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15 April 2004 H-bubbles in a perturbative setting: The finite-dimensional reduction method
Paolo Caldiroli, Roberta Musina
Duke Math. J. 122(3): 457-484 (15 April 2004). DOI: 10.1215/S0012-7094-04-12232-8

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 $\varepsilon\to 0$.

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Paolo Caldiroli. Roberta Musina. "H-bubbles in a perturbative setting: The finite-dimensional reduction method." Duke Math. J. 122 (3) 457 - 484, 15 April 2004. https://doi.org/10.1215/S0012-7094-04-12232-8

Information

Published: 15 April 2004
First available in Project Euclid: 22 April 2004

zbMATH: 1079.53012
MathSciNet: MR2057016
Digital Object Identifier: 10.1215/S0012-7094-04-12232-8

Subjects:
Primary: 53A10
Secondary: 49J10

Rights: Copyright © 2004 Duke University Press

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Vol.122 • No. 3 • 15 April 2004
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