Applications of certain maximum principles to gradient k-Yamabe solitons

In this short note, we deal with gradient k-Yamabe solitons whose Ricci curvature is non-positive in the direction of the gradient of its potential function. Firstly, assuming that the potential function has gradient converging to zero at infinity, we show that the gradient k-Yamabe soliton is trivial. Afterwards, supposing the boundedness of the gradient and Hessian of the potential function, if such a gradient k-Yamabe soliton has polynomial volume growth, we prove that it must be also trivial. Finally, for a stochastically complete gradient k-Yamabe soliton having potential function with bounded gradient, we also verify its triviality.