Sharp pinching theorems for complete submanifolds in the sphere

For every complete and minimally immersed submanifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:msup> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">→</m:mo> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mrow> </m:math> f Mⁿⁿ⁺ᵖ whose second fundamental form satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> A²≤ np/(2p-1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mn>4</m:mn> </m:msup> </m:math> S⁴ , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> Mⁿ . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>6</m:mn> </m:mrow> </m:math> n≤ 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.