  We prove that every complete, minimally immersed submanifold $f\: M^n \to
\mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le
np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or
a Veronese surface in $\mathbb{S}^4$, thereby extending the well-known results
by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete
$M^n$. We also obtain the corresponding result for complete hypersurfaces with
nonvanishing constant mean curvature, due to Alencar & do Carmo in the compact
case, under the optimal bound on the umbilicity tensor. In dimension $n \le 6$,
a pinching theorem for complete higher-codimensional submanifolds with
non-vanishing parallel mean curvature is proved, partly generalizing previous
work of Santos. Our approach is inspired by the conformal method of
Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.
