Duke Mathematical Journal

H-bubbles in a perturbative setting: The finite-dimensional reduction method

Paolo Caldiroli and Roberta Musina

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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 $\varepsilon\to 0$.

Article information

Source
Duke Math. J., Volume 122, Number 3 (2004), 457-484.

Dates
First available in Project Euclid: 22 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082665285

Digital Object Identifier
doi:10.1215/S0012-7094-04-12232-8

Mathematical Reviews number (MathSciNet)
MR2057016

Zentralblatt MATH identifier
1079.53012

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 49J10: Free problems in two or more independent variables

Citation

Caldiroli, Paolo; Musina, Roberta. H -bubbles in a perturbative setting: The finite-dimensional reduction method. Duke Math. J. 122 (2004), no. 3, 457--484. doi:10.1215/S0012-7094-04-12232-8. https://projecteuclid.org/euclid.dmj/1082665285


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