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.

Existence and multiplicity of entire radial spacelike graphs with prescribed mean curvature function in certain Friedmann–Lemaître–Robertson–Walker spacetimes

2017 ◽  
Vol 19 (02) ◽  
pp. 1650006 ◽  
Author(s):  
Cristian Bereanu ◽  
Daniel de la Fuente ◽  
Alfonso Romero ◽  
Pedro J. Torres
Keyword(s):  
Dirichlet Problem ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Euclidean Ball ◽  
Curvature Function ◽  
Prescribed Mean Curvature ◽  
Existence And Multiplicity ◽  
Homogeneous Dirichlet Problem ◽  
Whole Space

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann–Lemaître–Robertson–Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.

A Variational Approach for the Neumann Problem in Some FLRW Spacetimes

2019 ◽  
Vol 19 (2) ◽  
pp. 413-423
Author(s):  
Cristian Bereanu ◽  
Pedro J. Torres
Keyword(s):  
Neumann Problem ◽  
Mean Curvature ◽  
Hamiltonian Formulation ◽  
Main Tool ◽  
Curvature Function ◽  
Infinitely Many Solutions ◽  
Warping Function ◽  
Strongly Indefinite Functionals ◽  
Prescribed Mean Curvature ◽  
The Neumann Problem

AbstractIn this paper, we study, using critical point theory for strongly indefinite functionals, the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

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