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

Abstract The prescribed mean curvature problem is raised by Yau [16] as the following questions: is there an n-dimensional unit sphere $$X:{\mathbb {S}^{n}(1)}\longrightarrow {\mathbb {R}^{n+1}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>⟶</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> embedded in $$(n+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional Euclidean space whose mean curvature is a prescribed function H defined on $${\mathbb {R}^{n+1}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> ? 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.