6402:"Boundary regularity and the Dirichlet problem for harmonic maps"
7238:
Theorems on regularity and singularity of energy minimizing maps
4645:
are smooth and closed KĂ€hler manifolds, and if the curvature of
3637:
be smooth and closed Riemannian manifolds, and suppose that the
3078:. This coincides with the notion of harmonicity provided by the
255:. The definitions extend without modification to the setting of
7294:. Annals of Mathematics Studies. Vol. 102. Princeton, NJ:
1961:
From the perspective of local coordinates, as given above, the
5848:
5800:
4787:
between Riemannian manifolds is totally geodesic if, whenever
3601:
takes on its maximal possible value, which could be infinite.
1956:
97:
if its coordinate representatives satisfy a certain nonlinear
5548:
4222:
is harmonic; any harmonic map can be viewed as a constant-in-
4208:
This is also of interest in analyzing harmonic maps. Suppose
3785:
all harmonic maps are energy-minimizing and totally geodesic.
3653:
is nonpositive. Then for any continuously differentiable map
7098:. Universitext (Seventh edition of 1995 original ed.).
6281:(1981). "The existence of minimal immersions of 2-spheres".
5628:
Cao, Jianguo; Cheeger, Jeff; Rong, Xiaochun (January 2004).
5560:
5581:"Totally geodesic maps into manifolds with no focal points"
3904:
Modeled upon the fundamental works of Sacks and Uhlenbeck,
3566:
As a consequence of the uniqueness theorem, there exists a
5279:
4883:. The energy integrand is instead a function of the form
4670:, albeit in the restricted context of negative curvature.
4483:
by making use of Yau's theorem asserting that nonnegative
3778:
to the heat equation is nonincreasing for all time, thus:
3341:
Regularity. Any harmonic map heat flow is smooth as a map
3327:{\displaystyle {\frac {\partial f}{\partial t}}=\Delta f.}
3236:
is differentiable, and its derivative at a given value of
6852:; Lemaire, L. (1988). "Another report on harmonic maps".
6234:
Calculus of Variations and Partial Differential Equations
5487:
5151:
3949:; he and Yun Mei Chen also considered higher-dimensional
3726:
In particular, this shows that, under the assumptions on
218:
6604:"On the evolution of harmonic maps in higher dimensions"
5836:
4479:
noted that this reasoning can be extended to noncompact
7006:(Second edition of 1997 original ed.). Cambridge:
390:
matrix is symmetric and positive-definite. Denote the
168:
into harmonic maps. Their work was the inspiration for
5712:
4860:
4666:
Siu was able to prove a variant of the still-unsolved
4571:
has finite Dirichlet energy, then it must be constant.
7046:
Nonpositive curvature: geometric and analytic aspects
6068:(1964). "Harmonic mappings of Riemannian manifolds".
5871:
Chang, Kung-Ching; Ding, Wei Yue; Ye, Rugang (1992).
4892:
4837:
4793:
4761:
4524:
be smooth and complete Riemannian manifolds, and let
4235:
3995:
3908:
considered the case where no geometric assumption on
3841:
3821:
3795:
3291:
2751:
Due to the first variation formula, the Laplacian of
2549:
2298:
1982:
1788:
1347:
983:
481:
156:
The theory of harmonic maps was initiated in 1964 by
5824:
5688:
2739:
There is also a version for manifolds with boundary.
2388:, it is not necessary to place the restriction that
6892:
6810:; Lemaire, L. (1978). "A report on harmonic maps".
3956:
3858:
3409:Existence. Given a continuously differentiable map
7146:The analysis of harmonic maps and their heat flows
6999:Harmonic maps, conservation laws and moving frames
6232:(1994). "Equilibrium maps between metric spaces".
5700:
5676:
5078:
4849:
4823:
4779:
4402:
4197:
3847:
3827:
3807:
3326:
2727:
2352:
2188:
1899:
1460:
1307:
914:
4033:
3998:
2670:
2578:
903:
779:
690:
566:
144:in allocating each of its elements to a point of
7361:
4744:. In such a theory, harmonic maps correspond to
3576:, meaning that one has a harmonic map heat flow
6392:
6327:
5812:
5173:
5169:
937:, its second fundamental form defines for each
6663:. Springer Monographs in Mathematics. Berlin:
6661:Some nonlinear problems in Riemannian geometry
6276:
5627:
5157:
3604:
223:Here the geometry of a smooth mapping between
124:Informally, the Dirichlet energy of a mapping
7048:. Lectures in Mathematics ETH ZĂŒrich. Basel:
4456:and a single integration by parts shows that
4392:
4365:
4336:
4302:
4286:
4263:
4187:
4160:
4131:
4097:
4081:
4058:
2799:be smooth Riemannian manifolds. The notation
2446:for which there exists a precompact open set
2353:{\displaystyle E(f)=\int _{M}e(f)\,d\mu _{g}}
1857:
1834:
6848:
6806:
6755:
6705:
6114:
6060:
5842:
5782:
5754:
5730:
5533:
5529:
5509:
5505:
5477:
5473:
5469:
5437:
5433:
5429:
5405:
5401:
5397:
5377:
5373:
5345:
5341:
5317:
5301:
5297:
5285:
5269:
5265:
5245:
5241:
5237:
5217:
5213:
5145:
4704:
2770:
6855:Bulletin of the London Mathematical Society
6813:Bulletin of the London Mathematical Society
5870:
5666:
5585:Bulletin of the London Mathematical Society
4540:is positive and the sectional curvature of
3965:to the setting of a harmonic map heat flow
3770:is instead compact with nonempty boundary.
1957:Dirichlet energy and its variation formulas
195:'s work on four-dimensional manifolds, and
136:can be thought of as the total amount that
7096:Riemannian geometry and geometric analysis
5915:
5718:
4824:{\displaystyle \gamma :(a,b)\rightarrow M}
3758:Eells and Sampson's result was adapted by
3028:solves the geodesic differential equation.
1712:. This section is known as the hessian of
1635:, which is a section of the vector bundle
7190:
6621:
6444:
6419:
6355:
6195:
6035:
5890:
5806:
5596:
5453:
5413:
5361:
5045:
4980:
3570:harmonic map heat flow with initial data
3108:
3059:component functions are harmonic as maps
2711:
2614:
2336:
73:Learn how and when to remove this message
6944:Harmonic Maps of Manifolds with Boundary
6938:
6013:
5966:
5830:
5734:
5694:
5554:
5493:
5321:
5305:
5273:
5249:
5221:
4467:must be constant, and hence zero; hence
3773:Shortly after Eells and Sampson's work,
36:This article includes a list of general
7140:
6338:"A regularity theory for harmonic maps"
6172:
6024:Communications in Analysis and Geometry
5794:
5766:
5738:
5670:
5634:Communications in Analysis and Geometry
5566:
5513:
5449:
5357:
5253:
5197:
4429:is nonpositive, then this implies that
3055:is harmonic if and only if each of its
3014:is one-dimensional, then minimality of
7362:
7355:The Bibliography of Harmonic Morphisms
6992:
6598:
6548:
5706:
5682:
5578:
5441:
5349:
5189:
4536:. Suppose that the Ricci curvature of
219:Geometry of mappings between manifolds
7232:
6655:
5778:
5750:
5525:
5465:
5425:
5393:
5337:
5233:
5209:
5185:
3665:, the maximal harmonic map heat flow
7090:
7040:
6228:
5854:
5790:
5786:
5762:
5758:
5481:
5445:
5409:
5381:
5353:
5325:
5193:
3876:"finite-time blowup" even when both
3789:notes that every map from a product
2973:is a constant-speed immersion, then
2273:which is smooth and nonnegative. If
368:be a smooth real-valued function on
312:be a smooth real-valued function on
262:
22:
7282:
6498:
5818:
4861:Harmonic maps between metric spaces
1735:, which is a section of the bundle
1471:
176:. Harmonic maps and the associated
13:
6950:. Vol. 471. BerlinâNew York:
4445:is closed, then multiplication by
4268:
4237:
4063:
4022:
4009:
4005:
3315:
3303:
3295:
3282:. This is usually abbreviated as:
3142:be smooth Riemannian manifolds. A
2700:
2658:
2654:
2566:
2562:
2142:
2127:
2108:
2093:
1839:
1792:
1748:; this says that the laplacian of
1425:
1351:
1269:
1253:
1238:
1219:
1204:
1140:
1125:
1095:
1055:
1042:
1021:
984:
885:
867:
845:
827:
805:
787:
699:
672:
654:
632:
614:
592:
574:
486:
203:is significant in applications to
199:'s later discovery of bubbling of
42:it lacks sufficient corresponding
14:
7391:
7338:
6555:Commentarii Mathematici Helvetici
6455:Commentarii Mathematici Helvetici
3423:, there exists a positive number
3113:
7292:Seminar on Differential Geometry
6711:Selected topics in harmonic maps
6609:Journal of Differential Geometry
6407:Journal of Differential Geometry
6343:Journal of Differential Geometry
5878:Journal of Differential Geometry
4780:{\displaystyle u:M\rightarrow N}
4657:is appropriately negative, then
3957:The Bochner formula and rigidity
3859:Singularities and weak solutions
3764:Dirichlet boundary value problem
3755:subconverges to a harmonic map.
