Extrinsic Curvature

In General > s.a. curvature; Willmore Surface.
$ Def: Given an (n–1)-dimensional hypersurface 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 ,

K_ab:= q_a^m _mn_{b   n} q_ab = N^–1 (q^·_ab – _N q_ab) ,

or simply _a n_b if n^a is the unit tangent to the geodesics normal to .
* 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 variable in the initial-value formulation of gravity theories, and in boundary terms for the action; > s.a. action and initial-value formulation of general relativity.

Trace
$ Def: Defined by K:= q^abK_ab.
* Properties: It satisfies

_bdry(M) K dS = (/n) _bdry(M) dS .

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, 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 R³: Infinite minimal non-intersecting 2-surfaces; 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 R to admit a maximal slice is that the Cauchy slice admit a metric with non-negative scalar curvature, R 0; Most 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 R³, classification].
@ Lorentzian manifolds: Cantor et al CMP(76); Marsden & Tipler PRP(80); Brill in(83); Bartnik CMP(84), in(84), et al CMP(90); Chrusciel & Wald CMP(94) [stationary]; Burnett & Rendall CQG(96)gq/95 [spherically symmetric]; Aledo et al JGP(07) [2+1, positive definite K_ab]; López JGP(07) [constant mean curvature surfaces in Minkowski]; Witt a0908 [topological obstructions].
@ Spacetimes without extremal surfaces: Witt PRL(86) [vacuum].

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