Summary: Let \(M^n\) \((n\geq 3)\) be an \(n\)-dimensional complete bypersurface in a real space form \(N(c)\) \((c\geq 0)\). We prove that if the sectional curvature \(K_M\) of \(M\) satisfies the following pinching condition: \(c+\delta< K_M\leq c+1\), where \(\delta=\tfrac 15\) for \(n\geq 4\) and \(\delta =\frac 14\) for \(n=3\), then there are no stable currents (or stable varifolds) in \(M\). This is a positive answer to the well-known conjecture of \textit{H. B. Lawson jun.} and \textit{J. Simons} [Ann. Math. (2) 98, 427--450 (1973; Zbl 0283.53049)].