Harmonic map

In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional generalizing the Dirichlet energy (which is often itself called "Dirichlet energy"). As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds.

Informally, the Dirichlet energy of a mapping $f$ from a Riemannian manifold $M$ to a Riemannian manifold $N$ can be thought of as the total amount that $f$ "stretches" $M$ in allocating each of its elements to a point of $N$ . For instance, a rubber band which is stretched around a (smooth) stone can be mathematically formalized as a mapping from the points on the unstretched band to the surface of the stone. The unstretched band and stone are given Riemannian metrics as embedded submanifolds of three-dimensional Euclidean space; the Dirichlet energy of such a mapping is then a formalization of the notion of the total tension involved. Harmonicity of such a mapping means that, given any hypothetical way of physically deforming the given stretch, the tension (when considered as a function of time) has first derivative zero when the deformation begins.

The theory of harmonic maps was initiated in 1964 by James Eells and Joseph Sampson, who showed that in certain geometric contexts, arbitrary smooth maps could be deformed into harmonic maps. Their work was the inspiration for Richard Hamilton's first work on the Ricci flow. Harmonic maps and the associated harmonic map heat flow, in and of themselves, are among the most widely studied topics in the field of geometric analysis.

The discovery of the "bubbling" of sequences of harmonic maps, due to Jonathan Sacks and Karen Uhlenbeck, has been particularly influential, as the same phenomena has been found in many other geometric contexts. Notably, Uhlenbeck's parallel discovery of bubbling of Yang-Mills fields is important in Simon Donaldson's work on four-dimensional manifolds, and Mikhael Gromov's later discovery of bubbling of pseudoholomorphic curves is significant in applications to symplectic geometry and quantum cohomology. The techniques used by Richard Schoen 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.

Mathematical definition

Here the notion of the laplacian of a map is considered from three different perspectives. A map is called harmonic if its laplacian vanishes; it is called totally geodesic if its hessian vanishes.

Integral formulation

Let $(M, g)$ and $(N, h)$ be Riemannian manifolds. Given a smooth map $f$ from $M$ to $N$ , the pullback $f * h$ is a symmetric 2-tensor on $M$ ; the energy density $e (f)$ of $f$ is one-half of its $g$ -trace. If $M$ is oriented and $M$ is compact, the Dirichlet energy of $f$ is defined as

E(f)=\int _{M}e(f)\,d\mu _{g}

where $dμ g$ is the volume form on $M$ induced by $g$ . Even if $M$ is noncompact, the following definition is meaningful: the Laplacian or tension field $Δ f$ of $f$ is the vector field in $N$ along $f$ such that

\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}

for any one-parameter family of maps $f s : M \to N$ with $f 0 = f$ and for which there exists a precompact open set $K$ of $M$ such that $f s | M - K = f | M - K$ for all $s$ ; one supposes that the parametrized family is smooth in the sense that the associated map $(-ε, ε) \times M \to N$ given by $(s, p) \mapsto f s (p)$ is smooth.

In case $M$ is compact, the Laplacian of $f$ can then be thought of as the gradient of the Dirichlet energy functional.

Local coordinates

Let $U$ be an open subset of $ℝ m$ and let $V$ be an open subset of $ℝ n$ . For each $i$ and $j$ between 1 and $n$ , let $g ij$ be a smooth real-valued function on $U$ , such that for each $p$ in $U$ , one has that the $m \times m$ matrix $[g ij (p)]$ is symmetric and positive-definite. For each $α$ and $β$ between 1 and $m$ , let $h αβ$ be a smooth real-valued function on $V$ , such that for each $q$ in $V$ , one has that the $n \times n$ matrix $[h αβ (q)]$ is symmetric and positive-definite. Denote the inverse matrices by $[g ij (p)]$ and $[h αβ (q)]$ .

For each $i, j, k$ between 1 and $n$ and each $α, β, γ$ between 1 and $m$ define the Christoffel symbols $Γ(g) k ij : U \to ℝ$ and $Γ(h) γ αβ : V \to ℝ$

{\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}}

Given a smooth map $f$ from $U$ to $V$ , its hessian defines for each $i$ and $j$ between 1 and $m$ and for each $α$ between 1 and $n$ the real-valued function $\nabla(df) α ij$ on $U$ by

\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.