3716:subsequentially converge in the
1760:an element of the tangent space
235:. Such a mapping defines both a
27:
6183:Canadian Journal of Mathematics
6071:American Journal of Mathematics
5621:
5572:
5539:
4831:is a geodesic, the composition
1969:is the real-valued function on
1662:; this is to say that for each
1552:; this is to say that for each
1318:Its laplacian defines for each
148:. For instance, an unstretched
5070:
5064:
5042:
5030:
5005:
4999:
4977:
4974:
4968:
4959:
4953:
4947:
4918:
4912:
4909:
4903:
4850:{\displaystyle u\circ \gamma }
4815:
4812:
4800:
4771:
4280:
4271:
4255:
4249:
4075:
4066:
4047:
4041:
2611:
2598:
2333:
2327:
2308:
2302:
2180:
2158:
1894:
1868:
1851:
1842:
1799:
1789:
1731:to arrive at the laplacian of
1438:
1428:
1358:
1348:
1279:
1272:
1105:
1098:
997:
987:
709:
702:
496:
489:
1:
6715:American Mathematical Society
5123:
3427:and a harmonic map heat flow
3193:in such a way that, for each
99:partial differential equation
85:In the mathematical field of
6948:Lecture Notes in Mathematics
6761:Two reports on harmonic maps
6178:"On homotopic harmonic maps"
4608:is homotopic to a constant.
7:
6896:; Martinazzi, Luca (2012).
5646:10.4310/CAG.2004.v12.n1.a17
5579:Dibble, James (June 2019).
5174:Schoen & Uhlenbeck 1983
5170:Schoen & Uhlenbeck 1982
5111:
3722:topology to a harmonic map.
3605:Eells and Sampson's theorem
3174:a twice-differentiable map
2995:is harmonic if and only if
2943:is harmonic if and only if
257:pseudo-Riemannian manifolds
128:from a Riemannian manifold
105:of a functional called the
10:
7396:
7350:Harmonic Maps Bibliography
7296:Princeton University Press
7008:Cambridge University Press
5158:Sacks & Uhlenbeck 1981
5104:attached to each point of
3920:is made. In the case that
3750:, every continuous map is
3405:is geodesically complete.
15:
7304:10.1515/9781400881918-002
7250:10.1007/978-3-0348-9193-6
7200:Lectures on harmonic maps
7144:; Wang, Changyou (2008).
7108:10.1007/978-3-319-61860-9
7058:10.1007/978-3-0348-8918-6
6906:10.1007/978-88-7642-443-4
6803:Consists of reprints of:
6673:10.1007/978-3-662-13006-3
6037:10.4310/CAG.1995.v3.n4.a1
5923:Mathematische Zeitschrift
5667:Chang, Ding & Ye 1992
4705:Problems and applications
4471:must itself be constant.
3808:{\displaystyle W\times M}
3393:is a closed manifold and
3080:Laplace-Beltrami operator
2771:Examples of harmonic maps
2755:can be thought of as the
2746:second variation formula.
1326:the real-valued function
957:the real-valued function
211:. The techniques used by
132:to a Riemannian manifold
7016:10.1017/CBO9780511543036
5843:Gromov & Schoen 1992
5783:Eells & Sampson 1964
5755:Eells & Sampson 1964
5731:Eells & Sampson 1964
5534:Eells & Lemaire 1983
5530:Eells & Lemaire 1978
5510:Eells & Lemaire 1983
5506:Eells & Lemaire 1978
5478:Eells & Sampson 1964
5474:Eells & Lemaire 1983
5470:Eells & Lemaire 1978
5438:Eells & Sampson 1964
5434:Eells & Lemaire 1983
5430:Eells & Lemaire 1978
5406:Eells & Sampson 1964
5402:Eells & Lemaire 1983
5398:Eells & Lemaire 1978
5378:Eells & Lemaire 1983
5374:Eells & Lemaire 1978
5346:Eells & Lemaire 1983
5342:Eells & Lemaire 1978
5318:Eells & Lemaire 1978
5302:Eells & Sampson 1964
5298:Eells & Lemaire 1978
5286:Eells & Lemaire 1983
5270:Eells & Lemaire 1983
5266:Eells & Lemaire 1978
5246:Eells & Sampson 1964
5242:Eells & Lemaire 1983
5238:Eells & Lemaire 1978
5218:Eells & Lemaire 1983
5214:Eells & Lemaire 1978
5146:Eells & Sampson 1964
4677:for lattices in certain
2380:. Since any nonnegative
241:second fundamental form.
201:pseudoholomorphic curves
7345:MathWorld: Harmonic map
6759:; Lemaire, Luc (1995).
6709:; Lemaire, Luc (1983).
4528:be a harmonic map from
3937:were able to prove the
2535:first variation formula
1614:Levi-Civita connections
1560:, one has a linear map
1516:, one can consider its
231:and, equivalently, via
172:'s initial work on the
103:Euler-Lagrange equation
89:, a smooth map between
57:more precise citations.
6623:10.4310/jdg/1214442475
6421:10.4310/jdg/1214437663
6385:10.4310/jdg/1214437667
6357:10.4310/jdg/1214436923
6197:10.4153/cjm-1967-062-6
5892:10.4310/jdg/1214448751
5719:Chen & Struwe 1989
5080:
4851:
4825:
4781:
4574:
4404:
4199:
3849:
3829:
3809:
3762:to the setting of the
3724:
3328:
3144:harmonic map heat flow
3109:Harmonic map heat flow
3003:differential equation.
2729:
2372:is the volume form on
2354:
2190:
2076:
2055:
2034:
2013:
1939:-orthonormal basis of
1901:
1831:
1624:. So one may take the
1462:
1411:
1390:
1309:
1200:
1179:
1094:
916:
763:
550:
237:first fundamental form
178:harmonic map heat flow
7158:10.1142/9789812779533
6868:10.1112/blms/20.5.385
6773:10.1142/9789812832030
6505:Annals of Mathematics
6284:Annals of Mathematics
5970:Annals of Mathematics
5807:Schoen & Yau 1976
5454:Schoen & Yau 1997
5414:Schoen & Yau 1997
5362:Schoen & Yau 1997
5081:
4852:
4826:
4782:
4556:are both closed then
4498:
4485:subharmonic functions
4405:
4200:
3850:
3830:
3810:
3611:
3329:
2955:. As a special case:
2730:
2355:
2191:
2056:
2035:
2014:
1993:
1902:
1811:
1504:. Given a smooth map
1463:
1391:
1370:
1310:
1180:
1159:
1074:
917:
743:
530:
372:, such that for each
316:, such that for each
285:be an open subset of
87:differential geometry
6940:Hamilton, Richard S.
5797:, Proposition 1.5.2.
5695:Ding & Tian 1995
5516:, Proposition 1.6.2.
5468:, Proposition 10.2;
4890:
4835:
4791:
4759:
4730:quantum field theory
4697:to be replaced by a
4421:is positive and the
4233:
3993:
3986:. This formula says
3839:
3819:
3793:
3289:
3102:Riemannian manifolds
2547:
2296:
1980:
1786:
1626:covariant derivative
1592:. The vector bundle
1502:Riemannian manifolds
1345:
981:
479:
225:Riemannian manifolds
111:unit-speed geodesics
91:Riemannian manifolds
7370:Riemannian geometry
6826:10.1112/blms/10.1.1
5795:Lin & Wang 2008
5793:, Corollary 9.2.3;
5781:, Corollary 10.12;
5767:Lin & Wang 2008
5739:Lin & Wang 2008
5671:Lin & Wang 2008
5528:, Definition 10.3;
5514:Lin & Wang 2008
5450:Lin & Wang 2008
5428:, Definition 10.1;
5396:, Definition 10.1;
5358:Lin & Wang 2008
5340:, Definition 10.1;
5254:Lin & Wang 2008
5236:, Definition 10.2;
5198:Lin & Wang 2008
5063:
4998:
4726:theoretical physics
4441:is nonnegative. If
4423:sectional curvature
3939:energy quantization
3639:sectional curvature
3240:is, as a vector in
2847:, the constant map
2384:has a well-defined
2382:measurable function
1454:
1295:
1121:
1013:
925:Given a smooth map
725:
512:
435:Christoffel symbols
380:, one has that the
324:, one has that the
205:symplectic geometry
115:Riemannian geometry
7380:Analytic functions
7375:Harmonic functions
7148:. Hackensack, NJ:
6960:10.1007/BFb0087227
6894:Giaquinta, Mariano
6763:. River Edge, NJ:
6568:10.1007/BF02567432
6468:10.1007/BF02568161
6246:10.1007/BF01191341
6142:10.1007/bf02699433
5936:10.1007/BF01161997
5765:, Formula 9.2.13;
5761:, Formula 5.1.18;
5607:10.1112/blms.12241
5076:
5049:
4984:
4847:
4821:
4777:
4681:. Following this,
4400:
4195:
3845:
3825:
3805:
3686:with initial data
3324:
2725:
2397:variation formulas
2350:
2186:
1897:
1686:of tangent spaces
1458:
1437:
1305:
1278:
1104:
996:
912:
910:
708:
495:
227:is considered via
209:quantum cohomology
182:geometric analysis
119:harmonic functions
117:and the theory of
7298:. pp. 3â71.
