On Interpolating SesquiHarmonic Maps Between Riemannian Manifolds
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Abstract
Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesquiharmonic maps. In this article we initiate a rigorous mathematical treatment of this functional and study various basic aspects of its critical points.
Keywords
Interpolating sesquiharmonic maps Harmonic maps Biharmonic maps Bosonic string with extrinsic curvature termMathematics Subject Classification
58E20 31B301 Introduction and Results
In the literature that studies analytical aspects of biharmonic maps, one refers to (1.2) as the energy functional for intrinsic biharmonic maps.
For a survey on biharmonic maps between Riemannian manifolds, we refer to [10] and [26].
This functional appears at several places in the physics literature. In string theory it is known as bosonic string with extrinsic curvature term, see [17, 28].
On the mathematical side there have been several articles dealing with some particular aspect of (1.3). Up to the best knowledge of the author the first place where the functional (1.3) was mentioned is [13, pp.134–135] with \(\delta _2=1\) and \(\delta _1>0\). In that reference it is already shown that if the domain has dimension 2 or 3 and the target \(N\) negative sectional curvature then the critical points of (1.3) reduce to harmonic maps. Later it was shown in [20, p.191] that no critical points exist if one does not impose the curvature condition on \(N\) and also assumes that \(\deg \phi =1\). Some analytic questions related to critical points of (1.3) have been discussed in [19] assuming \(\delta _1=2,\delta _2=1\). For the sake of completeness we want to mention that the functional (1.3) with \(\delta _1>0\) and \(\delta _2=\frac{1}{2}\) is also presented in the survey article “A report on harmonic maps”, see [11, p.28, Example (6.30)].
In [21] the authors initiate an extensive study of (1.3) assuming \(\delta _2=1\) and \(\delta _1\in \mathbb {R}\) under the condition that \(\phi \) is an immersion. They consider variations of (1.3) that are normal to the image \(\phi (M)\subset N\). In this setup they call critical points of (1.3) biminimal immersions. They also point out possible applications of their model to the theory of elasticity.
Up to now there exist several results on biminimal immersions, see for example [9] for biminimal hypersurfaces into spheres, [23] for biminimal submanifolds in manifolds of nonpositive curvature and [24] for biminimal submanifolds of Euclidean space. Instead of investigating maps that are immersions, we here want to put the focus on arbitrary maps between Riemannian manifolds.
As in the case of biharmonic maps, it is obvious that harmonic maps solve (1.4). For this reason we are mostly interested in solutions of (1.4) that are not harmonic maps. However, we can expect that as in the case of biharmonic maps there may be many situations in which solutions of (1.4) will be harmonic maps. In particular, we can expect that this is the case if \(N\) has negative sectional curvature and \(\delta _1\delta _2>0\). This question will be dealt with in Sect. 4. On the other hand, if \(\delta _1\) and \(\delta _2\) have opposite sign, we might expect a different behavior of solutions of (1.4) since in this case the two terms in the energy functional (1.3) are competing with each other and the energy functional can become unbounded from above and below.
This article is organized as follows: In Sect. 2 we study basic features of interpolating sesquiharmonic maps. Afterwards, in Sect. 3, we derive several explicit solutions of the interpolating sesquiharmonic map equation and in the last section we provide several results that characterize the qualitative behavior of interpolating sesquiharmonic maps.
Throughout this paper we will make use of the following conventions. Whenever choosing local coordinates we will use Greek letters to denote indices on the domain \(M\) and Latin letters for indices on the target \(N\). We will choose the following convention for the curvature tensor \(R(X,Y)Z:=[\nabla _X,\nabla _Y]Z\nabla _{[X,Y]}Z\) such that the sectional curvature is given by \(K(X,Y)=R(X,Y,Y,X)\). For the Laplacian acting on functions \(f\in C^\infty (M)\) we choose the convention \(\Delta f={\text {div}}\,{\text {grad}}\,f\), and for sections in the vector bundle \(\phi ^{*}TN\) we make the choice \(\Delta ^{\phi ^*TN}={\text {Tr}}(\nabla ^{\phi ^*TN}\nabla ^{\phi ^*TN})\). Note that the connection Laplacian on \(\phi ^*TN\) is defined by \(\Delta :=\nabla _{e_\alpha }\nabla _{e_\alpha }\nabla _{\nabla _{e_\alpha }e_\alpha }\).
