scholarly journals A constant rank theorem for level sets of immersed hypersurfaces in ℝn+1with prescribed mean curvature

2010 ◽  
Vol 245 (2) ◽  
pp. 255-271 ◽  
Author(s):  
Changqing Hu ◽  
Xi-Nan Ma ◽  
Qianzhong Ou
Keyword(s):  
Mean Curvature ◽  
Level Sets ◽  
Constant Rank ◽  
Prescribed Mean Curvature ◽  
Rank Theorem ◽  
Constant Rank Theorem

PRESCRIBING MEAN CURVATURE ON 𝔹n

2010 ◽  
Vol 21 (09) ◽  
pp. 1157-1187 ◽  
Author(s):  
WAEL ABDELHEDI ◽  
HICHEM CHTIOUI
Keyword(s):  
Critical Points ◽  
Mean Curvature ◽  
Level Sets ◽  
Curvature Function ◽  
Number Of Solutions ◽  
Total Contribution ◽  
Prescribed Mean Curvature ◽  
The Difference ◽  
Zero Scalar Curvature ◽  
Lagrange Functional

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

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