The radial coordinate of a completion and Hamiltonians linear at infinity #
Let M be a completion of the Liouville domain W ⊆ M (TauCeti.IsLiouvilleCompletion), with
Liouville field Z and Liouville flow ψ. Every point x of M outside W is ψ_t(y) for a
unique point y of the boundary ∂W and a unique time t > 0
(TauCeti.IsLiouvilleCompletion.existsUnique_flowOf_eq). In the model W ∪ (∂W × [1, ∞)) of the
completion, where Z = r ∂_r on the cylindrical end, this point is (y, e^t): its radial
coordinate is r = e^t. A Hamiltonian on the completion is linear at infinity with slope
a when it equals a r + b outside a compact set. Symplectic homology is the direct limit of the
Floer homologies of such Hamiltonians as the slope tends to infinity.
Here the radial coordinate is the function TauCeti.liouvilleRadialCoordinate Z W, sending x
to e^τ, where τ ≥ 0 is the least time for which the backward Liouville flow carries x into
W. It is 1 exactly on W, and on the cylindrical end it is the coordinate r above:
r(ψ_t y) = e^tfory ∈ ∂Wandt ≥ 0(TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_flowOf), and more generallyr(ψ_t x) = e^t r(x)forxoutside the interior ofWandt ≥ 0, and even fort < 0as long aslog r(x) + t ≥ 0(TauCeti.IsLiouvilleCompletion.log_liouvilleRadialCoordinate_flowOf);- conversely
ψ_{-log r(x)}(x)is the boundary point on the trajectory through such anx(TauCeti.IsLiouvilleCompletion.flowOf_neg_log_liouvilleRadialCoordinate_mem_frontier).
Its sublevel sets are images of W under the flow: for T ≥ 0, r(x) ≤ e^T exactly when
ψ_{-T}(x) ∈ W (TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_le_exp_iff), and for
T > 0, r(x) < e^T exactly when ψ_{-T}(x) lies in the interior of W. Hence r is
continuous and its sublevel sets {r ≤ c} are compact, so that "outside a compact set" and
"where r is large" mean the same thing. For c ≥ 1 these truncations {r ≤ c} of the
completion are again Liouville domains, containing W in their interior when c > 1.
Smoothness of r on the cylindrical end is not treated here.
A function F : M → ℝ is linear at infinity with slope a (TauCeti.IsLinearAtInfinity) when
F = a r + b outside a compact set; on a completion this holds exactly when F = a r + b on some
region {r ≥ R} (TauCeti.IsLiouvilleCompletion.isLinearAtInfinity_iff), and the slope is then
determined by F as soon as W ≠ M (TauCeti.IsLinearAtInfinity.unique).
Main definitions #
TauCeti.liouvilleRadialCoordinate: the radial coordinaterof the cylindrical end, equal to1on the Liouville domain.TauCeti.IsLinearAtInfinity: functions equal toa r + boutside a compact set.
Main results #
TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_le_exp_iff,TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_lt_exp_iff: forT ≥ 0, resp.T > 0, the sublevel set{r ≤ e^T}, resp.{r < e^T}, is the preimage ofW, resp. of its interior, under the backward Liouville flowψ_{-T}.TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_flowOf_of_notMem_interior:r(ψ_t x) = e^t r(x)on the cylindrical end, fort ≥ 0.TauCeti.IsLiouvilleCompletion.log_liouvilleRadialCoordinate_flowOf:log r(ψ_t x) = log r(x) + ton the cylindrical end, also for timest < 0that stay on it.TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_flowOf_of_notMem: the same identity for allt, as long as bothxandψ_t xlie outside the Liouville domain.TauCeti.IsLiouvilleCompletion.flowOf_notMem_of_one_lt_exp_mul: the Liouville flowψ_tkeeps a pointxof the cylindrical end outside the Liouville domain whene^t r(x) > 1.TauCeti.IsLiouvilleCompletion.continuous_liouvilleRadialCoordinate,TauCeti.IsLiouvilleCompletion.isCompact_setOf_liouvilleRadialCoordinate_le:ris continuous with compact sublevel sets.TauCeti.IsLiouvilleCompletion.isLiouvilleDomain_setOf_liouvilleRadialCoordinate_le: the truncations{r ≤ c},c ≥ 1, of a completion are Liouville domains.TauCeti.IsLiouvilleCompletion.isLinearAtInfinity_iff: linearity at infinity in terms of the region whereris large.TauCeti.IsLinearAtInfinity.unique: the slope at infinity is unique.