7242:BirkhÀuser Verlag
7167:978-981-277-952-6
7117:978-3-319-61859-3
7050:BirkhÀuser Verlag
6969:978-3-540-07185-3
6915:978-88-7642-442-7
6649:Books and surveys
6508:. Second Series.
6287:. Second Series.
5973:. Second Series.
5857:, Definition 1.1.
5789:, Theorem 5.1.2;
5484:, Formula 9.1.13.
5074:
4675:rigidity theorems
4560:must be constant.
4016:
3848:{\displaystyle M}
3828:{\displaystyle N}
3389:Now suppose that
3310:
3098:harmonic morphism
3018:is equivalent to
2874:The identity map
2665:
2573:
2386:Lebesgue integral
2269:is a function on
2156:
2122:
1991:
1612:induced from the
1267:
1233:
1154:
1069:
899:
859:
819:
741:
686:
646:
606:
528:
343:positive-definite
263:Local coordinates
229:local coordinates
83:
82:
75:
18:harmonic function
7387:
7333:
7279:
7229:
7187:
7150:World Scientific
7137:
7087:
7037:
6994:Hélein, Frédéric
6989:
6935:
6887:
6845:
6802:
6765:World Scientific
6752:
6723:10.1090/cbms/050
6702:
6643:
6625:
6595:
6545:
6495:
6441:
6423:
6398:Uhlenbeck, Karen
6389:
6388:
6377:
6359:
6334:Uhlenbeck, Karen
6324:
6273:
6225:
6199:
6169:
6111:
6062:Eells, James Jr.
6057:
6039:
6030:(3â4): 543â554.
6010:
5963:
5912:
5894:
5858:
5852:
5846:
5840:
5834:
5828:
5822:
5816:
5810:
5804:
5798:
5776:
5770:
5769:, Theorem 1.5.1.
5748:
5742:
5728:
5722:
5716:
5710:
5704:
5698:
5692:
5686:
5680:
5674:
5664:
5658:
5657:
5625:
5619:
5618:
5600:
5576:
5570:
5564:
5558:
5552:
5546:
5543:
5537:
5523:
5517:
5503:
5497:
5491:
5485:
5463:
5457:
5423:
5417:
5391:
5385:
5371:
5365:
5335:
5329:
5315:
5309:
5295:
5289:
5283:
5277:
5263:
5257:
5231:
5225:
5207:
5201:
5183:
5177:
5167:
5161:
5155:
5149:
5143:
5099:
5098:
5085:
5083:
5082:
5077:
5075:
5073:
5062:
5057:
5029:
5028:
5019:
5018:
5008:
4997:
4992:
4946:
4945:
4936:
4935:
4925:
4902:
4901:
4878:
4857:is a geodesic.
4856:
4854:
4853:
4848:
4830:
4828:
4827:
4822:
4786:
4784:
4783:
4778:
4738:Dirichlet energy
4736:is given by the
4696:
4668:Hodge conjecture
4664:
4660:
4656:
4644:
4632:
4618:KĂ€hler manifolds
4607:
4603:
4599:
4587:
4570:
4566:
4559:
4555:
4551:
4544:is nonpositive.
4543:
4539:
4535:
4531:
4527:
4523:
4511:
4493:
4482:
4470:
4466:
4455:
4444:
4440:
4428:
4420:
4409:
4407:
4406:
4401:
4396:
4395:
4389:
4388:
4379:
4378:
4369:
4368:
4359:
4358:
4346:
4345:
4340:
4339:
4329:
4328:
4316:
4315:
4306:
4305:
4296:
4295:
4290:
4289:
4267:
4266:
4245:
4244:
4225:
4221:
4204:
4202:
4201:
4196:
4191:
4190:
4184:
4183:
4174:
4173:
4164:
4163:
4154:
4153:
4141:
4140:
4135:
4134:
4124:
4123:
4111:
4110:
4101:
4100:
4091:
4090:
4085:
4084:
4062:
4061:
4037:
4036:
4030:
4029:
4017:
4015:
4004:
4002:
4001:
3985:
3951:closed manifolds
3923:
3919:
3899:
3887:
3874:
3854:
3852:
3851:
3846:
3834:
3832:
3831:
3826:
3814:
3812:
3811:
3806:
3769:
3760:Richard Hamilton
3749:
3737:
3721:
3715:
3704:
3700:
3696:
3689:
3685:
3664:
3660:
3656:
3652:
3636:
3624:
3600:
3596:
3575:
3561:
3559:
3543:
3537:
3521:
3519:
3504:
3496:
3472:
3468:
3462:
3456:
3445:
3438:on the interval
3437:
3426:
3422:
3418:
3414:
3404:
3392:
3384:
3359:
3333:
3331:
3330:
3325:
3311:
3309:
3301:
3293:
3281:
3263:
3239:
3235:
3215:
3200:
3196:
3192:
3173:
3161:
3158:assigns to each
3157:
3141:
3129:
3091:KĂ€hler manifolds
3077:
3058:
3054:
3027:
3023:
3017:
3013:
2998:
2994:
2972:
2954:
2946:
2942:
2912:
2893:
2870:
2866:
2846:
2842:
2835:
2814:totally geodesic
2807:
2798:
2786:
2765:Banach manifolds
2754:
2744:There is also a
2734:
2732:
2731:
2726:
2724:
2723:
2710:
2706:
2696:
2695:
2686:
2685:
2674:
2673:
2666:
2664:
2653:
2643:
2642:
2627:
2626:
2610:
2609:
2594:
2593:
2582:
2581:
2574:
2572:
2561:
2559:
2558:
2528:
2503:
2492:
2488:
2453:
2449:
2445:
2432:
2413:
2409:
2391:
2379:
2375:
2371:
2359:
2357:
2356:
2351:
2349:
2348:
2323:
2322:
2288:
2283:Dirichlet energy
2281:is compact, the
2280:
2277:is oriented and
2276:
2272:
2268:
2257:
2253:
2249:
2243:
2241:
2240:
2237:
2234:
2226:
2206:
2202:
2195:
2193:
2192:
2187:
2173:
2172:
2157:
2155:
2154:
2153:
2140:
2139:
2138:
2125:
2123:
2121:
2120:
2119:
2106:
2105:
2104:
2091:
2089:
2088:
2075:
2070:
2054:
2049:
2033:
2028:
2012:
2007:
1992:
1984:
1972:
1968:
1952:
1938:
1927:
1906:
1904:
1903:
1898:
1893:
1892:
1880:
1879:
1867:
1866:
1861:
1860:
1838:
1837:
1830:
1825:
1807:
1806:
1778:
1759:
1755:
1752:assigns to each
1751:
1747:
1743:
1734:
1730:
1722:
1715:
1711:
1685:
1669:
1665:
1661:
1657:
1634:
1623:
1619:
1607:
1591:
1570:
1559:
1555:
1551:
1547:
1524:
1515:
1511:
1507:
1499:
1487:
1472:Bundle formalism
1467:
1465:
1464:
1459:
1453:
1448:
1424:
1423:
1410:
1405:
1389:
1384:
1366:
1365:
1337:
1333:
1325:
1321:
1314:
1312:
1311:
1306:
1294:
1289:
1268:
1266:
1265:
1264:
1251:
1250:
1249:
1236:
1234:
1232:
1231:
1230:
1217:
1216:
1215:
1202:
1199:
1194:
1178:
1173:
1155:
1153:
1152:
1151:
1138:
1137:
1136:
1123:
1120:
1115:
1093:
1088:
1070:
1068:
1067:
1066:
1054:
1053:
1040:
1039:
1038:
1029:
1028:
1018:
1012:
1007:
973:
969:
956:
952:
948:
944:
940:
936:
932:
928:
921:
919:
918:
913:
911:
907:
906:
900:
898:
897:
896:
883:
882:
881:
865:
860:
858:
857:
856:
843:
842:
841:
825:
820:
818:
817:
816:
803:
802:
801:
785:
783:
782:
776:
775:
762:
757:
742:
734:
724:
719:
694:
693:
687:
685:
684:
683:
670:
669:
668:
652:
647:
645:
644:
643:
630:
629:
628:
612:
607:
605:
604:
603:
590:
589:
588:
572:
570:
569:
563:
562:
549:
544:
529:
521:
511:
506:
471:
453:
432:
428:
414:
410:
392:inverse matrices
389:
379:
375:
371:
367:
356:
352:
348:
333:
323:
319:
315:
311:
300:
296:
292:
288:
284:
279:
270:
170:Richard Hamilton
147:
143:
139:
135:
131:
127:
107:Dirichlet energy
78:
71:
67:
64:
58:
53:this article by
44:inline citations
31:
30:
23:
7395:
7394:
7390:
