Proper submanifolds in product manifolds

Abstract

The authors obtain various versions of the Omori-Yau’s maximum principle on complete properly immersed submanifolds with controlled mean curvature in certain product manifolds, in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the generalized maximum principle, an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N ₁ in the product manifold N ₁ × N ₂ is given. Other applications of the generalized maximum principle are also given.

Keywords

Calabi-Chern problem / Omori-Yau maximum principle / Properly immersed submanifold / Mean curvature flow / Stochastic completeness

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