A remark on hyperbolic space form (Q2767474)

In this paper, the authors first give a simplified proof of their previous result in [\textit{Z. Chen} and \textit{C. Zhuo}, Chin. Ann. Math., Ser. B 20, 51-56 (1999; Zbl 0969.53014)], which is stated as follows: An \(n\)-dimensional complete Riemannian manifold \((M, g)\) is isometric to the hyperbolic space form \(H^n(-c)\) \((c >0)\) if and only if there is a non-trivial function \(\varphi\) and a constant \(c>0\) such that \( D^2 \varphi = c^2 \varphi g\) and \(\varphi\) attains its extremum on \(M\). NEWLINENEWLINENEWLINEBy a similar argument, the authors assert that the following statement is true. NEWLINENEWLINENEWLINETheorem: Let \((M, g)\) be a compact Riemannian manifold with smooth boundary. Suppose that there is a non-trivial solution \(\varphi\) to the following equation NEWLINE\[NEWLINE D^2\varphi = \varphi g \quad (\text{on } M), \qquad {\partial \varphi \over \partial n } + \lambda \varphi =0, \quad (\text{on } \partial M),NEWLINE\]NEWLINE where \(\lambda <-1\) is a constant. Then \((M, g)\) is isometric to the metric ball of radius \(r= {1\over 2} \ln { \lambda-1\over \lambda+1} \) in \(H^n(-1)\). When \(\lambda \geq -1\), the above equation has only a trivial solution. NEWLINENEWLINENEWLINEThere are several typing mistakes in this paper.