Title | H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method |

Publication Type | Journal Article |

Year of Publication | 2004 |

Authors | Caldiroli, P, Musina, R |

Journal | Duke Math. J. 122 (2004), no. 3, 457--484 |

Abstract | Given a regular function $H\\\\colon\\\\mathbb{R}^{3}\\\\to\\\\mathbb{R}$, we look for $H$-bubbles, that is, regular surfaces in $\\\\mathbb{R}^{3}$ parametrized on the sphere $\\\\mathbb{S}+^{2}$ with mean curvature $H$ at every point. Here we study the case of $H(u)=H_{0}+\\\\varepsilon H_{1}(u)=:H_{\\\\varepsilon}(u)$, where $H_{0}$ is a nonzero constant, $\\\\varepsilon$ is the smallness parameter, and $H_{1}$ is any $C^{2}$-function. We prove that if $\\\\bar p\\\\in\\\\mathbb{R}^{3}$ is a ``good\\\'\\\' stationary point for the Melnikov-type function $\\\\Gamma(p)=-\\\\int_{|q-p|<|H_{0}|^{-1}}H_{1}(q)\\\\,dq$, then for $|\\\\varepsilon|$ small there exists an $H_{\\\\varepsilon}$-bubble $\\\\omega^{\\\\varepsilon}$ that converges to a sphere of radius $|H_{0}|^{-1}$ centered at $\\\\bar p$, as $\\\\varepsilon\\\\to 0$. |

URL | http://hdl.handle.net/1963/1607 |

DOI | 10.1215/S0012-7094-04-12232-8 |

## H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method

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