Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let ( M , g 0 ) be a smooth compact manifold of dimension n ≥ 3 with boundary. Given any smooth functions f in M and h on ∂ M , does there exist a conformal metric of g 0 such that its scalar curv...

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Bibliographic Details
Published in:Annals of global analysis and geometry 2018, Vol.53 (1), p.121-150
Main Authors: Chen, Xuezhang, Ho, Pak Tung, Sun, Liming
Format: Article
Language:English
Subjects:
Analysis
Curvature
Differential Geometry
Geometry
Global Analysis and Analysis on Manifolds
Invariants
Manifolds (mathematics)
Mathematical analysis
Mathematical Physics
Mathematics
Mathematics and Statistics
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Summary:Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let ( M , g 0 ) be a smooth compact manifold of dimension n ≥ 3 with boundary. Given any smooth functions f in M and h on ∂ M , does there exist a conformal metric of g 0 such that its scalar curvature equals f and boundary mean curvature equals h ? Assume that f and h are negative and the conformal invariant Q ( M , ∂ M ) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.
ISSN:0232-704X
1572-9060
DOI:10.1007/s10455-017-9570-4