The maximal principle for properly immersed submanifolds and its applications

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In this note we consider the Liouville type theorem for a properly immersed submanifold M in a complete Riemmanian manifold N. Assume that the sectional curvature Kⁿ of N satisfies Kⁿ≥-L(1+distₙ(·,q₀)²)α2 for some L>0, 2>α≥ 0 and q₀∈ N. (i) If Δ|H|²ᵖ⁻²≥ k|H|²ᵖ(p>1) for some constant k>0, then we prove that M is minimal. (ii) Let u be a smooth nonnegative function on M satisfying Δu≥ kuᵃ for some constant k>0 and a>1. If |H|≤ C(1+distₙ(·,q₀)²)β2 for some C>0, 0≤β<1, then u=0 on M. As applications we get some nonexistence result for p-biharmonic submanifolds.