Harmonic maps between Riemannian manifolds are critical points of the energy functional \(\int_M |d\phi|^2 dV\) for maps \(\phi: (M, g) \to (N, h)\), and biharmonic maps are critical points of the bienergy functional \(\int_M |\tau(\phi)|^2dV\), where \(\tau(\phi)\) is the torsion of \(\phi\).
In this paper, the author studies an action functional that interpolates between energy functional and bienergy functional defined by \[ E_{\delta_1, \delta_2}(\phi) = \delta_1 \int_M |d\phi|^2 dV + \delta_2 \int_M |\tau(\phi)|^2 dV\tag{1} \] with \(\delta_1, \delta_2 \in\mathbb{R}\). There have been several articles dealing with some particular aspect of (1) with restrictive \(\delta_1, \delta_2\) or under immersion of maps \(\phi\) [S. Cao, J. Geom. Phys. 61, No. 12, 2378–2383 (2011; Zbl 1226.53056); J. Eells and L. Lemaire, Bull. Lond. Math. Soc. 10, 1–68 (1978; Zbl 0401.58003); H. I. Eliasson, in: Global Analysis Appl., internat. Sem. Course Trieste 1972, Vol. II, 113–135 (1974; Zbl 0324.49034); T. Lamm, Calc. Var. Partial Differ. Equ. 22, No. 4, 421–445 (2005; Zbl 1070.58017); L. Lemaire, Lect. Notes Math. 838, 187–193 (1981; Zbl 0437.58005); E. Loubeau and S. Montaldo, Proc. Edinb. Math. Soc., II. Ser. 51, No. 2, 421–437 (2008; Zbl 1144.58010); Y. Luo, Differ. Geom. Appl. 35, 1–8 (2014; Zbl 1295.53065); S. Maeta, J. Geom. Phys. 62, No. 11, 2288–2293 (2012; Zbl 1252.53004)]. In this paper, the author wants to put the focus on arbitrary maps between Riemannian manifolds. The Euler-Lagrange equation for (1) is \[ \delta_2 \Delta \tau(\phi) = \delta_2 R^N(d\phi(e_\alpha, \tau(\phi))d\phi(e_\alpha) +\delta_1 \tau(\phi)\tag{2} \] and the critical points of (1) are called interpolating sesqui-harmonic maps. As in the case of biharmonic maps, it is obvious that harmonic maps are interpolating sesqui-harmonic maps solving (2).
First, the author derives the energy-momentum tensor associated to (1) as an explicit form and shows that it is divergence-free as harmonic maps and biharmonic maps. Then the author discusses a conservation law for solutions of the interpolating sesqui-harmonic map equation in the case that the target manifold admits Killing vector fields. The author also derives explicit solutions and investigates several properties in cases of flat and \(3\)-dimensional sphere.
Second, the author studies the qualitative behavior of interpolating sesqui-harmonic maps. Suppose that \((M, g)\) is a compact Riemannian manifold and let \(\phi: M \to N\) be a smooth solution of (2). The author proves that, if \(N\) has nonpositive sectional curvature and \(\delta_1, \delta_2\) have the same sign, then \(\phi\) is harmonic. When \(M\) is noncompact, the author shows that, if \(\phi: M \to N\) is a Riemannian immersion that solves (2) with \(|\tau(\phi)| = \mathrm{const}.\), and if \(N\) has nonpositive sectional curvature and and \(\delta_1, \delta_2\) have the same sign, then \(\phi\) must be harmonic. The author also proves a unique continuation theorem for interpolating sesqui-harmonic maps as for harmonic maps.
Related to vanishing of energy-momentum tensor, the author shows harmonicity and triviality of interpolating sesqui-harmonic maps. Let \(\phi : M\to N\) be a smooth map with vanishing energy-momentum tensor. Then the followings hold:

(i)

If \(\dim(M) = 2\), then \(\phi\) is harmonic.

(ii)

If either \(\dim(M) = 3\) and \(\delta_1\delta_2 <0\), or \(\dim(M) \ge 5\) and \(\delta_1\delta_2 >0\), then \(\phi\) is trivial.

(iii)

If \(\dim(M) = 4\), then \(\phi\) is trivial.

Finally, the author studies a Liouville-type property for interpolating sesqui-harmonic maps between complete Riemannian manifolds. Let \((M, g)\) be a complete noncompact Riemannian manifold and \((N, h)\) a manifold with nonpositive sectional curvature. The author proves that, if \(\phi: M\to N\) is a smooth solution of (2) with \(\delta_1 \delta_2 >0\) and \(\mathrm{vol}(M, g) = \infty\), and if \(\int_M |\tau(\phi)|^p dV <\infty\) for \(2 \le p< \infty\), then \(\phi\) must be constant.