3
$\begingroup$

Let $X: M^n \to N^{n+1}$ be a Riemannian immersion. Write $g, A, \nu, H$ for the first fundamental form, second fundamental form, Gauss map and mean curvature of $X$ respectively. Consider the normal pertubation $X_u: M \to N$ define by $X_u := X + u\nu$, where $u$ is smooth. For small enough $u$, we have that $X_u$ is an immersion.

In the special case where the dimension of the hypersurface is $n=2$ and the ambient manifold is Euclidean $N = \mathbb{R}^{n+1}$, Nicolaos Kapouleas gives a calculation of the quadratic and higher order terms mean curvature of $X_u$ in 'Complete Constant Mean Curvature Surfaces in Euclidean Three-Space', Appendix C.

Is there a reference or calculation of the mean curvature of $X_u$ (or its linear/quadratic terms) in the more general cases of when the dimension $n$ is not equal to $2$ and/or when the ambient manifold $N$ is not Euclidean?

Improve this question
$\endgroup$
1
  • 4
    $\begingroup$ The linearized perturbation is pretty standard, see Y. Choquet-Bruhat, Maximal submanifolds and submanifolds with constant mean extrinsic curvature of a Lorentzian manifold, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 3, 361–376 (even though the paper is about Lorentzian manifolds, the computation is the same for all pseudo Riemannian manifolds). The computation is also reproduced for higher codimension in the appendix to my paper arxiv.org/pdf/1404.0223v3.pdf $\endgroup$
    – Willie Wong
    Commented Jan 13, 2021 at 16:12

0

Reset to default

You must log in to answer this question.