View source for Biharmonic map

{{MOS|article|MOS:BBB - ℝ|date=July 2023}}
In the mathematical field of [[Differential geometry|differential geometry]], a '''biharmonic map''' is a map between [[Riemannian manifold|Riemannian]] or [[Pseudo-Riemannian manifold|pseudo-Riemannian manifold]]s which satisfies a certain fourth-order [[Partial differential equation|partial differential equation]]. A '''biharmonic submanifold''' refers to an embedding or immersion into a Riemannian or pseudo-Riemannian manifold which is a biharmonic map when the domain is equipped with its induced metric. The problem of understanding biharmonic maps was posed by [[Biography:James Eells|James Eells]] and Luc Lemaire in 1983.{{sfnm|1a1=Eells|1a2=Lemaire|1y=1983|1loc=(8.7) and (8.8)}} The study of [[Harmonic map|harmonic map]]s, of which the study of biharmonic maps is an outgrowth (any harmonic map is also a biharmonic map), had been (and remains) an active field of study for the previous twenty years.{{sfnm|1a1=Eells|1a2=Sampson|1y=1964}} A simple case of biharmonic maps is given by [[Biharmonic equation|biharmonic functions]].

==Definition==
Given Riemannian or pseudo-Riemannian manifolds {{math|(''M'', ''g'')}} and {{math|(''N'', ''h'')}}, a map {{mvar|f}} from {{mvar|M}} to {{mvar|N}} which is differentiable at least four times is called a ''biharmonic map'' if
:<math>\Delta\Delta f+\sum_{i=1}^m R^h\big(\Delta f,df(e_i),df(e_i)\big)=0;</math>
given any point {{mvar|p}} of {{mvar|M}}, each side of this equation is an element of the [[Tangent space|tangent space]] to {{mvar|N}} at {{math|''f''(''p'')}}.{{sfnm|1a1=Jiang|1y=1986|1loc=Definition 5|2a1=Chen|2y=2011|2loc=eq. (7.64)}} In other words, the above equation is an equality of sections of the [[Vector bundle|vector bundle]] {{math|''f''<sup> *</sup>''TN'' → ''M''}}. In the equation, {{math|''e''<sub>1</sub>, ..., ''e''<sub>''m''</sub>}} is an arbitrary {{mvar|g}}-orthonormal basis of the [[Tangent space|tangent space]] to {{mvar|M}} and {{math|''R''<sup>''h''</sup>}} is the [[Riemann curvature tensor]], following the convention {{math|''R''(''u'', ''v'', ''w'') {{=}} ∇<sub>''u''</sub>∇<sub>''v''</sub>''w'' − ∇<sub>''v''</sub>∇<sub>''u''</sub>''w'' − ∇<sub>[''u'', ''v'']</sub>w}}. The quantity {{math|∆''f''}} is the "tension field" or "Laplacian" of {{mvar|f}}, as was introduced by Eells and Sampson in the study of harmonic maps.{{sfnm|1a1=Eells|1a2=Sampson|1y=1964|1p=116}}