7389:
7388:
7386:
7385:
7384:
7360:
7359:
7341:
7336:
7314:
7288:Yau, Shing-Tung
7284:Yau, Shing Tung
7260:
7210:
7168:
7118:
7068:
7026:
6970:
6952:Springer-Verlag
6916:
6783:
6733:
6683:
6665:Springer-Verlag
6646:
6600:Struwe, Michael
6550:Struwe, Michael
6518:10.2307/1971321
6450:Yau, Shing Tung
6446:Schoen, Richard
6394:Schoen, Richard
6379:(Erratum:
6378:
6330:Schoen, Richard
6328:
6297:10.2307/1971131
6174:Hartman, Philip
6120:Schoen, Richard
6116:Gromov, Mikhail
6084:10.2307/2373037
5983:10.2307/2946567
5918:Struwe, Michael
5916:Chen, Yun Mei;
5861:
5853:
5849:
5841:
5837:
5829:
5825:
5817:
5813:
5805:
5801:
5777:
5773:
5753:, Lemma 10.11;
5749:
5745:
5729:
5725:
5717:
5713:
5705:
5701:
5693:
5689:
5681:
5677:
5665:
5661:
5626:
5622:
5577:
5573:
5565:
5561:
5553:
5549:
5544:
5540:
5524:
5520:
5504:
5500:
5492:
5488:
5464:
5460:
5424:
5420:
5392:
5388:
5372:
5368:
5336:
5332:
5316:
5312:
5296:
5292:
5284:
5280:
5264:
5260:
5232:
5228:
5208:
5204:
5184:
5180:
5168:
5164:
5156:
5152:
5144:
5135:
5126:
5114:
5100:is a family of
5097:
5092:
5091:
5090:
5058:
5053:
5024:
5020:
5014:
5010:
5009:
4993:
4988:
4941:
4937:
4931:
4927:
4926:
4924:
4897:
4893:
4891:
4888:
4887:
4866:
4863:
4836:
4833:
4832:
4792:
4789:
4788:
4760:
4757:
4756:
4707:
4686:
4662:
4658:
4646:
4634:
4622:
4605:
4601:
4589:
4577:
4568:
4564:
4557:
4553:
4549:
4541:
4537:
4533:
4529:
4525:
4513:
4501:
4489:
4480:
4468:
4457:
4446:
4442:
4430:
4426:
4418:
4415:Ricci curvature
4391:
4390:
4384:
4380:
4374:
4370:
4364:
4363:
4354:
4350:
4341:
4335:
4334:
4333:
4324:
4320:
4311:
4307:
4301:
4300:
4291:
4285:
4284:
4283:
4262:
4261:
4240:
4236:
4234:
4231:
4230:
4223:
4209:
4186:
4185:
4179:
4175:
4169:
4165:
4159:
4158:
4149:
4145:
4136:
4130:
4129:
4128:
4119:
4115:
4106:
4102:
4096:
4095:
4086:
4080:
4079:
4078:
4057:
4056:
4032:
4031:
4025:
4021:
4008:
4003:
3997:
3996:
3994:
3991:
3990:
3976: : 0 <
3975:
3966:
3963:Bochner formula
3959:
3947:Euclidean space
3921:
3909:
3889:
3877:
3864:
3861:
3840:
3837:
3836:
3820:
3817:
3816:
3794:
3791:
3790:
3767:
3739:
3727:
3717:
3714:
3706:
3702:
3698:
3691:
3687:
3676: : 0 <
3675:
3666:
3662:
3658:
3654:
3642:
3626:
3614:
3607:
3598:
3587: : 0 <
3586:
3577:
3571:
3555:
3545:
3542:
3533:
3531:
3523:
3515:
3510: : 0 <
3509:
3500:
3498:
3487: : 0 <
3486:
3477:
3476:Uniqueness. If
3473:decreases to 0.
3470:
3464:
3458:
3455:
3447:
3439:
3436:
3428:
3424:
3420:
3416:
3410:
3394:
3390:
3378:
3361:
3342:
3302:
3294:
3292:
3290:
3287:
3286:
3280:
3274:
3265:
3259:
3253:
3241:
3237:
3229:
3217:
3202:
3198:
3194:
3183:
3175:
3163:
3159:
3147:
3146:on an interval
3131:
3119:
3116:
3111:
3087:holomorphic map
3075:
3060:
3056:
3052:
3034:
3025:
3019:
3015:
3011:
3010:Recall that if
2996:
2984:
2974:
2959:
2952:
2944:
2918:
2900:
2875:
2868:
2848:
2844:
2840:
2817:
2806:
2800:
2788:
2776:
2773:
2761:global analysis
2752:
2719:
2715:
2691:
2687:
2675:
2669:
2668:
2667:
2657:
2652:
2651:
2647:
2638:
2634:
2622:
2618:
2605:
2601:
2583:
2577:
2576:
2575:
2565:
2560:
2554:
2550:
2548:
2545:
2544:
2522:
2505:
2494:
2490:
2487:
2473:
2463:
2455:
2451:
2447:
2440:
2434:
2423:
2415:
2411:
2410:as the mapping
2400:
2389:
2377:
2373:
2370:
2364:
2344:
2340:
2318:
2314:
2297:
2294:
2293:
2286:
2278:
2274:
2270:
2259:
2255:
2251:
2238:
2235:
2232:
2231:
2229:
2228:
2212:
2204:
2200:
2165:
2161:
2149:
2145:
2141:
2134:
2130:
2126:
2124:
2115:
2111:
2107:
2100:
2096:
2092:
2090:
2081:
2077:
2071:
2060:
2050:
2039:
2029:
2018:
2008:
1997:
1983:
1981:
1978:
1977:
1970:
1966:
1959:
1948:
1940:
1937:
1929:
1926:
1917:
1911:
1888:
1884:
1875:
1871:
1862:
1856:
1855:
1854:
1833:
1832:
1826:
1815:
1802:
1798:
1787:
1784:
1783:
1774:
1761:
1757:
1753:
1749:
1745:
1736:
1732:
1728:
1727:the hessian of
1720:
1713:
1708:
1700:
1692:
1687:
1684:
1674:
1667:
1663:
1659:
1636:
1628:
1621:
1617:
1593:
1588:
1580:
1575:
1569:
1561:
1557:
1553:
1549:
1533:
1520:
1513:
1509:
1505:
1489:
1477:
1474:
1449:
1441:
1416:
1412:
1406:
1395:
1385:
1374:
1361:
1357:
1346:
1343:
1342:
1335:
1327:
1323:
1319:
1290:
1282:
1260:
1256:
1252:
1245:
1241:
1237:
1235:
1226:
1222:
1218:
1211:
1207:
1203:
1201:
1195:
1184:
1174:
1163:
1147:
1143:
1139:
1132:
1128:
1124:
1122:
1116:
1108:
1089:
1078:
1062:
1058:
1049:
1045:
1041:
1034:
1030:
1024:
1020:
1019:
1017:
1008:
1000:
982:
979:
978:
971:
968:
958:
954:
950:
946:
942:
938:
934:
930:
926:
909:
908:
902:
901:
892:
888:
884:
874:
870:
866:
864:
852:
848:
844:
834:
830:
826:
824:
812:
808:
804:
794:
790:
786:
784:
778:
777:
768:
764:
758:
747:
733:
726:
720:
712:
696:
695:
689:
688:
679:
675:
671:
661:
657:
653:
651:
639:
635:
631:
621:
617:
613:
611:
599:
595:
591:
581:
577:
573:
571:
565:
564:
555:
551:
545:
534:
520:
513:
507:
499:
482:
480:
477:
476:
465:
455:
447:
437:
430:
416:
412:
398:
381:
377:
373:
369:
366:
358:
354:
350:
346:
325:
321:
317:
313:
310:
302:
298:
294:
290:
286:
282:
277:
268:
265:
221:
193:Simon Donaldson
189:Karen Uhlenbeck
145:
141:
137:
133:
129:
125:
79:
68:
62:
59:
49:Please help to
48:
32:
28:
21:
12:
11:
5:
7393:
7383:
7382:
7377:
7372:
7358:
7357:
7352:
7347:
7340:
7339:External links
7337:
7335:
7334:
7312:
7280:
7258:
7230:
7208:
7188:
7166:
7138:
7116:
7100:Springer, Cham
7088:
7066:
7038:
7024:
6990:
6968:
6936:
6914:
6890:
6889:
6888:
6862:(5): 385â524.
6846:
6781:
6753:
6731:
6703:
6681:
6657:Aubin, Thierry
6652:
6645:
6644:
6616:(3): 485â502.
6596:
6562:(4): 558â581.
6546:
6496:
6462:(3): 333â341.
6442:
6414:(2): 253â268.
6390:
6350:(2): 307â335.
6325:
6274:
6240:(2): 173â204.
6226:
6170:
6112:
6078:(1): 109â160.
6066:Sampson, J. H.
6058:
6014:Ding, Weiyue;
6011:
5977:(1): 165â182.
