# Stability Analysis for Variational Problems for Surfaces with Constraint

## Abstract

Surfaces with constant mean curvature (CMC surfaces) are critical points of the area functional among surfaces enclosing the same volume. Therefore, they are a simple example of solutions of variational problem with constraint. A CMC surface is said to be stable if the second variation of the area is nonnegative for all volume-preserving variations satisfying the given boundary condition. The purpose of this article is to show some fundamental methods to study the stability for CMC surfaces. Especially, we give a criterion on the stability for compact CMC surfaces with prescribed boundary. Another concept that is closely related to the stability for CMC surfaces is the so-called bifurcation. We give sufficient conditions on a one-parameter family of CMC surfaces so that there exists a bifurcation. Moreover, we give a criterion for CMC surfaces in the bifurcation branch to be stable.

### Keywords

Bifurcation Constant mean curvature Pitchfork bifurcation Stability Symmetry breaking Variational problem## Notes

### Acknowledgments

The author is partially supported by Grant-in-Aid for Scientific Research (B) No. 25287012 of the Japan Society for the Promotion of Science, and the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development.

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