Its laplacian or tension field defines for each $α$ between 1 and $n$ the real-valued function $(∆ f) α$ on $U$ by

(\Delta f)^{\alpha }=\sum _{i=1}^{m}\sum _{j=1}^{m}g^{ij}\nabla (df)_{ij}^{\alpha }.

The energy density of $f$ is the real-valued function on $U$ given by

\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).

Bundle formalism

Let $(M, g)$ and $(N, h)$ be Riemannian manifolds. Given a smooth map $f$ from $M$ to $N$ , one can consider its differential $df$ as a section of the vector bundle $T * M \otimes f * TN$ over $M$ ; all this says is that for each $p$ in $M$ , one has a linear map $df p$ as $T p M \to T f(p) N$ . The Riemannian metrics on $M$ and $N$ induce a bundle metric on $T * M \otimes f * TN$ , and so one may define $1 / 2 | df | 2$ as a smooth function on $M$ , known as the energy density.

The bundle $T * M \otimes f * TN$ also has a metric-compatible connection induced from the Levi-Civita connections on $M$ and $N$ . So one may take the covariant derivative $\nabla(df)$ , which is a section of the vector bundle $T * M \otimes T * M \otimes f * TN$ over $M$ ; this says that for each $p$ in $M$ , one has a bilinear map $(\nabla(df)) p$ as $T p M \times T p M \to T f(p) N$ . This section is known as the hessian of $f$ .

Using $g$ , one may trace the hessian of $f$ to arrive at the laplacian or tension field of $f$ , which is a section of the bundle $f * TN$ over $M$ ; this says that the laplacian of $f$ assigns to each $p$ in $M$ an element of $T f (p) N$ . It is defined by

(\Delta f)_{p}=\sum _{i=1}^{m}{\big (}\nabla (df){\big )}_{p}(e_{i},e_{i})

where $e 1, ..., e m$ is a $g p$ -orthonormal basis of $T p M$ .

Examples of harmonic maps

Let $(M, g)$ and $(N, h)$ be smooth Riemannian manifolds. The notation $g stan$ is used to refer to the standard Riemannian metric on Euclidean space.

Every totally geodesic map (M, g) → (N, h) is harmonic; this follows directly from the above definitions. As special cases:
- For any $q$ in $N$ , the constant map $(M, g) \to (N, h)$ valued at $q$ is harmonic.
- The identity map $(M, g) \to (M, g)$ is harmonic.
If f : M → N is an immersion, then f : (M, f^*h) → (N, h) is harmonic if and only if f is minimal relative to h. As a special case:
- If $f : ℝ \to (N, h)$ is a constant-speed immersion, then $f : (ℝ, g stan) \to (N, h)$ is harmonic if and only if $f$ solves the geodesic differential equation.

Recall that if

M

is one-dimensional, then minimality of

f

is equivalent to

f

being geodesic, although this does not imply that it is a constant-speed parametrization, and hence does not imply that

f

solves the geodesic differential equation.

A smooth map $f : (M, g) \to (ℝ n, g stan)$ is harmonic if and only if each of its $n$ component functions are harmonic as maps $(M, g) \to (ℝ, g stan)$ . This coincides with the notion of harmonicity provided by the Laplace-Beltrami operator.
Every holomorphic map between Kähler manifolds is harmonic.
Every harmonic morphism between Riemannian manifolds is harmonic.

Harmonic map heat flow

Let $(M, g)$ and $(N, h)$ be smooth Riemannian manifolds. A harmonic map heat flow on an interval $(a, b)$ assigns to each $t$ in $(a, b)$ a twice-differentiable map $f t : M \to N$ in such a way that, for each $p$ in $M$ , the map $(a, b) \to N$ given by $t \mapsto f t (p)$ is differentiable, and its derivative at a given value of $t$ is, as a vector in $T f t (p) N$ , equal to $(∆ f t) p$ . This is usually abbreviated as:

{\frac {\partial f}{\partial t}}=\Delta f.

Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties:

Regularity. Any harmonic map heat flow is smooth as a map $(a, b) \times M \to N$ given by $(t, p) \mapsto f t (p)$ .

Now suppose that $M$ is a closed manifold and $(N, h)$ is geodesically complete.

Existence. Given a continuously differentiable map $f$ from $M$ to $N$ , there exists a positive number $T$ and a harmonic map heat flow $f t$ on the interval $(0, T)$ such that $f t$ converges to $f$ in the $C 1$ topology as $t$ decreases to 0.[1]
Uniqueness. If ${f t : 0 < t < T}$ and ${f t : 0 < t < T}$ are two harmonic map heat flows as in the existence theorem, then $f t = f t$ whenever $0 < t < min(T, T)$ .

As a consequence of the uniqueness theorem, there exists a maximal harmonic map heat flow with initial data $f$ , meaning that one has a harmonic map heat flow ${f t : 0 < t < T}$ as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that $T$ takes on its maximal possible value, which could be infinite.

Eells and Sampson's theorem

The primary result of Eells and Sampson's 1964 paper is the following:

Let $(M, g)$ and $(N, h)$ be smooth and closed Riemannian manifolds, and suppose that the sectional curvature of $(N, h)$ is nonpositive. Then for any continuously differentiable map $f$ from $M$ to $N$ , the maximal harmonic map heat flow ${f t : 0 < t < T}$ with initial data $f$ has $T = \infty$ , and as $t$ increases to $\infty$ , the maps $f t$ subsequentially converge in the $C \infty$ topology to a harmonic map.

In particular, this shows that, under the assumptions on $(M, g)$ and $(N, h)$ , every continuous map is homotopic 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. In 1967, Philip Hartman extended their methods to study uniqueness of harmonic maps within homotopy classes, additionally showing that the convergence in the Eells-Sampson theorem is strong, without the need to select a subsequence. Eells and Sampson's result was adapted to the setting of the Dirichlet boundary value problem, when $M$ is instead compact with nonempty boundary, by Richard Hamilton in 1975.

For many years after Eells and Sampson's work, it was unclear to what extent the sectional curvature assumption on $(N, h)$ 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 "finite-time blowup" even when both $(M, g)$ and $(N, h)$ 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 Chang-Ding-Ye result is considered to be indicative of the general character of the flow.

The Bochner formula and rigidity

The main computational point in the proof of Eells and Sampson's theorem is an adaptation of the Bochner formula to the setting of a harmonic map heat flow ${f t : 0 < t < T}$ . This formula says

{\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 )}.

This is also of interest in analyzing harmonic maps themselves; suppose $f : M \to N$ is harmonic. Any harmonic map can be viewed as a constant-in- $t$ solution of the harmonic map heat flow, and so one gets from the above formula that

\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 )}.

If the Ricci curvature of $g$ is positive and the sectional curvature of $h$ is nonpositive, then this implies that $∆ e (f)$ is nonnegative. If $M$ is closed, then multiplication by $e (f)$ and a single integration by parts shows that $e (f)$ must be constant, and hence zero; hence $f$ must itself be constant. Richard Schoen & Shing-Tung Yau (1976) note that this can be extended to noncompact $M$ by making use of Yau's theorem asserting that nonnegative subharmonic functions which are $L 2$ -bounded must be constant. In summary, according to Eells & Sampson (1964) and Schoen & Yau (1976), one has:

Let $(M, g)$ and $(N, h)$ be smooth and complete Riemannian manifolds, and let $f$ be a harmonic map from $M$ to $N$ . Suppose that the Ricci curvature of $g$ is positive and the sectional curvature of $h$ is nonpositive.

If $M$ and $N$ are both closed then $f$ must be constant.

If $N$ is closed and $f$ has finite Dirichlet energy, then it must be constant.

In combination with the Eells-Sampson theorem, this shows (for instance) that if $(M, g)$ is a closed Riemannian manifold with positive Ricci curvature and $(N, h)$ is a closed Riemannian manifold with nonpositive sectional curvature, then every continuous map from $M$ to $N$ is homotopic to a constant.