5964:
5913:
5885:(2): 507â515.
5867:
5860:
5859:
5847:
5835:
5823:
5811:
5799:
5785:, Section 3C;
5771:
5757:, Section 3C;
5743:
5741:, Lemma 5.3.3.
5733:, Section 8A;
5723:
5711:
5699:
5687:
5675:
5673:, Section 6.3.
5659:
5640:(1): 389â415.
5620:
5591:(3): 443â458.
5571:
5559:
5547:
5538:
5518:
5498:
5486:
5480:, Section 2B;
5458:
5440:, Section 1A;
5418:
5408:, Section 1A;
5386:
5366:
5330:
5310:
5304:, Section 3B;
5290:
5278:
5258:
5248:, Section 2B;
5226:
5202:
5178:
5162:
5150:
5148:, Section 11A.
5132:
5125:
5122:
5121:
5120:
5118:Geometric flow
5113:
5110:
5093:
5087:
5086:
5072:
5069:
5066:
5061:
5056:
5052:
5048:
5044:
5041:
5038:
5035:
5032:
5027:
5023:
5017:
5013:
5007:
5004:
5001:
4996:
4991:
4987:
4983:
4979:
4976:
4973:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4944:
4940:
4934:
4930:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4900:
4896:
4862:
4859:
4846:
4843:
4840:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4776:
4773:
4770:
4767:
4764:
4753:
4752:
4749:
4740:is known as a
4722:
4719:twistor theory
4715:
4706:
4703:
4683:Mikhael Gromov
4573:
4572:
4567:is closed and
4561:
4477:Shing-Tung Yau
4473:Richard Schoen
4411:
4410:
4399:
4394:
4387:
4383:
4377:
4373:
4367:
4362:
4357:
4353:
4349:
4344:
4338:
4332:
4327:
4323:
4319:
4314:
4310:
4304:
4299:
4294:
4288:
4282:
4279:
4276:
4273:
4270:
4265:
4260:
4257:
4254:
4251:
4248:
4243:
4239:
4206:
4205:
4194:
4189:
4182:
4178:
4172:
4168:
4162:
4157:
4152:
4148:
4144:
4139:
4133:
4127:
4122:
4118:
4114:
4109:
4105:
4099:
4094:
4089:
4083:
4077:
4074:
4071:
4068:
4065:
4060:
4055:
4052:
4049:
4046:
4043:
4040:
4035:
4028:
4024:
4020:
4014:
4011:
4007:
4000:
3971:
3958:
3955:
3926:weak solutions
3906:Michael Struwe
3860:
3857:
3844:
3824:
3804:
3801:
3798:
3787:
3786:
3783:
3775:Philip Hartman
3710:
3671:
3606:
3603:
3582:
3564:
3563:
3538:
3527:
3505:
3482:
3474:
3451:
3432:
3387:
3386:
3374:
3335:
3334:
3323:
3320:
3317:
3314:
3308:
3305:
3300:
3297:
3276:
3270:
3249:
3245:
3225:
3179:
3115:
3114:Well-posedness
3112:
3110:
3107:
3106:
3105:
3094:
3083:
3073:
3050:
3030:
3029:
3007:
3006:
3005:
3004:
2982:
2897:
2896:
2895:
2872:
2804:
2772:
2769:
2749:
2748:
2741:
2740:
2737:
2736:
2735:
2722:
2718:
2714:
2709:
2705:
2702:
2699:
2694:
2690:
2684:
2681:
2678:
2672:
2663:
2660:
2656:
2650:
2646:
2641:
2637:
2633:
2630:
2625:
2621:
2617:
2613:
2608:
2604:
2600:
2597:
2592:
2589:
2586:
2580:
2571:
2568:
2564:
2557:
2553:
2539:
2538:
2518:
2479:
2465:
2459:
2438:
2419:
2366:
2361:
2360:
2347:
2343:
2339:
2335:
2332:
2329:
2326:
2321:
2317:
2313:
2310:
2307:
2304:
2301:
2289:is defined as
2197:
2196:
2185:
2182:
2179:
2176:
2171:
2168:
2164:
2160:
2152:
2148:
2144:
2137:
2133:
2129:
2118:
2114:
2110:
2103:
2099:
2095:
2087:
2084:
2080:
2074:
2069:
2066:
2063:
2059:
2053:
2048:
2045:
2042:
2038:
2032:
2027:
2024:
2021:
2017:
2011:
2006:
2003:
2000:
1996:
1990:
1987:
1963:energy density
1958:
1955:
1944:
1933:
1922:
1915:
1908:
1907:
1896:
1891:
1887:
1883:
1878:
1874:
1870:
1865:
1859:
1853:
1850:
1847:
1844:
1841:
1836:
1829:
1824:
1821:
1818:
1814:
1810:
1805:
1801:
1797:
1794:
1791:
1765:
1706:
1698:
1690:
1680:
1586:
1578:
1573:tangent spaces
1565:
1473:
1470:
1469:
1468:
1457:
1452:
1447:
1444:
1440:
1436:
1433:
1430:
1427:
1422:
1419:
1415:
1409:
1404:
1401:
1398:
1394:
1388:
1383:
1380:
1377:
1373:
1369:
1364:
1360:
1356:
1353:
1350:
1322:between 1 and
1316:
1315:
1304:
1301:
1298:
1293:
1288:
1285:
1281:
1277:
1274:
1271:
1263:
1259:
1255:
1248:
1244:
1240:
1229:
1225:
1221:
1214:
1210:
1206:
1198:
1193:
1190:
1187:
1183:
1177:
1172:
1169:
1166:
1162:
1158:
1150:
1146:
1142:
1135:
1131:
1127:
1119:
1114:
1111:
1107:
1103:
1100:
1097:
1092:
1087:
1084:
1081:
1077:
1073:
1065:
1061:
1057:
1052:
1048:
1044:
1037:
1033:
1027:
1023:
1016:
1011:
1006:
1003:
999:
995:
992:
989:
986:
964:
953:between 1 and
945:between 1 and
923:
922:
905:
895:
891:
887:
880:
877:
873:
869:
863:
855:
851:
847:
840:
837:
833:
829:
823:
815:
811:
807:
800:
797:
793:
789:
781:
774:
771:
767:
761:
756:
753:
750:
746:
740:
737:
732:
729:
727:
723:
718:
715:
711:
707:
704:
701:
698:
697:
692:
682:
678:
674:
667:
664:
660:
656:
650:
642:
638:
634:
627:
624:
620:
616:
610:
602:
598:
594:
587:
584:
580:
576:
568:
561:
558:
554:
548:
543:
540:
537:
533:
527:
524:
519:
516:
514:
510:
505:
502:
498:
494:
491:
488:
485:
484:
461:
443:
429:between 1 and
411:between 1 and
362:
353:between 1 and
306:
297:between 1 and
264:
261:
233:linear algebra
220:
217:
213:Richard Schoen
197:Mikhael Gromov
162:Joseph Sampson
81:
80:
35:
33:
26:
9:
6:
4:
3:
2:
7392:
7381:
7378:
7376:
7373:
7371:
7368:
7367:
7365:
7356:
7353:
7351:
7348:
7346:
7343:
7342:
7331:
7327:
7323:
7319:
7315:
7313:9781400881918
7309:
7305:
7301:
7297:
7293:
7289:
7285:
7281:
7277:
7273:
7269:
7265:
7261:
7259:3-7643-5397-X
7255:
7251:
7247:
7243:
7239:
7235:
7231:
7227:
7223:
7219:
7215:
7211:
7209:1-57146-002-0
7205:
7201:
7197:
7193:
7189:
7185:
7181:
7177:
7173:
7169:
7163:
7159:
7155:
7151:
7147:
7143:
7139:
7135:
7131:
7127:
7123:
7119:
7113:
7109:
7105:
7101:
7097:
7093:
7089:
7085:
7081:
7077:
7073:
7069:
7067:3-7643-5736-3
7063:
7059:
7055:
7051:
7047:
7043:
7039:
7035:
7031:
7027:
7025:0-521-81160-0
7021:
7017:
7013:
7009:
7005:
7001:
7000:
6995:
6991:
6987:
6983:
6979:
6975:
6971:
6965:
6961:
6957:
6953:
6949:
6945:
6941:
6937:
6933:
6929:
6925:
6921:
6917:
6911:
6907:
6903:
6899:
6895:
6891:
6885:
6881:
6877:
6873:
6869:
6865:
6861:
6857:
6856:
6851:
6847:
6843:
6839:
6835:
6831:
6827:
6823:
6819:
6815:
6814:
6809:
6805:
6804:
6800:
6796:
6792:
6788:
6784:
6782:981-02-1466-9
6778:
6774:
6770:
6766:
6762:
6758:
6754:
6750:
6746:
6742:
6738:
6734:
6732:0-8218-0700-5
6728:
6724:
6720:
6716:
6712:
6708:
6704:
6700:
6696:
6692:
6688:
6684:
6682:3-540-60752-8
6678:
6674:
6670:
6666:
6662:
6658:
6654:
6653:
6651:
6650:
6641:
6637:
6633:
6629:
6624:
6619:
6615:
6611:
6610:
6605:
6601:
6597:
6593:
6589:
6585:
6581:
6577:
6573:
6569:
6565:
6561:
6557:
6556:
6551:
6547:
6543:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6512:(1): 73â111.
