The study of surfaces with constant mean curvature (CMC) is one
of the main topics in classical differential geometry. Moreover,
CMC surfaces are important mathematical models for the physics of
interfaces in the absence of gravity, where they separate two
different media or for capillary phenomena. Further, as most
techniques used in the theory of CMC surfaces not only involve
geometric methods but also PDE and complex analysis, the theory is
also of great interest for many other mathematical fields.
While minimal surfaces and CMC surfaces in general have already
been treated in the literature, the present work is the first to
present a comprehensive study of compact surfaces with boundaries,
narrowing its focus to a geometric view. Basic issues include the
discussion whether the symmetries of the curve inherit to the
surface; the possible values of the mean curvature, area and
volume; stability; the circular boundary case and the existence of
the Plateau problem in the non-parametric case. The exposition
provides an outlook on recent research but also a set of techniques
that allows the results to be expanded to other ambient spaces.
Throughout the text, numerous illustrations clarify the results and
their proofs.
The book is intended for graduate students and researchers in
the field of differential geometry and especially theory of
surfaces, including geometric analysis and geometric PDEs. It
guides readers up to the state-of-the-art of the theory and
introduces them to interesting open problems."
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