Homology groups in CR-warped products of complex space forms

<h2>Abstract</h2> In the present paper, by using the result of Lawson-Simons [15], we adopt a different technique to show that there are no stable integral <i>q</i>-currents and a non-trivial homology group in a compact oriented CR-warped product submanifold <mml:math><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math> in a complex space forms space <mml:math><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mo>\(</mml:mo><mml:mn>4</mml:mn><mml:mi>c</mml:mi><mml:mo>\)</mml:mo></mml:math> by imposing some restrictions on the squared norm of gradient and the Laplacian of warped function. Finally, we show that the same approach can be given by using Dirichlet energy, positive eigenvalue, and Hamiltonian.