The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that m = 4 and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is R^4. In the second part of the paper we carry out a stability study with respect to equivariant variations (equivariant stability). In particular, we prove that: (i) the inverse of the stereographic projection from the open 4-dimensional Euclidean ball to the hyperbolic space is equivariant stable (ii) the inverse of the stereographic projection from the closed 4-dimensional Euclidean ball to the sphere is equivariant stable with respect to variations which preserve the boundary data.

Rotationally symmetric biharmonic maps between models

MONTALDO, STEFANO;RATTO, ANDREA
2015

Abstract

The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that m = 4 and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is R^4. In the second part of the paper we carry out a stability study with respect to equivariant variations (equivariant stability). In particular, we prove that: (i) the inverse of the stereographic projection from the open 4-dimensional Euclidean ball to the hyperbolic space is equivariant stable (ii) the inverse of the stereographic projection from the closed 4-dimensional Euclidean ball to the sphere is equivariant stable with respect to variations which preserve the boundary data.
Biharmonic maps; Conformal diffeomorphisms; Equivariant theory; Hamiltonian; Second variation; Analysis; Applied mathematics
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11584/176242
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