9
$\begingroup$

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space has non-positive sectional curvature.

If $f$ is minimal, i.e. $$ \vec{H} = 0 $$ where $\vec{H}$ is the mean curvature vector of $f$ (with respect to the induced metric $f^*g$), then $f$ is not closed. (The only proof I know of this fact uses the maximum principle. A geometric proof would be nice, if anybody has a reference. Perhaps using Cartan-Hadamard?)

Clearly then, for $f$ closed with genus zero, an inequality of the form $$ \int_\Sigma |\vec{H}|^2d\mu > c_g > 0 $$ should hold, for some $c_g$ depending only on $(N,g)$.

My question is: what are the known values of $c_g$?

  • 0
    Incidentally, the argument below should also give you a proof that there are no genus zero closed minimal surfaces in a Hadamard manifold which does not use the maximum principle.2012-10-03

1 Answers 1