The general idea of deforming a general object to a harmonic map, and then showing that the any such harmonic map must automatically be of a highly restricted class, has found many applications. For instance, Yum-Tong Siu (1980) found an important complex-analytic version of the Bochner formula, asserting that a harmonic map between Kähler manifolds must be holomorphic, provided that the target manifold has appropriately negative curvature. As an application, by making use of the Eells-Sampson existence theorem for harmonic maps, he was able to show that if $(M, g)$ and $(N, h)$ are smooth and closed Kähler manifolds, and if the curvature of $(N, h)$ is appropriately negative, then $M$ and $N$ 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, Siu was able to prove a variant of the still-unsolved Hodge conjecture, albeit in the restricted context of negative curvature.

Kevin Corlette (1992) gave a significant extension of Siu's Bochner formula, and used it to prove new rigidity theorems for lattices in certain Lie groups. Following this, Mikhael Gromov and Richard Schoen extended much of the theory of harmonic maps to allow $(N, h)$ to be replaced by a metric space. By an extension of the Eells-Sampson theorem together with an extension of the Siu-Corlette Bochner formula, they were able to prove new rigidity theorems for lattices.

Problems and applications

If, after applying the rubber M onto the marble N via some map $\phi$ , one "releases" it, it will try to "snap" into a position of least tension. This "physical" observation leads to the following mathematical problem: given a homotopy class of maps from M to N, does it contain a representative that is a harmonic map?
Existence results on harmonic maps between manifolds has consequences for their curvature.
Once existence is known, how can a harmonic map be constructed explicitly? (One fruitful method uses twistor theory.)
In theoretical physics, a quantum field theory whose action is given by the Dirichlet energy is known as a sigma model. In such a theory, harmonic maps correspond to instantons.
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.

Harmonic maps between metric spaces

The energy integral can be formulated in a weaker setting for functions u : M → N between two metric spaces (Jost 1995). The energy integrand is instead a function of the form

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)}}

in which μ^ε
_x is a family of measures attached to each point of M.

gollark: It's probably fine, as long as technology keeps improving.

gollark: Solution: achieve immortality.

gollark: "Immortality" is the most convenient descriptor for that.

gollark: I mean, I don't want to live *forever*, but an arbitrarily long (chosen by me) time.

gollark: ......

References

This means that, relative to any local coordinate charts, one has uniform convergence on compact sets of the functions and their first partial derivatives.

