Spectral geometry of harmonic maps into warped product manifolds. II (Q5957225)

Let \((M,g)\) and \((N,h)\) be Riemannian manifolds; assume \((M,g)\) is closed. Let \(\varphi\) be a smooth map from \(M\) to \(N\). The Jacobi operator \(J_\varphi\) is an operator of Laplace type on \(C^\infty(\varphi^*TN)\) over \(M\) which appears in the second variational formula for the energy functional \(E\) associated to the map \(\varphi\); NEWLINE\[NEWLINEJ_\varphi=- \text{Tr}(\widetilde\nabla^2)-\sum R_N(\varphi_*e_i,\cdot)\varphi_*(e_i)NEWLINE\]NEWLINE where \(\widetilde\nabla\) is the induced connection on the pull-back bundle \(\varphi^*TN\) and \(\{e_i\}\) is a local orthonormal frame field on \(TM\). As \(t\downarrow 0\), there is a complete asymptotic expansion of the heat trace NEWLINE\[NEWLINE\text{Tr}_{L^2}(e^{-tJ_\varphi})\sim\sum_{n\geq 0}a_n(J_\varphi)t^{(n-m)/2}NEWLINE\]NEWLINE where \(m=\dim(M)\). The heat trace coefficients \(a_n\) are locally computable invariants. NEWLINENEWLINENEWLINEThe author assumes the target manifold \(N=N_{m_1}(c_1)\times_fN_{m_2}(c_2)\) is a warped product of two space forms of constant sectional curvatures \(c_i\). The author determines the heat trace coefficients \(a_n(J_\varphi)\) for \(n=0,2,4\) in this setting, expressing them in terms of geometrical quantities and the warping function \(f\) for quite general \(\varphi\) and then specializes them to the case that \(\varphi\) is harmonic. Several corollaries are obtained, including the following: \smallbreak\noindent Corollary: Let \(\varphi,\psi:(M,g)\rightarrow N_{m_1}(c_1)\times N_{m_2}(c_2)\) be two isometric minimal immersions. Suppose that \(c_1\neq 0\) or that \(c_2\neq 0\) and that either \(m_1\) or \(m_2\) is greater than \(1\). If \(J_\varphi\) and \(J_\psi\) are isospectral, then \(\varphi\) and \(\psi\) have the same total energy. NEWLINENEWLINENEWLINE[For part I of this paper see the author, Far East J. Math. Sci. (FJMS) 2, No. 2, 295-312 (2000; Zbl 0953.58030)].