The Prescribed Mean Curvature Problem for Embedding Sphere–A New Existence Theorem of Yau Problem

The prescribed mean curvature problem is raised by Yau [16] as the following questions: is there an n-dimensional unit sphere X:Sⁿ(1) Rⁿ⁺¹ X : S n ( 1 ) ⟶ R n + 1 embedded in $$(n+1)$$ ( n + 1 ) -dimensional Euclidean space whose mean curvature is a prescribed function H defined on Rⁿ⁺¹ R n + 1 ? Stimulating by Bakelman, Kantor [1], Treibergs and Wei [14], the author established a new existence theorem for this classical problem by observing some new appropriate conditions which are natural and simple.