2 Interpolating SesquiHarmonic Maps
In this section we analyze the basic features of the action functional (1.3) and start by calculating its critical points.
Proposition 2.1
Proof
Solutions of (1.4) will be called interpolating sesquiharmonic maps.^{1}
Remark 2.2
Choosing \(\delta _1=2,\delta _2=1\) and \(M=S^4,N=S^k\) solutions of (1.4) were called quasibiharmonic maps in [30]. These arise when considering a sequence of weakly intrinsic biharmonic maps in dimension four. When taking the limit, one finds that quasibiharmonic spheres separate at finitely many points as in many conformally invariant variational problems.
Remark 2.3
Remark 2.4
In order to highlight the analytical structure of (1.4) we take a look at the case of a spherical target. For biharmonic maps this was carried out in [18] making use of a different method.
Proposition 2.5
Proof
By varying (1.3) with respect to the domain metric, we obtain the energymomentum tensor. Since the energymomentum tensor for both harmonic and biharmonic maps is well known in the literature, we can directly give the desired result.
Proposition 2.6
Proof
It can be directly seen that the energymomentum tensor (2.2) is symmetric. For the sake of completeness we prove the following:
Proposition 2.7
The energymomentum tensor (2.2) is divergence free.
Proof
2.1 Conservation Laws for Targets with Symmetries
In this subsection we discuss how to obtain a conservation law for solutions of the interpolating sesquiharmonic map equation in the case that the target manifold has a certain amount of symmetry, more precisely, if it possesses Killing vector fields. A similar discussion has been performed in [4].
Definition 2.8
Note that if \(X\) generates an isometry then \(\mathcal {L}_Xg=0\) such that we have to require the existence of Killing vector fields on the target.
In the following we will make use of the following facts:
Lemma 2.9
Lemma 2.10
Proof
Proposition 2.11
Proof
Remark 2.12
In the physics literature the vector field (2.5) is usually called Noether current.
3 Explicit Solutions of the Interpolating SesquiHarmonic Map Equation
In this section we want to derive several explicit solutions to the Euler–Lagrange equation (1.4). We can confirm that solutions may have a different behavior than biharmonic or harmonic maps.
Let us start in the most simple setup possible.
Example 3.1
Example 3.2
 (1)
\(\delta _1=0\), that is \(f\) is biharmonic. In this case \(\alpha ^2+\beta ^2=0\) and we have to choose \(\alpha ,\beta \in {\mathbb {C}}\).
 (2)
\(\delta _2=0\), that is \(f\) is harmonic. In this case there are no restrictions on \(\alpha ,\beta \).
 (3)
If \(\delta _1\delta _2>0\) we have to impose the condition \(\alpha ^2+\beta ^2>0\) meaning that \(\alpha ,\beta \in \mathbb {R}\).
 (4)
If \(\delta _1\delta _2<0\) we find that \(\alpha ^2+\beta ^2<0\) meaning that \(\alpha ,\beta \in {\mathbb {C}}\).
3.1 Interpolating SesquiHarmonic Functions in Flat Space
In this section we study interpolating sesquiharmonic functions in flat space.
We cannot expect to find a unique solution to (3.1) since we can always add a harmonic function once we have constructed a solution to (3.1). Since we are considering \(\mathbb {R}^n\) instead of a curved manifold at the moment, all curvature terms in (1.4) vanish and we are dealing with a linear problem.