Eells, James, Jr.; Sampson, J.H. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160. doi:10.2307/2373037, JSTOR 2373037
Hartman, Philip. On homotopic harmonic maps. Canadian J. Math. 19 (1967), 673–687.
Schoen, Richard; Yau, Shing Tung. Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51 (1976), no. 3, 333–341.
Hildebrandt, Stéfan; Kaul, Helmut; Widman, Kjell-Ove. An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138 (1977), no. 1-2, 1–16.
Schoen, R.; Yau, Shing Tung. Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. of Math. (2) 110 (1979), no. 1, 127–142.
Siu, Yum Tong. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math. (2) 112 (1980), no. 1, 73–111.
Baird, P.; Eells, J. A conservation law for harmonic maps. Geometry Symposium, Utrecht 1980 (Utrecht, 1980), pp. 1–25, Lecture Notes in Math., 894, Springer, Berlin-New York, 1981.
Sacks, J.; Uhlenbeck, K. The existence of minimal immersions of 2-spheres. Ann. of Math. (2) 113 (1981), no. 1, 1–24.
Giaquinta, Mariano; Giusti, Enrico. On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31–46.
Schoen, Richard; Uhlenbeck, Karen. A regularity theory for harmonic maps. J. Differential Geom. 17 (1982), no. 2, 307–335.
Schoen, Richard; Uhlenbeck, Karen. Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom. 18 (1983), no. 2, 253–268.
Simon, Leon. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. (2) 118 (1983), no. 3, 525–571.
Schoen, Richard M. Analytic aspects of the harmonic map problem. Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), 321–358, Math. Sci. Res. Inst. Publ., 2, Springer, New York, 1984.
Struwe, Michael. On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60 (1985), no. 4, 558–581.
Brezis, Haïm; Coron, Jean-Michel; Lieb, Elliott H. Harmonic maps with defects. Comm. Math. Phys. 107 (1986), no. 4, 649–705.
Struwe, Michael. On the evolution of harmonic maps in higher dimensions. J. Differential Geom. 28 (1988), no. 3, 485–502.
Chen, Yun Mei; Struwe, Michael. Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201 (1989), no. 1, 83–103.
Uhlenbeck, Karen. Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differential Geom. 30 (1989), no. 1, 1–50.
Evans, Lawrence C. Partial regularity for stationary harmonic maps into spheres. Arch. Rational Mech. Anal. 116 (1991), no. 2, 101–113.
Hélein, Frédéric. Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591–596.
Chang, Kung-Ching; Ding, Wei Yue; Ye, Rugang. Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differential Geom. 36 (1992), no. 2, 507–515.
Corlette, Kevin. Archimedean superrigidity and hyperbolic geometry. Ann. of Math. (2) 135 (1992), no. 1, 165–182.
Gromov, Mikhail; Schoen, Richard. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. No. 76 (1992), 165–246.
Bethuel, Fabrice. On the singular set of stationary harmonic maps. Manuscripta Math. 78 (1993), no. 4, 417–443.
Korevaar, Nicholas J.; Schoen, Richard M. Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659.
Jost, Jürgen (1994), "Equilibrium maps between metric spaces", Calculus of Variations and Partial Differential Equations, 2 (2): 173–204, doi:10.1007/BF01191341, ISSN 0944-2669, MR 1385525.
Ding, Weiyue; Tian, Gang. Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3 (1995), no. 3-4, 543–554.
Dorfmeister, J.; Pedit, F.; Wu, H. Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), no. 4, 633–668.

Books and surveys

Aubin, Thierry. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xviii+395 pp. ISBN 3-540-60752-8
Eells, J.; Lemaire, L. A report on harmonic maps. Bull. London Math. Soc. 10 (1978), no. 1, 1–68.
Eells, James; Lemaire, Luc. Selected topics in harmonic maps. CBMS Regional Conference Series in Mathematics, 50. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. v+85 pp. ISBN 0-8218-0700-5
Eells, J.; Lemaire, L. Another report on harmonic maps. Bull. London Math. Soc. 20 (1988), no. 5, 385–524. doi:10.1112/blms/20.5.385
Eells, James; Lemaire, Luc. Two reports on harmonic maps. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. xii+216 pp. ISBN 981-02-1466-9 [Release of Eells & Lemaire (1978, 1988) in book form]
Giaquinta, Mariano; Martinazzi, Luca. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Second edition. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 11. Edizioni della Normale, Pisa, 2012. xiv+366 pp. ISBN 978-88-7642-442-7, 978-88-7642-443-4
Hamilton, Richard S. Harmonic maps of manifolds with boundary. Lecture Notes in Mathematics, Vol. 471. Springer-Verlag, Berlin-New York, 1975. i+168 pp.
Hélein, Frédéric. Harmonic maps, conservation laws and moving frames. Translated from the 1996 French original. With a foreword by James Eells. Second edition. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002. xxvi+264 pp. ISBN 0-521-81160-0
Jost, Jürgen. Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997. viii+108 pp. ISBN 3-7643-5736-3
Jost, Jürgen. Riemannian geometry and geometric analysis. Seventh edition. Universitext. Springer, Cham, 2017. xiv+697 pp. ISBN 978-3-319-61859-3, 978-3-319-61860-9
Schoen, R.; Yau, S.T. Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology, II. International Press, Cambridge, MA, 1997. vi+394 pp. ISBN 1-57146-002-0
Simon, Leon. Theorems on regularity and singularity of energy minimizing maps. Based on lecture notes by Norbert Hungerbühler. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996. viii+152 pp. ISBN 3-7643-5397-X
Yau, Shing Tung. Survey on partial differential equations in differential geometry. Seminar on Differential Geometry, pp. 3–71, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982.

External links

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] This means that, relative to any local coordinate charts, one has uniform convergence on compact sets of the functions and their first partial derivatives.