The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity

As a consequence of the Collar lemma, a geodesic transverse to such a long geodesic shrinks, leading to the development of a δ-thin region.

“As a consequence of the Collar lemma [19], a geodesic transverse to such long geodesic shrinks (again relative to hyperbolic length) leading to the development of a δ -thin region.”

The Dirichlet energy of the identity harmonic map is a proper function on Teichmüller space.

“Note that the conformal and diffeomorphism invariance of E [ id ; g, ρ ] allow it to be a function on the Teichmüller space of Σ p and in particular a proper function (i.e., the inverse images of the compact sets are compact)[19, 35].”

The transverse-traceless tensor kTT models the tangent space at γ.

“Now, M -1 /D 0 is precisely the Teichmüller space of Σ p and following [19], the transverse-traceless tensor k TT models the tangent space at γ .”

M_-1/D0 is precisely the Teichmüller space of Σp.

“Now, M -1 /D 0 is precisely the Teichmüller space of Σ p and following [19], the transverse-traceless tensor k TT models the tangent space at γ .”

Using the collar lemma, there is a non-trivial geodesic transverse to l_i^n with length approximately e^{-l_i^n}; the nontrivial geodesic γ2 collapses while the hyperbolic length l1 of γ1 approaches infinity.

“Using the collar lemma (see [19] for the detailed proof of the collar lemma), we immediately obtain that there is a non-trivial geodesic transverse to l n i with length ≈ lim n →∞ e -l n i . Note that the nontrivial (in π 1 (Σ p ) ) geodesic γ 2 collapses while the hyperbolic length l 1 of γ 1 approaches infinity.”