 Suppose that \(n=3\), then the solution of (3.2) is given byAs in the onedimensional case both limits \(\delta _1\rightarrow 0\) and \(\delta _2\rightarrow 0\) do not exist.$$\begin{aligned} f(r)=c_1\frac{e^{\sqrt{\frac{\delta _1}{\delta _2}}r}}{r}+c_2 \frac{e^{\sqrt{\frac{\delta _1}{\delta _2}}r}}{\sqrt{\delta _1/\delta _2}r}\frac{\delta _2}{\delta _1}\frac{1}{r}. \end{aligned}$$
 Suppose that \(n=4\) and that \(\frac{\delta _1}{\delta _2}>0\). Then the solution of (3.2) is given byIf \(\frac{\delta _1}{\delta _2}<0\), then we obtain the solution by an analytic continuation.$$\begin{aligned} f(r)=&\,c_1\frac{J_1\left( \sqrt{\frac{\delta _1}{\delta _2}}r\right) }{r} +c_2\frac{Y_1\left( \sqrt{\frac{\delta _1}{\delta _2}}r\right) }{r} \\&+\frac{\pi }{2\sqrt{\delta _1/\delta _2}r}\left( J_1\left( \sqrt{\frac{\delta _1}{\delta _2}}r\right) Y_0\left( \sqrt{\frac{\delta _1}{\delta _2}}r\right) J_0\left( \sqrt{\frac{\delta _1}{\delta _2}}r\right) Y_1\left( \sqrt{\frac{\delta _1}{\delta _2}}r\right) \right) . \end{aligned}$$

The qualitative behavior of solutions to (3.2) for \(n\ge 5\) seems to be the same as above.
3.2 Interpolating SesquiHarmonic Curves on the ThreeDimensional Sphere
In this subsection we study interpolating sesquiharmonic curves on threedimensional spheres with the round metric, where we follow the ideas from [7].
To this end let \((N,g)\) be a threedimensional Riemannian manifold with constant sectional curvature \(K\). Moreover, let \(\gamma :I\rightarrow N\) be a smooth curve that is parametrized by arc length. Let \(\{T,N,B\}\) be an orthonormal frame field of \(TN\) along the curve \(\gamma \). Here, \(T=\gamma '\) is the unit tangent vector of \(\gamma \), \(N\) the unit normal field, and \(B\) is chosen such that \(\{T,N,B\}\) forms a positive oriented basis.
Lemma 3.3
Proof
Corollary 3.4
We directly obtain the following characterization of interpolating sesquiharmonic curves:
Proposition 3.5
 (1)
Let \(\gamma :I\rightarrow N\) be a curve in a threedimensional Riemannian manifold. If \(K\le \frac{\delta _1}{\delta _2}\), then any interpolating sesquiharmonic curve is a geodesic.
 (2)
To obtain a nongeodesic interpolating sesquiharmonic curve \(\gamma :I\rightarrow S^3\), we have to demand that \(\delta _2>\delta _1\).
Proposition 3.6
Proof
Proposition 3.7
Proof
Remark 3.8
Note that it is required in (3.6) that \(1\delta _1+\delta _2>1\), which is equivalent to \(\delta _2>\delta _1\). This is consistent with the assumption \(k_g^2=\delta _2\delta _1\).
Theorem 3.9
Proof
Remark 3.10
Remark 3.11
 (1)
If \(k_g^2=\delta _2\delta _1\), then \(\gamma \) is a circle of radius \(\frac{1}{\sqrt{1+k_g^2}}\).
 (2)If \(\delta _2\delta _1>k_g^2>0\), then \(\gamma \) is a geodesic of the rescaled Clifford torus$$\begin{aligned}&S^1\left( \frac{\sqrt{(1+\delta _1\delta _2+\sqrt{(1+\delta _1\delta _2)^2 +4k_g^2})}}{\sqrt{2}((1+\delta _1\delta _2)^2+4k_g^2)^\frac{1}{4}}\right) \\&\quad \times S^1\left( \frac{\sqrt{1\delta _1+\delta _2+\sqrt{(1+\delta _1\delta _2)^2 +4k_g^2}}}{\sqrt{2}((1+\delta _1\delta _2)^2+4k_g^2)^\frac{1}{4}}\right) . \end{aligned}$$
We want to close this subsection by mentioning that it is possible to generalize the results obtained from above to higherdimensional spheres as was done for biharmonic curves in [8].