6511:
6507:
6506:
6501:
6500:Siu, Yum Tong
6497:
6493:
6489:
6485:
6481:
6477:
6473:
6469:
6465:
6461:
6457:
6456:
6451:
6447:
6443:
6439:
6435:
6431:
6427:
6422:
6417:
6413:
6409:
6408:
6403:
6399:
6395:
6391:
6386:
6382:
6375:
6371:
6367:
6363:
6358:
6353:
6349:
6345:
6344:
6339:
6335:
6331:
6326:
6322:
6318:
6314:
6310:
6306:
6302:
6298:
6294:
6290:
6286:
6285:
6280:
6279:Uhlenbeck, K.
6275:
6271:
6267:
6263:
6259:
6255:
6251:
6247:
6243:
6239:
6235:
6231:
6227:
6223:
6219:
6215:
6211:
6207:
6203:
6198:
6193:
6189:
6185:
6184:
6179:
6175:
6171:
6167:
6163:
6159:
6155:
6151:
6147:
6143:
6139:
6135:
6131:
6130:
6125:
6121:
6117:
6113:
6109:
6105:
6101:
6097:
6093:
6089:
6085:
6081:
6077:
6073:
6072:
6067:
6063:
6059:
6055:
6051:
6047:
6043:
6038:
6033:
6029:
6025:
6021:
6017:
6012:
6008:
6004:
6000:
5996:
5992:
5988:
5984:
5980:
5976:
5972:
5971:
5965:
5961:
5957:
5953:
5949:
5945:
5941:
5937:
5933:
5930:(1): 83â103.
5929:
5925:
5924:
5919:
5914:
5910:
5906:
5902:
5898:
5893:
5888:
5884:
5880:
5879:
5874:
5869:
5868:
5866:
5865:
5856:
5851:
5844:
5839:
5832:
5831:Corlette 1992
5827:
5820:
5815:
5808:
5803:
5796:
5792:
5788:
5784:
5780:
5775:
5768:
5764:
5760:
5756:
5752:
5747:
5740:
5737:, p.128-130;
5736:
5735:Hamilton 1975
5732:
5727:
5720:
5715:
5708:
5703:
5696:
5691:
5684:
5679:
5672:
5668:
5663:
5655:
5651:
5647:
5643:
5639:
5635:
5631:
5624:
5616:
5612:
5608:
5604:
5599:
5594:
5590:
5586:
5582:
5575:
5568:
5563:
5556:
5555:Hamilton 1975
5551:
5542:
5535:
5531:
5527:
5522:
5515:
5511:
5507:
5502:
5495:
5494:Hamilton 1975
5490:
5483:
5479:
5475:
5471:
5467:
5462:
5455:
5451:
5447:
5443:
5439:
5435:
5431:
5427:
5422:
5415:
5412:, p.490-491;
5411:
5407:
5403:
5399:
5395:
5390:
5383:
5379:
5375:
5370:
5363:
5359:
5355:
5351:
5347:
5343:
5339:
5334:
5327:
5323:
5322:Hamilton 1975
5319:
5314:
5307:
5306:Hamilton 1975
5303:
5299:
5294:
5287:
5282:
5275:
5274:Hamilton 1975
5271:
5267:
5262:
5255:
5251:
5250:Hamilton 1975
5247:
5243:
5239:
5235:
5230:
5223:
5222:Hamilton 1975
5219:
5215:
5211:
5206:
5199:
5195:
5191:
5187:
5182:
5175:
5171:
5166:
5159:
5154:
5147:
5142:
5140:
5138:
5133:
5131:
5130:
5119:
5116:
5115:
5109:
5107:
5103:
5096:
5067:
5059:
5054:
5050:
5046:
5039:
5036:
5033:
5025:
5021:
5015:
5011:
5002:
4994:
4989:
4985:
4981:
4971:
4965:
4962:
4956:
4950:
4942:
4938:
4932:
4928:
4921:
4915:
4906:
4898:
4894:
4886:
4885:
4884:
4882:
4881:metric spaces
4877:
4873:
4869:
4858:
4844:
4841:
4838:
4818:
4809:
4806:
4803:
4797:
4794:
4774:
4768:
4765:
4762:
4750:
4747:
4743:
4739:
4735:
4731:
4727:
4723:
4720:
4716:
4713:
4709:
4708:
4702:
4700:
4694:
4690:
4684:
4680:
4676:
4671:
4669:
4654:
4650:
4642:
4638:
4630:
4626:
4619:
4615:
4609:
4597:
4593:
4585:
4581:
4562:
4547:
4546:
4545:
4521:
4517:
4509:
4505:
4497:
4495:
4492:
4486:
4478:
4474:
4464:
4460:
4453:
4449:
4438:
4434:
4424:
4416:
4397:
4385:
4381:
4375:
4371:
4360:
4355:
4351:
4347:
4342:
4330:
4325:
4321:
4317:
4312:
4308:
4297:
4292:
4277:
4274:
4258:
4252:
4246:
4241:
4229:
4228:
4227:
4220:
4216:
4212:
4192:
4180:
4176:
4170:
4166:
4155:
4150:
4146:
4142:
4137:
4125:
4120:
4116:
4112:
4107:
4103:
4092:
4087:
4072:
4069:
4053:
4050:
4044:
4038:
4026:
4018:
4012:
3989:
3988:
3987:
3983:
3979:
3974:
3970:
3964:
3954:
3952:
3948:
3943:
3940:
3936:
3932:
3927:
3917:
3913:
3907:
3902:
3897:
3893:
3885:
3881:
3872:
3868:
3856:
3842:
3822:
3802:
3799:
3796:
3784:
3781:
3780:
3779:
3776:
3771:
3765:
3761:
3756:
3753:
3747:
3743:
3735:
3731:
3723:
3720:
3713:
3709:
3701:increases to
3694:
3683:
3679:
3674:
3670:
3650:
3646:
3640:
3634:
3630:
3622:
3618:
3610:
3602:
3594:
3590:
3585:
3581:
3574:
3569:
3558:
3553:
3549:
3541:
3536:
3530:
3526:
3518:
3513:
3508:
3503:
3494:
3490:
3485:
3481:
3475:
3467:
3461:
3457:converges to
3454:
3450:
3443:
3435:
3431:
3413:
3408:
3407:
3406:
3402:
3398:
3382:
3377:
3373:
3369:
3365:
3358:
3354:
3350:
3346:
3340:
3339:
3338:
3321:
3318:
3312:
3306:
3298:
3285:
3284:
3283:
3279:
3273:
3269:
3262:
3257:
3252:
3248:
3244:
3233:
3228:
3224:
3220:
3214:
3210:
3206:
3191:
3187:
3182:
3178:
3171:
3167:
3155:
3151:
3145:
3139:
3135:
3127:
3123:
3103:
3099:
3095:
3092:
3088:
3084:
3081:
3072:
3068:
3064:
3049:
3045:
3041:
3037:
3033:A smooth map
3032:
3031:
3022:
3009:
3008:
3002:
2992:
2988:
2981:
2977:
2970:
2966:
2963: : â â (
2962:
2957:
2956:
2950:
2940:
2936:
2932:
2929:
2925:
2921:
2916:
2911:
2907:
2903:
2898:
2891:
2887:
2883:
2879:
2873:
2864:
2860:
2856:
2852:
2838:
2837:
2833:
2829:
2825:
2821:
2815:
2811:
2810:
2809:
2803:
2796:
2792:
2784:
2780:
2768:
2766:
2762:
2758:
2747:
2743:
2742:
2738:
2720:
2716:
2712:
2707:
2703:
2697:
2692:
2688:
2682:
2679:
2676:
2661:
2648:
2644:
2639:
2635:
2631:
2628:
2623:
2619:
2615:
2606:
2602:
2595:
2590:
2587:
2584:
2569:
2555:
2551:
2543:
2542:
2541:
2540:
2536:
2532:
2531:
2530:
2526:
2521:
2517:
2513:
2509:
2502:
2498:
2486:
2482:
2477:
2472:
2468:
2462:
2458:
2444:
2437:
2431:
2427:
2422:
2418:
2407:
2403:
2398:
2393:
2387:
2383:
2369:
2345:
2341:
2337:
2330:
2324:
2319:
2315:
2311:
2305:
2299:
2292:
2291:
2290:
2284:
2266:
2262:
2247:
2225:
2222:
2218:
2215:
2210:
2209:bundle metric
2183:
2177:
2174:
2169:
2166:
2162:
2150:
2146:
2135:
2131:
2116:
2112:
2101:
2097:
2085:
2082:
2078:
2072:
2067:
2064:
2061:
2057:
2051:
2046:
2043:
2040:
2036:
2030:
2025:
2022:
2019:
2015:
2009:
2004:
2001:
1998:
1994:
1988:
1985:
1976:
1975:
1974:
1965:of a mapping
1964:
1954:
1951:
1947:
1943:
1936:
1932:
1925:
1921:
1914:
1889:
1885:
1881:
1876:
1872:
1863:
1848:
1845:
1827:
1822:
1819:
1816:
1812:
1808:
1803:
1795:
1782:
1781:
1780:
1777:
1772:
1768:
1764:
1742:
1739:
1726:
1717:
1710:
1702:
1694:
1683:
1678:
1673:
1656:
1653:
