Extrinsic Curvature |

**In General**
> s.a. curvature; Willmore Surface.

$ __Def__: Given an \((n-1)\)-dimensional
hypersurface Σ in an *n*-dimensional manifold *M*, its extrinsic
curvature or second fundamental form, is the rate of change of the unit normal \(n^a\) to Σ,

*K*_{ab}:=
*q*_{a}^{m}
∇_{m}*n*_{b}
≡ \({1\over2}{\cal L}\)_{n}
*q*_{ab} = \(1\over2\)*N*^{−1}
(\(\dot q\)_{ab} − \(\cal L\)_{N}
*q*_{ab}) ,

or simply ∇_{a} *n*_{b}
if *n*^{a} is the unit tangent
to the geodesics normal to Σ.

* __Meaning__: The tensor \(K_{ab}\)
has information on the the metric intrinsic to the surface, as well as on the curvature
due to the embedding of the surface; It is like the "acceleration" of a surface.

* __Applications__: Used as a dynamical
variable in the initial-value formulation of gravity theories, and in boundary terms for the action;
> s.a. Gauss-Codazzi Equations; action
and initial-value formulation of general relativity.

> __Online resources__:
see Wikipedia page.

**Trace of the Extrinsic Curvature**
> s.a. time in gravity [York time].

$ __Def__: Defined by *K*:=
*q*^{ab}*K*_{ab}.

* __Properties__: It satisfies

∫_{∂(M)}
*K* d*S* = (∂/∂*n*)
∫_{∂(M)} d*S* .

**Generalizations** > s.a. embeddings.

* __Higher codimension__:
For a submanifold *N* of codimension *k* > 1, it can be generalized to

*K*_{ab}^{c}:=
*K*_{ab}^{(i)}
*e*_{(i)}^{c} ,

where *e*_{(i)}^{c},
with *i* = 1, ..., *k*, are orthonormal vectors normal to *N*.

**Extremal Surface** > s.a. Bubbles;
foliations; Hypersurface.

$ __Def__: A hypersurface in a
manifold such that tr *K* = 0, with *K* the extrinsic curvature.

* __In R__^{3}:
Infinite minimal non-intersecting 2-surfaces, solutions to the Plateau problem (finding
the surface of least area that spans a given rigid boundary curve); The plane, catenoid,
and helicoid have been known for a long time; A new, countably infinite family has been
found by D Hoffman and W Meeks, with surfaces that are homeomorphic to a sphere with
three holes and one or more handles (computers were used for visualization).

* __Applications__: Soap bubbles.

* __Lorentzian manifolds__: A necessary
condition for a globally hyperbolic spacetime \(\mathbb R\) × Σ to admit
a maximal slice is that the Cauchy slice Σ admit a metric with non-negative
scalar curvature, *R* ≥ 0; Most Σ do not admit such metrics, for
topological reasons.

@ __General references__: Kanigel ThSc(93)may.

@ __Riemannian manifolds__: Pitts 81;
Colding & Minicozzi AM(04),
AM(04),
Meeks et al JDG(04) [3D, finite genus];
Collin et al JDG(04);
Meeks & Rosenberg JDG(04);
Cecil JCP(05) [numerical, arbitrary dimension];
Frohman & Meeks AM(08) [in \(\mathbb R\)^{3}, classification];
Tråsdahl & Rønquist JCP(11) [high-order numerical approximations];
Meeks & Pérez BAMS(11) [classical theory of minimal surfaces];
Mahadevan PRS(12) [minimal surfaces bounded by elastic lines].

@ __Lorentzian manifolds__: Cantor et al CMP(76);
Marsden & Tipler PRP(80);
Brill in(83);
Bartnik CMP(84),
in(84),
et al CMP(90);
Chruściel & Wald CMP(94) [stationary];
Burnett & Rendall CQG(96)gq/95 [spherically symmetric];
Aledo et al JGP(07)
[2+1, positive definite *K*_{ab}];
Witt a0908 [topological obstructions];
Anciaux 10.

@ __Maximal slicings of spacetimes__:
Cordero-Carrión et al JPCS(10)-a1003 [spherically symmetric, local existence].

@ __Spacetimes without extremal surfaces__:
Witt PRL(86) [vacuum].

**Constant-Mean-Curvature Surfaces**

@ __References__:
López 13
[Riemannian, surfaces of constant mean curvature with boundary];
López JGP(07) [in Minkowski spacetime];
Dilts & Holst a1710 [spacetimes, existence results].

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