Extrinsic Curvature| 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:= qam ∇mnb ≡ \({1\over2}{\cal L}\)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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 29 oct 2019