1649:
1646:
1642:
1639:
1632:
1627:
1615:
1611:
1606:
1603:
1599:
1596:
1590:
1582:
1574:
1568:
1564:
1546:
1543:
1539:
1536:
1532:
1531:vector bundle
1528:
1523:
1519:
1503:
1497:
1493:
1485:
1481:
1455:
1450:
1445:
1442:
1434:
1431:
1420:
1417:
1413:
1407:
1402:
1399:
1396:
1392:
1386:
1381:
1378:
1375:
1371:
1367:
1362:
1354:
1341:
1340:
1339:
1331:
1302:
1299:
1296:
1291:
1286:
1283:
1275:
1261:
1257:
1246:
1242:
1227:
1223:
1212:
1208:
1196:
1191:
1188:
1185:
1181:
1175:
1170:
1167:
1164:
1160:
1156:
1148:
1144:
1133:
1129:
1117:
1112:
1109:
1101:
1090:
1085:
1082:
1079:
1075:
1071:
1063:
1059:
1050:
1046:
1035:
1031:
1025:
1014:
1009:
1004:
1001:
993:
990:
977:
976:
975:
967:
962:
949:and for each
893:
889:
878:
875:
871:
861:
853:
849:
838:
835:
831:
821:
813:
809:
798:
795:
791:
772:
769:
765:
759:
754:
751:
748:
744:
738:
735:
730:
728:
721:
716:
713:
705:
680:
676:
665:
662:
658:
648:
640:
636:
625:
622:
618:
608:
600:
596:
585:
582:
578:
559:
556:
552:
546:
541:
538:
535:
531:
525:
522:
517:
515:
508:
503:
500:
492:
475:
474:
473:
469:
464:
459:
451:
446:
441:
436:
427:
423:
419:
409:
405:
401:
395:
393:
388:
384:
365:
361:
344:
340:
336:
332:
328:
309:
305:
280:
274:
260:
258:
254:
250:
249:tension field
247:(also called
246:
242:
238:
234:
230:
226:
216:
214:
210:
206:
202:
198:
194:
190:
185:
183:
179:
175:
171:
167:
163:
159:
154:
151:
122:
120:
116:
112:
108:
104:
100:
96:
92:
88:
77:
74:
66:
56:
52:
46:
45:
39:
34:
25:
24:
19:
7291:
7237:
7199:
7145:
7142:Lin, Fanghua
7095:
7092:Jost, JĂŒrgen
7045:
7042:Jost, JĂŒrgen
6998:
6943:
6897:
6859:
6853:
6817:
6811:
6760:
6757:Eells, James
6710:
6707:Eells, James
6660:
6648:
6647:
6613:
6607:
6559:
6553:
6509:
6503:
6459:
6453:
6411:
6405:
6347:
6341:
6288:
6282:
6237:
6233:
6230:Jost, JĂŒrgen
6187:
6181:
6133:
6127:
6075:
6069:
6027:
6023:
5974:
5968:
5927:
5921:
5882:
5876:
5863:
5862:
5850:
5838:
5826:
5814:
5809:, p.336-337.
5802:
5774:
5746:
5726:
5714:
5702:
5690:
5678:
5662:
5637:
5633:
5623:
5588:
5584:
5574:
5569:, Theorem B.
5567:Hartman 1967
5562:
5557:, p.157-161.
5550:
5541:
5521:
5501:
5489:
5461:
5421:
5389:
5384:, p.490-491.
5369:
5333:
5313:
5293:
5281:
5261:
5229:
5205:
5181:
5165:
5153:
5128:
5127:
5105:
5094:
5088:
4879:between two
4875:
4871:
4867:
4864:
4754:
4699:metric space
4692:
4688:
4672:
4652:
4648:
4640:
4636:
4628:
4624:
4614:Yum-Tong Siu
4610:
4595:
4591:
4583:
4579:
4575:
4519:
4515:
4507:
4503:
4499:
4490:
4462:
4458:
4451:
4447:
4436:
4432:
4412:
4218:
4214:
4210:
4207:
3981:
3977:
3972:
3968:
3960:
3944:
3938:
3930:
3915:
3911:
3903:
3895:
3891:
3883:
3879:
3870:
3866:
3862:
3788:
3772:
3757:
3745:
3741:
3733:
3729:
3725:
3718:
3711:
3707:
3692:
3681:
3677:
3672:
3668:
3648:
3644:
3632:
3628:
3620:
3616:
3612:
3608:
3592:
3588:
3583:
3579:
3572:
3567:
3565:
3556:
3551:
3547:
3539:
3534:
3528:
3524:
3516:
3511:
3506:
3501:
3492:
3488:
3483:
3479:
3469:topology as
3465:
3459:
3452:
3448:
3441:
3433:
3429:
3411:
3400:
3396:
3388:
3380:
3375:
3371:
3367:
3363:
3356:
3352:
3348:
3344:
3336:
3277:
3271:
3267:
3260:
3255:
3250:
3246:
3242:
3231:
3226:
3222:
3218:
3212:
3208:
3204:
3189:
3185:
3180:
3176:
3169:
3165:
3153:
3149:
3143:
3137:
3133:
3125:
3121:
3117:
3104:is harmonic.
3093:is harmonic.
3070:
3066:
3062:
3047:
3043:
3039:
3035:
3020:
2990:
2986:
2979:
2978: : (â,
2975:
2968:
2964:
2960:
2951:relative to
2938:
2934:
2930:
2927:
2923:
2919:
2909:
2905:
2901:
2894:is harmonic.
2889:
2885:
2881:
2877:
2871:is harmonic.
2862:
2858:
2854:
2850:
2831:
2827:
2823:
2819:
2801:
2794:
2790:
2782:
2778:
2774:
2750:
2745:
2534:
2524:
2519:
2515:
2511:
2507:
2500:
2496:
2484:
2480:
2475:
2470:
2466:
2460:
2456:
2442:
2435:
2429:
2425:
2420:
2416:
2405:
2401:
2396:
2394:
2367:
2362:
2282:
2264:
2260:
2245:
2223:
2220:
2216:
2213:
2198:
1962:
1960:
1949:
1945:
1941:
1934:
1930:
1923:
1919:
1912:
1909:
1775:
1770:
1766:
1762:
1740:
1737:
1718:
1704:
1696:
1688:
1681:
1676:
1672:bilinear map
1670:, one has a
1654:
1651:
1647:
1644:
1640:
1637:
1630:
1604:
1601:
1597:
1594:
1584:
1576:
1566:
1562:
1544:
1541:
1537:
1534:
1521:
1518:differential
1495:
1491:
1483:
1479:
1475:
1329:
1317:
965:
960:
924:
467:
462:
457:
449:
444:
439:
425:
421:
417:
407:
403:
399:
396:
386:
382:
363:
359:
330:
326:
307:
303:
266:
252:
248:
244:
240:
236:
222:
186:
177:
155:
123:
94:
84:
69:
60:
41:
7234:Simon, Leon
7004:James Eells
6820:(1): 1â68.
6291:(1): 1â24.
6277:Sacks, J.;
6190:: 673â687.
6136:: 165â246.
5707:Struwe 1988
5683:Struwe 1985
5442:HĂ©lein 2002
5350:HĂ©lein 2002
5190:HĂ©lein 2002
4742:sigma model
3705:, the maps
3264:, equal to
2999:solves the
2529:is smooth.
2376:induced by
433:define the
345:. For each
289:. For each
273:open subset
158:James Eells
150:rubber band
63:August 2020
55:introducing
7364:Categories
7330:0478.53001
7276:0864.58015
7226:0886.53004
7196:Yau, S. T.
7192:Schoen, R.
7184:1203.58004
7134:1380.53001
7084:0896.53002
7034:1010.58010
6986:0308.35003
6932:1262.35001
6884:0669.58009
6842:0401.58003
6799:0836.58012
6749:0515.58011
6699:0896.53003
6640:0631.58004
6592:0595.58013
6542:0517.53058
6492:0361.53040
6438:0547.58020
6374:0521.58021
6321:0462.58014
6270:0798.58021
6222:0148.42404
6166:0896.58024
6108:0122.40102
6054:0855.58016
6016:Tian, Gang
6007:0768.53025
5960:0652.58024
5909:0765.53026
5779:Aubin 1998
5751:Aubin 1998
5598:1807.08236
5526:Aubin 1998
5466:Aubin 1998
5426:Aubin 1998
5394:Aubin 1998
5338:Aubin 1998
5234:Aubin 1998
5210:Aubin 1998
5186:Aubin 1998
5124:References
5089:in which Ό
4746:instantons
4679:Lie groups
4487:which are
3446:such that
3201:, the map
2867:valued at
2495:(âΔ, Δ) Ă
2454:such that
1723:, one may
1610:connection
394:by and .
