No-hair theorems for non-canonical self-gravitating static multiple scalar fields

We prove under certain assumptions no-hair theorems for non-canonical self-gravitating static multiple scalar fields in spherically symmetric spacetimes. It is shown that the only static, spherically symmetric and asymptotically flat black hole solutions consist of the Schwarzschild metric and a constant multi-scalar map. We also prove that there are no static, horizonless, asymptotically flat, spherically symmetric solutions with static scalar fields and a regular center. The last theorem shows that the static, asymptotically flat, spherically symmetric reflecting compact objects with Neumann boundary conditions can not support a non-trivial self-gravitating non-canonical (and canonical) multi-scalar map in their exterior spacetime regions. In order to prove the no-hair theorems we derive a new divergence identity.