## $$H$$-bubbles in a perturbative setting: the finite-dimensional reduction method.(English)Zbl 1079.53012

The authors prove the existence of embedded spheres in $${\mathbb R}^3$$ having prescribed mean curvature of the form $$H(u) = H_0 + \varepsilon H_1(u)$$ where $$H_0$$ is a non-zero constant, $$H_1:{\mathbb R}^3\to{\mathbb R}$$ is a $$C^2$$ function, and $$\varepsilon$$ is of sufficiently small absolute value. Solutions are obtained as critical points of an appropriate energy functional. The finite-dimensional reduction method mentioned in the title refers to the fact that a set of solutions of the unperturbed problem defines a finite dimensional manifold $$Z$$ of critical points for the unperturbed energy functional. The manifold $$Z$$ is sufficiently well understood so that the authors can construct a $$3$$-dimensional manifold that serves as a natural constraint for the perturbed energy functional.
The final section of the paper contains a non-existence result.

### MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49J10 Existence theories for free problems in two or more independent variables
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