4 The Qualitative Behavior of Solutions
In this section we study the qualitative behavior of interpolating sesquiharmonic maps.
Proposition 4.1
Proof
As already stated in the introduction, it is obvious that harmonic maps solve (1.4). We will give several conditions under which interpolating sesquiharmonic maps must be harmonic generalizing several results from [19, 27]. To achieve these results we will frequently make use of the following Bochner formula:
Lemma 4.2
Proof
This follows by a direct calculation. \(\square \)
Proposition 4.3
 (1)
If \(N\) has nonpositive curvature \(K^N\le 0\) and \(\delta _1,\delta _2\) have the same sign, then \(\phi \) is harmonic.
 (2)
If \(\mathrm{{d}}\phi ^2\le \frac{\delta _1}{R^N_{L^\infty }\delta _2}\) and \(\delta _1,\delta _2\) have the same sign, then \(\phi \) is harmonic.
Proof
If we do not require \(M\) to be compact, we can give the following result.
Proposition 4.4
Let \(\phi :M\rightarrow N\) be a Riemannian immersion that solves (1.4) with \(\tau (\phi )=const\). If \(N\) has nonpositive curvature \(K^N\le 0\) and \(\delta _1,\delta _2\) have the same sign, then \(\phi \) must be harmonic.
Proof
In the case that \(\dim M=\dim N1\) the assumption of \(N\) having negative sectional curvature can be replaced by demanding negative Ricci curvature.
Theorem 4.5
Let \(\phi :M\rightarrow N\) be a Riemannian immersion. Suppose that \(M\) is compact and \(\dim M=\dim N1\). If \(N\) has nonpositive Ricci curvature and \(\delta _1,\delta _2\) have the same sign, then \(\phi \) is interpolating sesquiharmonic if and only if it is harmonic.
Proof
As for harmonic maps ([29, Theorem 2]) we can prove a unique continuation theorem for interpolating sesquiharmonic maps. To obtain this result we recall the following ([1, p.248]):
Theorem 4.6
Making use of this result we can prove the following:
Proposition 4.7
Let \(\phi \in C^4(M,N)\) be an interpolating sesquiharmonic map. If \(\phi \) is harmonic on a connected open set \(W\) of \(M\), then it is harmonic on the whole connected component of \(M\) which contains \(W\).
Proof
4.1 Interpolating SesquiHarmonic Maps with Vanishing EnergyMomentum Tensor
In this section we study the qualitative behavior of solutions to (1.4) under the additional assumption that the energymomentum tensor (2.2) vanishes similar to [22]. Such an assumption is partially motivated from physics: In physics one usually also varies the action functional (1.3) with respect to the metric on the domain and the resulting Euler–Lagrange equation yields the vanishing of the energymomentum tensor.
Proposition 4.8
Let \(\gamma :S^1\rightarrow N\) be a curve with vanishing energymomentum tensor. If \(\delta _1\delta _2>0\), then \(\gamma \) maps to a point.
Proof
Proposition 4.9
Suppose that \((M,h)\) is a Riemannian surface. Let \(\phi :M\rightarrow N\) be a smooth map with vanishing energymomentum tensor. Then \(\phi \) is harmonic.
Proof
Since \(\dim M=2\), we obtain from (4.4) that \(\tau (\phi )^2=0\) yielding the claim. \(\square \)
For a higherdimensional domain we have the following result.
Proposition 4.10
 (1)
If \(\dim M=3\) and \(\delta _1\delta _2<0\), then \(\phi \) is trivial.
 (2)
If \(\dim M=4\), then \(\phi \) is trivial.