In terms of the [[Trace (linear algebra)|trace]], [[Interior product|interior product]], and [[Pullback (differential geometry)|pullback]] operations, the biharmonic map equation can be written as
:<math>\Delta\Delta f+\operatorname{tr}_g\Big(f^\ast\big(\iota_{\Delta f}R^h\big)\Big)=0.</math>
In terms of local coordinates {{math|''x''<sup>''i''</sup>}} for {{mvar|M}} and local coordinates {{math|''y''<sup>α</sup>}} for {{mvar|N}}, the biharmonic map equation is written as
:<math>g^{ij}\left(\frac{\partial}{\partial x^i}\left(\frac{\partial(\Delta f)^\alpha}{\partial x^j}+\frac{\partial f^\beta}{\partial x^j}\Gamma_{\beta\gamma}^\alpha(\Delta f)^\gamma\right)-\Gamma_{ij}^k\left(\frac{\partial(\Delta f)^\alpha}{\partial x^k}+\frac{\partial f^\beta}{\partial x^k}\Gamma_{\beta\gamma}^\alpha(\Delta f)^\gamma\right)+\frac{\partial f^\delta}{\partial x^i}\Gamma_{\delta\epsilon}^\alpha\left(\frac{\partial(\Delta f)^\epsilon}{\partial x^j}+\frac{\partial f^\beta}{\partial x^j}\Gamma_{\beta\gamma}^\epsilon(\Delta f)^\gamma\right)\right)+g^{ij}R_{\beta\gamma\delta}^\alpha(\Delta f)^\beta\frac{\partial f^\gamma}{\partial x^i}\frac{\partial f^\delta}{\partial x^j}=0,</math>
in which the Einstein summation convention is used with the following definitions of the [[Christoffel symbols]], [[Riemann curvature tensor]], and [[Harmonic map|tension field]]:
:<math>\begin{align}
\Gamma_{ij}^k&=\frac{1}{2}g^{kl}\Big(\frac{\partial g_{jl}}{\partial x^i}+\frac{\partial g_{il}}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^l}\Big)\\
\Gamma_{\beta\gamma}^\alpha&=\frac{1}{2}h^{\alpha\delta}\Big(\frac{\partial h_{\gamma\delta}}{\partial y^\beta}+\frac{\partial h_{\beta\delta}}{\partial y^\gamma}-\frac{\partial h_{\beta\gamma}}{\partial y^\delta}\Big)\\
R_{\beta\gamma\delta}^\alpha&=\frac{\partial\Gamma_{\gamma\delta}^\alpha}{\partial y^\beta}-\frac{\partial\Gamma_{\beta\delta}^\alpha}{\partial y^\gamma}+\Gamma_{\beta\rho}^\alpha\Gamma_{\gamma\delta}^\rho-\Gamma_{\gamma\rho}^\alpha\Gamma_{\beta\delta}^\rho\\
(\Delta f)^\alpha&=g^{ij}\Big(\frac{\partial^2f^\alpha}{\partial x^i\partial x^j}-\Gamma_{ij}^k\frac{\partial f^\alpha}{\partial x^k}+\frac{\partial f^\beta}{\partial x^i}\Gamma_{\beta\gamma}^\alpha\frac{\partial f^\gamma}{\partial x^j}\Big).
\end{align}</math>
It is clear from any of these presentations of the equation that any harmonic map is automatically biharmonic. For this reason, a ''proper biharmonic map'' refers to a biharmonic map which is not harmonic.

In the special setting where {{mvar|f}} is a (pseudo-)Riemannian immersion, meaning that it is an [[Immersion (mathematics)|immersion]] and that {{mvar|g}} is equal to the [[Induced metric|induced metric]] {{math|''f''<sup> *</sup>''h''}}, one says that one has a ''biharmonic submanifold'' instead of a biharmonic map. Since the [[Mean curvature|mean curvature vector]] of {{mvar|f}} is equal to the laplacian of {{math|''f'' : (''M'', ''f''<sup> *</sup>''h'') → (''N'', ''h'')}}, one knows that an immersion is minimal if and only if it is harmonic. In particular, any minimal immersion is automatically a biharmonic submanifold. A ''proper biharmonic submanifold'' refers to a biharmonic submanifold which is not minimal.

