Résume | Constant mean curvature (CMC) surfaces are critical points of the
area functional for volume-preserving variations. The index of a CMC surface is
a natural variational quantity which, in essence, is the cardinality of a maximal
set of independent volume-preserving variations which decrease surface area to
second degree.
There is a rich relationship between the index and the geometry/topology of
CMC surfaces. In this talk we will use blow-up arguments to explore this
relationship for closed CMC surfaces embedded in closed 3-manifolds. More
specifically, we will show that in a closed 3-manifold the index together with
the area of a CMC surface control its genus, and in a spherical 3-manifold the
index of a CMC surface controls its area and its genus. |