 (3)
If \(\dim M\ge 5\) and \(\delta _1\delta _2>0\), then \(\phi \) is trivial.
Proof
As a next step we rewrite the condition on the vanishing of the energymomentum tensor.
Proposition 4.11
Proof
This allows us to give the following:
Proposition 4.12
Let \(\phi :M\rightarrow N\) be a smooth map with vanishing energymomentum tensor. Suppose that \(\dim M>2\) and \({\text {rank}}\,\phi \le n1\). Then \(\phi \) is harmonic.
Proof
Fix a point \(p\in M\). By assumption \({\text {rank}}\,\phi \le n1\) and hence there exists a vector \(X_p\in \ker \mathrm{{d}}\phi _p\). For \(X=Y=X_p\) we can infer from (4.5) that \(\tau (\phi )=0\) yielding the claim. \(\square \)
If the domain manifold \(M\) is noncompact we can give the following variant of the previous results.
Proposition 4.13
Let \(\phi :M\rightarrow N\) be a smooth Riemannian immersion with vanishing energymomentum tensor. If \(\dim M=2\), then \(\phi \) is harmonic, and if \(\dim M=4\), then \(\phi \) is trivial.
Proof
4.2 Conformal Construction of Interpolating SesquiHarmonic Maps
In [3] the authors present a powerful construction method for biharmonic maps. Instead of trying to directly solve the fourthorder equation for biharmonic maps they assume the existence of a harmonic map and then perform a conformal transformation of the metric on the domain to render this map biharmonic. In particular, they call a metric that renders the identity map biharmonic, a biharmonic metric. In this section we will discuss if the same approach can also be used to construct interpolating sesquiharmonic maps.
Proposition 4.14
Proof
In the following we will call a metric that renders the identity map interpolating sesquiharmonic an interpolating sesquiharmonic metric.
Corollary 4.15
We now rewrite this as an equation for \(\nabla u\).
Proposition 4.16
Proof
Proposition 4.17
Let \((M,h)\) be a compact manifold of strictly negative Ricci curvature with \(\dim M>2\) and assume that \(\delta _1\delta _2>0\). Then there does not exist an interpolating sesquiharmonic metric that is conformally related to \(h\) except a constant multiple of \(h\).
Proof
Remark 4.18
In contrast to the case of biharmonic maps (4.6) contains also a term involving \(u\) on the righthand side. This reflects the fact that both harmonic and biharmonic maps on its own have a nice behavior under conformal deformations of the domain metric, whereas interpolating sesquiharmonic maps do not. This prevents us from making a connection between interpolating sesquiharmonic metrics and isoparametric functions as was done in [3] for biharmonic maps.
4.3 A LiouvilleType Theorem for Interpolating SesquiHarmonic Maps Between Complete Manifolds
In this section we will prove a Liouvilletype theorem for solutions of (1.4) between complete Riemannian manifolds generalizing a similar result for biharmonic maps from [25]. For more Liouvilletype theorems for biharmonic maps see [5, 6] and references therein.
To this end we will make use of the following result due to Gaffney [14]:
Theorem 4.19
Theorem 4.20
 (1)If \(\delta _1\delta _2>0\) andthen \(\phi \) must be harmonic.$$\begin{aligned} \int _M\tau (\phi )^p{\mathrm{{d}}V}<\infty ,\qquad \int _M\mathrm{{d}}\phi ^2{\mathrm{{d}}V}<\infty , \end{aligned}$$
 (2)If \(\delta _1\delta _2>0\), \({{\text {vol}}}(M,h)=\infty \) andthen \(\phi \) must be harmonic.$$\begin{aligned} \int _M\tau (\phi )^p{\mathrm{{d}}V}<\infty , \end{aligned}$$
Proof
Footnotes
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The author gratefully acknowledges the support of the Austrian Science Fund (FWF) through the project P 30749N35 “Geometric variational problems from string theory.”
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