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,

Kab:= qammnb ≡ \({1\over2}{\cal L}\)n qab = \(1\over2\)N−1 (\(\dot q\)ab − \(\cal L\)N qab) ,

or simply ∇a nb if na 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:= qabKab.
* Properties: It satisfies

∂(M) K dS = (∂/∂n) ∂(M) dS .

Generalizations > s.a. embeddings.
* Higher codimension: For a submanifold N of codimension k > 1, it can be generalized to

Kabc:= Kab(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 R3: 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 Kab]; Witt a0908 [topological obstructions]; Anciaux 10; > s.a. schwarzschild-de sitter spacetime.
@ 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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