References #
- P. Seidel, A biased view of symplectic cohomology, Current Developments in Mathematics 2006, Section 3 (completions and Hamiltonians linear at infinity).
- K. Cieliebak, Y. Eliashberg, From Stein to Weinstein and Back, AMS Colloquium Publications 59, 2012, Section 11.1.
- J.-F. Barraud, M. Damian, V. Humilière, A. Oancea, Floer homology with DG coefficients. Applications to cotangent bundles, arXiv:2404.07953.
The radial coordinate of the cylindrical end of a completion of the Liouville domain W
with Liouville field Z: it sends x to e^τ, where τ ≥ 0 is the least time for which the
backward flow of Z carries x into W. It equals 1 on W, and e^t at the point reached
from the boundary of W by flowing for time t ≥ 0
(TauCeti.IsLiouvilleCompletion.liouvilleRadialCoordinate_flowOf).
Instances For
The radial coordinate is e^τ for the least time τ ≥ 0 at which the backward flow carries
the point into W.
The radial coordinate is at least 1.
A function F on a completion of the Liouville domain W with Liouville field Z is
linear at infinity with slope a when F = a r + b outside a compact set, for a constant b,
where r is the radial coordinate TauCeti.liouvilleRadialCoordinate Z W.
Instances For
The defining property of a function linear at infinity.
Constant functions are linear at infinity with slope 0.
The radial coordinate is linear at infinity with slope 1.
The sum of functions linear at infinity is linear at infinity, and the slopes add.
A multiple of a function linear at infinity is linear at infinity, with the slope multiplied by the same constant.
The sublevel sets of the radial coordinate. For T ≥ 0, r(x) ≤ e^T exactly when the
backward Liouville flow ψ_{-T} carries x into the Liouville domain.
The radial coordinate equals 1 exactly on the Liouville domain.
The radial coordinate is greater than 1 exactly outside the Liouville domain.
The strict sublevel sets of the radial coordinate. For T > 0, r(x) < e^T exactly
when the backward Liouville flow ψ_{-T} carries x into the interior of the Liouville
domain.
The radial coordinate on a Liouville trajectory from the boundary. If y lies on the
boundary of the Liouville domain and t ≥ 0, then r(ψ_t y) = e^t.
The Liouville flow rescales the radial coordinate. If x lies outside the interior of the
Liouville domain and t ≥ 0, then r(ψ_t x) = e^t r(x).
The Liouville flow translates the logarithm of the radial coordinate. If x lies outside
the interior of the Liouville domain and 0 ≤ log r(x) + t, then log r(ψ_t x) = log r(x) + t.
Here t may be negative, as long as the trajectory through x does not enter the interior of the
domain before time t: log r is the time coordinate of the cylindrical end.
The Liouville flow multiplies the radial coordinate by e^t between two points of the
cylindrical end, also for negative times.
The Liouville flow ψ_t keeps a point x of the cylindrical end outside the Liouville domain
as long as e^t r(x) > 1, also for negative times.
The boundary point of a Liouville trajectory. If x lies outside the interior of the
Liouville domain, then flowing x backwards for time log r(x) lands on the boundary.
The radial coordinate of a completion is continuous.
The truncations of a completion are Liouville domains. For c ≥ 1, the sublevel set
{r ≤ c} of the radial coordinate of a completion of the Liouville domain W, which is the image
of W under the Liouville flow for time log c, is a Liouville domain for the same form and
primitive.
The sublevel sets {r ≤ c} of the radial coordinate of a completion are compact.
For c > 1, a Liouville domain lies in the interior of the truncation {r ≤ c} of its
completion.
Linearity at infinity on a completion. A function is linear at infinity with slope a
exactly when it equals a r + b on the region {r ≥ R} where the radial coordinate is large.
The slope at infinity is unique. On a completion of a Liouville domain W ≠ M, a function
linear at infinity has only one slope.