The motivation for the biharmonic map equation is from the ''bienergy functional''
: <math>E_2(f) = \frac{1}{2}\,\int_M |\Delta f|_h^2\, dv_g,</math>
in the setting where {{mvar|M}} is [[Closed manifold|closed]] and {{mvar|g}} and {{mvar|h}} are both Riemannian; {{math|''dv''<sub>''g''</sub>}} denotes the volume [[Measure (mathematics)|measure]] on <math>M</math> induced by {{mvar|g}}. Eells & Lemaire, in 1983, suggested the study of [[Critical point (mathematics)|critical points]] of this functional.{{sfnm|1a1=Eells|1a2=Lemaire|1y=1983|1loc=(8.7)}} Guo Ying Jiang, in 1986, calculated its first variation formula, thereby finding the above biharmonic map equation as the corresponding Euler-Lagrange equation.{{sfnm|1a1=Jiang|1y=1986|1loc=Theorem 3}} Harmonic maps correspond to critical points for which the bioenergy functional takes on its minimal possible value of zero.

==Examples and classification==
A number of examples of biharmonic maps, such as inverses of [[Stereographic projection|stereographic projection]]s in the special case of four dimensions, and inversions of punctured [[Euclidean space]], are known.{{sfnm|1a1=Montaldo|1a2=Oniciuc|1y=2006|1loc=Sections 5−7}} There are many examples of biharmonic submanifolds, such as (for any {{mvar|k}}) the generalized [[Clifford torus]]
:<math>\Big\{x\in\mathbb{R}^{n+2}:x_1^2+\cdots+x_{k+1}^2=x_{k+2}^2+\cdots+x_{n+2}^2=\frac{1}{2}\Big\},</math>
as a submanifold of the {{math|(''n'' + 1)}}-sphere.{{sfnm|1a1=Jiang|1y=1986|1loc=Example 12}} It is minimal if and only if {{mvar|n}} is even and equal to {{math|2''k''}}.

The biharmonic curves in three-dimensional [[Space form|space form]]s can be studied via the Frenet equations. It follows easily that every constant-speed biharmonic curve in a three-dimensional space form of nonpositive curvature must be geodesic.{{sfnm|1a1=Caddeo|1a2=Montaldo|1a3=Oniciuc|1y=2001|1loc=Proposition 3.1}} Any constant-speed biharmonic curves in the round three-dimensional sphere {{math|''S''<sup>3</sup>}} can be viewed as the solution of a certain [[Linear differential equation#Homogeneous equation with constant coefficients|constant-coefficient fourth-order linear ordinary differential equation]] for a {{math|ℝ<sup>4</sup>}}-valued function.{{sfnm|1a1=Caddeo|1a2=Montaldo|1a3=Oniciuc|1y=2001|1loc=Proposition 3.2}} As such the situation can be completely analyzed, with the result that any such curve is, up to an isometry of the sphere:
* a constant-speed parametrization of the intersection of {{math|''S''<sup>3</sup> ⊂ ℝ<sup>4</sup>}} with the two-dimensional linear subspace {{math|ℝ × ℝ × {0} × {0}}}
* a constant-speed parametrization of the intersection of {{math|''S''<sup>3</sup> ⊂ ℝ<sup>4</sup>}} with the two-dimensional affine subspace {{math|ℝ × ℝ × {''d''<sub>1</sub>} × {''d''<sub>2</sub>}}}, for any choice of {{math|(''d''<sub>1</sub>, ''d''<sub>2</sub>)}} which is on the circle of radius {{math|2<sup>−1/2</sup>}} around the origin in {{math|ℝ<sup>2</sup>}}
* a constant-speed reparametrization of
::<math>t\mapsto \Big(\frac{\cos at}{\sqrt{2}},\frac{\sin at}{\sqrt{2}},\frac{\cos bt}{\sqrt{2}},\frac{\sin bt}{\sqrt{2}}\Big)</math>
:for any {{math|(''a'', ''b'')}} on the circle of radius {{math|2<sup>1/2</sup>}} around the origin in {{math|ℝ<sup>2</sup>}}.
In particular, every constant-speed biharmonic curve in {{math|''S''<sup>3</sup>}} has constant [[Geodesic curvature|geodesic curvature]].