174:Ricci flow
140:stretches
93:is called
38:references
6850:Eells, J.
6808:Eells, J.
6584:122295509
6484:120845708
6262:122184265
6158:118023776
5855:Jost 1994
5791:Jost 2017
5787:Jost 1997
5763:Jost 2017
5759:Jost 1997
5654:1944-9992
5615:0024-6093
5482:Jost 2017
5448:, p.491;
5446:Jost 2017
5410:Jost 2017
5382:Jost 2017
5356:, p.489;
5354:Jost 2017
5326:Jost 2017
5212:, p.349;
5196:, p.489;
5194:Jost 2017
5129:Footnotes
5060:ϵ
5051:μ
5012:∫
4995:ϵ
4986:μ
4929:∫
4899:ϵ
4845:γ
4842:∘
4816:→
4795:γ
4772:→
4712:curvature
4376:∗
4361:
4348:−
4326:∗
4269:∇
4238:Δ
4171:∗
4156:
4121:∗
4093:−
4064:∇
4054:−
4023:Δ
4019:−
4010:∂
4006:∂
3935:Gang Tian
3800:×
3752:homotopic
3697:, and as
3550:< min(
3544:whenever
3360:given by
3316:Δ
3304:∂
3296:∂
3216:given by
3038: : (
2922: : (
2915:immersion
2717:μ
2701:Δ
2659:∂
2655:∂
2636:∫
2632:−
2620:μ
2567:∂
2563:∂
2552:∫
2537:says that
2504:given by
2342:μ
2316:∫
2207:induce a
2175:∘
2170:β
2167:α
2143:∂
2136:β
2128:∂
2109:∂
2102:α
2094:∂
2062:β
2058:∑
2041:α
2037:∑
2016:∑
1995:∑
1973:given by
1840:∇
1813:∑
1793:Δ
1451:α
1426:∇
1393:∑
1372:∑
1363:α
1352:Δ
1297:∘
1292:α
1287:γ
1284:β
1270:Γ
1254:∂
1247:γ
1239:∂
1220:∂
1213:β
1205:∂
1186:γ
1182:∑
1165:β
1161:∑
1141:∂
1134:α
1126:∂
1096:Γ
1076:∑
1072:−
1056:∂
1043:∂
1036:α
1022:∂
1010:α
985:∇
894:δ
886:∂
879:β
876:α
868:∂
862:−
854:β
846:∂
839:δ
836:α
828:∂
814:α
806:∂
799:δ
796:β
788:∂
773:δ
770:γ
749:δ
745:∑
722:γ
717:β
714:α
700:Γ
681:ℓ
673:∂
655:∂
649:−
633:∂
626:ℓ
615:∂
593:∂
586:ℓ
575:∂
560:ℓ
536:ℓ
532:∑
487:Γ
415:and each
397:For each
339:symmetric
245:Laplacian
7236:(1996).
7198:(1997).
7094:(2017).
7044:(1997).
6996:(2002).
6942:(1975).
6659:(1998).
6602:(1988).
6400:(1983).
6336:(1982).
6214:13381249
6176:(1967).
6122:(1992).
6018:(1995).
5952:11210055
5864:Articles
5819:Siu 1980
5532:, p.11;
5512:, p.28;
5508:, p.10;
5496:, p.135.
5476:, p.14;
5472:, p.11;
5436:, p.13;
5432:, p.10;
5404:, p.13;
5400:, p.10;
5380:, p.13;
5376:, p.10;
5348:, p.13;
5344:, p.10;
5328:, p.494.
5272:, p.13;
5244:, p.15;
5220:, p.15;
5112:See also
5102:measures
4874:→
4870: :
4494:-bounded
4337:⟩
4303:⟨
4213: :
4132:⟩
4098:⟨
3855:-fiber.
3184: :
3100:between
3089:between
3069:) â (â,
3046:) â (â,
3001:geodesic
2904: :
2839:For any
2757:gradient
2489:for all
2424: :
1571:between
466: :
448: :
281:and let
253:harmonic
166:deformed
95:harmonic
7322:0645729
7290:(ed.).
7268:1399562
7218:1474501
7176:2431658
7126:3726907
7076:1451625
6978:0482822
6924:3099262
6876:0956352
6834:0495450
6791:1363513
6741:0703510
6691:1636569
6632:0965226
6576:0826871
6534:0584075
6526:1971321
6476:0438388
6430:0710054
6366:0664498
6313:0604040
6305:1971131
6254:1385525
6206:0214004
6150:1215595
6100:0164306
6092:2373037
6046:1371209
5999:1147961
5991:2946567
5944:0990191
5901:1180392
5536:, p.14.
5452:, p.1;
5444:, p.7;
5360:, p.1;
5352:, p.7;
5324:, p.4;
5320:, p.9;
5300:, p.8;
5268:, p.8;
5252:, p.4;
5240:, p.9;
5216:, p.9;
5192:, p.6;
5188:, p.6;
4413:If the
3766:, when
3568:maximal
3546:0 <
3463:in the
2949:minimal
2917:, then
2242:
2230:
1928:is any
1918:, ...,
1529:of the
1527:section
51:improve
7328:
7320:
7310:
7274:
7266:
7256:
7224:
7216:
7206:
7182:
7174:
7164:
7132:
7124:
7114:
7082:
7074:
7064:
7032:
7022:
6984:
6976:
6966:
6930:
6922:
6912:
6882:
6874:
6840:
6832:
6797:
6789:
6779:
6747:
6739:
6729:
6697:
6689:
6679:
6638:
6630:
6590:
6582:
6574:
6540:
6532:
6524:
6490:
6482:
6474:
6436:
6428:
6372:
6364:
6319:
6311:
6303:
6268:
6260:
6252:
6220:
6212:
6204:
6164:
6156:
6148:
6106:
6098:
6090:
6052:
6044:
6005:
5997:
5989:
5958:
5950:
5942:
5907:
5899:
5652:
5613:
5456:, p.2.
5416:, p.1.
5364:, p.1.
5308:, p.4.
5288:, p.4.
5276:, p.3.
5256:, p.3.
5224:, p.4.
5200:, p.2.
4755:A map
4734:action
4732:whose
3931:bubble
3096:Every
3085:Every
2913:is an
2812:Every
2363:where
1910:where
1719:Using
1608:has a
357:, let
335:matrix
301:, let
271:be an
40:, but
6580:S2CID
6522:JSTOR
6480:S2CID
6301:JSTOR
6258:S2CID
6210:S2CID
6154:S2CID
6088:JSTOR
5987:JSTOR
5948:S2CID
5593:arXiv
3980:<
3815:into
3680:<
3657:from
3591:<
3514:<
3491:<
3415:from
2985:) â (
2933:) â (
2884:) â (
2857:) â (
2826:) â (
2433:with
1744:over
1725:trace
1658:over
1548:over
1525:as a
1508:from
929:from
7308:ISBN
7254:ISBN
7204:ISBN
7162:ISBN
7112:ISBN
7062:ISBN
7020:ISBN
6964:ISBN
6910:ISBN
6777:ISBN
6727:ISBN
6677:ISBN
5650:ISSN
5611:ISSN
4728:, a
4661:and
4633:and
4552:and
4512:and
4500:Let
4475:and
4352:scal
4147:scal
3888:and
3738:and
3690:has
3625:and
3613:Let
3497:and
3440:(0,
3370:) âŠ
3351:) Ă
3211:) â
3130:and
3118:Let
3074:stan
3051:stan
2983:stan
2816:map
2805:stan
2787:and
2775:Let
2763:and
2533:The
2514:) âŠ
2395:The
2203:and
1707:f(p)
1620:and
1587:f(p)
1488:and
1476:Let
941:and
454:and
349:and
341:and
293:and
267:Let
243:The
239:and
207:and
160:and
7326:Zbl
7300:doi
7272:Zbl
7246:doi
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7180:Zbl
7154:doi
7130:Zbl
7104:doi
7080:Zbl
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7030:Zbl
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6982:Zbl
6956:doi
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6864:doi
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6822:doi
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6769:doi
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6080:doi
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6032:doi
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5979:doi
5975:135
5956:Zbl
5932:doi
5928:201
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5887:doi
5642:doi
5603:doi
4724:In
4604:to
4563:If
4548:If
4532:to
4425:of
4417:of
4309:Ric
4104:Ric
3695:= â
3661:to
3641:of
3419:to
3197:in
3162:in
2958:If
2947:is
2899:If
2843:in
2450:of
2285:of
2250:on
2211:on
1756:in
1675:(â(
1666:in
1616:on
1556:in
1512:to
1500:be
1338:by
1334:on
974:by
970:on
933:to
472:by
470:â â
452:â â
376:in
337:is
320:in
275:of
113:in
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7318:MR
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