As a consequence of the purely local study of the Gauss-Codazzi equations and the biharmonic map equation, any connected biharmonic surface in {{math|''S''<sup>3</sup>}} must have constant mean curvature.{{sfnm|1a1=Caddeo|1a2=Montaldo|1a3=Oniciuc|1y=2001|1loc=Theorem 4.5}} If it is nonzero (so that the surface is not minimal) then the [[Second fundamental form|second fundamental form]] must have constant length equal to {{math|2<sup>1/2</sup>}}, as follows from the biharmonic map equation. Surfaces with such strong geometric conditions can be completely classified, with the result that any connected biharmonic surface in {{math|''S''<sup>3</sup>}} must be either locally (up to isometry) part of the hypersphere
:<math>\left\{\Big((w,x,y,\frac{1}{\sqrt{2}}\Big):w^2+x^2+y^2=\frac{1}{2}\right\},</math>
or minimal.{{sfnm|1a1=Caddeo|1a2=Montaldo|1a3=Oniciuc|1y=2001|1loc=Theorem 4.8}} In a similar way, any biharmonic hypersurface of [[Euclidean space]] which has constant mean curvature must be minimal.{{sfnm|1a1=Chen|1y=2011|1loc=Corollary 2.10}}

Guo Ying Jiang showed that if {{mvar|g}} and {{mvar|h}} are Riemannian, and if {{mvar|M}} is closed and {{mvar|h}} has nonpositive [[Sectional curvature|sectional curvature]], then a map from {{math|(''M'', ''g'')}} to {{math|(''N'', ''h'')}} is biharmonic if and only if it is harmonic.{{sfnm|1a1=Jiang|1y=1986|1loc=Proposition 7}} The proof is to show that, due to the sectional curvature assumption, the Laplacian of {{math|{{!}}∆''f''{{!}}<sup>2</sup>}} is nonnegative, at which point the [[Maximum principle|maximum principle]] applies. This result and proof can be compared to Eells & Sampson's vanishing theorem, which says that if additionally the [[Ricci curvature]] of {{mvar|g}} is nonnegative, then a map from {{math|(''M'', ''g'')}} to {{math|(''N'', ''h'')}} is harmonic if and only if it is totally geodesic.{{sfnm|1a1=Eells|1a2=Sampson|1y=1964|1p=124}} As a special case of Jiang's result, a closed submanifold of a Riemannian manifold of nonpositive sectional curvature is biharmonic if and only if it is minimal. Partly based on these results, it was conjectured by R. Caddeo, [[Biography:Stefano Montaldo|S. Montaldo]] and C. Oniciuc that ''every'' biharmonic submanifold of a Riemannian manifold of nonpositive sectional curvature must be minimal.{{sfnm|1a1=Caddeo|1a2=Montaldo|1a3=Oniciuc|1y=2001|1p=869}} This, however, is now known to be false.{{sfnm|1a1=Chen|1y=2011|1p=147}} The special case of submanifolds of Euclidean space is an older conjecture of [[Biography:Bang-Yen Chen|Bang-Yen Chen]].{{sfnm|1a1=Chen|1y=1991|1loc=Conjecture 3|2a1=Chen|2y=1996|2loc=Conjecture 25.B.6}} Chen's conjecture has been proven in a number of geometrically special cases.{{sfnm|1a1=Chen|1y=1996|1loc=Theorems 15.4, 15.6−15.8, 15.10, 15.12−15.13}}

==References==
{{refbegin}}
'''Footnotes'''
{{reflist|30em}}
'''Books and surveys'''
*{{cite book|first1=Bang-Yen|last1=Chen|title=Pseudo-Riemannian geometry, δ-invariants and applications|others=With a foreword by Leopold Verstraelen|publisher=[[Company:World Scientific|World Scientific]]|location=Hackensack, NJ|year=2011|isbn=978-981-4329-63-7|mr=2799371|doi=10.1142/9789814329644|zbl=1245.53001}}
*{{cite book|first1=Bang-Yen|last1=Chen|title=Total mean curvature and submanifolds of finite type|edition=Second edition of 1984 original|others=With a foreword by Leopold Verstraelen|series=Series in Pure Mathematics|volume=27|publisher=[[Company:World Scientific|World Scientific]]|location=Hackensack, NJ|year=2015|isbn=978-981-4616-69-0|zbl=1326.53004|mr=3362186|doi=10.1142/9237}}
*{{cite book|last1=Eells|first1=James|last2=Lemaire|first2=Luc|title=Selected topics in harmonic maps|series=CBMS Regional Conference Series in Mathematics|volume=50|publisher=American Mathematical Society|location=Providence, RI|year=1983|isbn=0-8218-0700-5|mr=0703510|doi=10.1090/cbms/050|zbl=0515.58011}}
'''Articles'''
*{{cite journal|first1=R.|last1=Caddeo|first2=S.|last2=Montaldo|first3=C.|last3=Oniciuc|title=Biharmonic submanifolds of {{math|''S''<sup>3</sup>}}|mr=1863283|journal=[[International Journal of Mathematics]]|volume=12|year=2001|issue=8|pages=867–876|doi=10.1142/S0129167X01001027|zbl=1111.53302}}
*{{cite journal|first1=R.|last1=Caddeo|first2=S.|last2=Montaldo|first3=C.|last3=Oniciuc|title=Biharmonic submanifolds in spheres|mr=1919374|journal=[[Israel Journal of Mathematics]]|volume=130|year=2002|pages=109–123|doi=10.1007/BF02764073|zbl=1038.58011}}
*{{cite journal|first1=Bang-Yen|last1=Chen|url=https://www.researchgate.net/publication/264782362_Some_Open_Problems_and_Conjectures_on_Submanifolds_of_Finite_Type|mr=1143504|title=Some open problems and conjectures on submanifolds of finite type|zbl=0749.53037|journal=Soochow Journal of Mathematics|volume=17|year=1991|issue=2|pages=169–188}}
*{{cite journal|first1=Bang-Yen|last1=Chen|url=https://www.researchgate.net/publication/264741832_A_report_on_submanifolds_of_finite_type|title=A report on submanifolds of finite type|zbl=0867.53001|mr=1391469|journal=Soochow Journal of Mathematics|volume=22|year=1996|issue=2|pages=117–337}}
*{{cite journal |first1=James Jr. |last1=Eells |first2=J. H. |last2=Sampson |year=1964 |title=Harmonic mappings of Riemannian manifolds |journal=[[American Journal of Mathematics]] |volume=86 |issue=1 |pages=109–160 |doi=10.2307/2373037 |mr=0164306 |zbl=0122.40102|author-link2=Joseph H. Sampson}}
*{{wikicite|ref={{sfnRef|Jiang|1986}}|reference={{cite journal|ref=none|first1=Guo Ying|last1=Jiang|title=2-harmonic maps and their first and second variational formulas|journal=Chinese Annals of Mathematics, Series A|language=Chinese|volume=7|year=1986|issue=4|pages=389–402|zbl=0628.58008|mr=0886529}}<br>{{cite journal|author-mask=1|first1=Guo Ying|last1=Jiang|title=2-harmonic maps and their first and second variational formulas|translator-first1=Hajime|translator-last1=Urakawa|journal=Note di Matematica|volume=28|year=2009|issue=1|pages=209–232|doi=10.1285/i15900932v28n1supplp209|doi-access=free|mr=2640582|zbl=1200.58015}}}}
*{{cite journal|first1=S.|last1=Montaldo|first2=C.|last2=Oniciuc|mr=2301373|url=https://inmabb.criba.edu.ar/revuma/pdf/v47n2/v47n2a02.pdf|title=A short survey on biharmonic maps between Riemannian manifolds|journal=Revista de la Unión Matemática Argentina|volume=47|year=2006|issue=2|pages=1–22|zbl=1140.58004}}
{{refend}}

[[Category:Riemannian geometry]]

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