The Liouville energy of the curvature equation #
Let M be a closed surface with a Riemannian metric g, volume dV and Gaussian curvature
K = Scal^g / 2. The metric e^(2u) g is hyperbolic exactly when u solves the curvature
equation Δ⁺u + K + e^(2u) = 0, where Δ⁺ = -div grad is the nonnegative Laplacian
(Bundle.ContMDiffRiemannianMetric.exists_hyperbolicMetric_metric_eq_rescale_iff). This equation is
the Euler–Lagrange equation of the Liouville energy
J(u) = ∫ (|∇u|² / 2 + K u + e^(2u) / 2) dV
(TauCeti.Manifold.liouvilleEnergy), and this file sets up the variational treatment of the
equation through J. The function K is arbitrary here; nothing below uses the dimension.
The first variation of J at u in the direction v is
∫ (⟪∇v, ∇u⟫ + K v + e^(2u) v) dV, which Green's identity turns into ∫ v (Δ⁺u + K + e^(2u)) dV
on a closed manifold. Hence a C² function is a critical point of J exactly when it solves the
equation. The first-variation and Euler–Lagrange arguments adapt the proof pattern of the
Euler–Lagrange equation of the Yamabe quotient,
TauCeti.Yamabe.forall_hasDerivAt_yamabeQuotient_add_smul_iff_exists_conformalLaplacian_eq:
differentiate under the integral sign, apply Green's identity, and conclude by the fundamental
lemma of the calculus of variations. The integrand of J is convex in (u, ∇u), so J lies
above its tangent lines at a C² function, and a solution is a global minimizer of J; conversely
a C² minimizer is a solution.
For existence by minimization, the decisive property is coercivity. Suppose M is connected and
∫ K dV < 0; when K is the Gaussian curvature of a closed surface, Gauss–Bonnet identifies
∫ K dV with 2π χ(M), so this says that M has negative Euler characteristic. Writing
u = ū + w with ū = ⨍ u the mean, Jensen's inequality bounds ∫ e^(2u) dV below by
Vol(M) e^(2ū), and the Poincaré–Wirtinger inequality bounds ∫ K w dV by the Dirichlet
energy. This gives a constant C with
(∫ |∇u|² dV) / 4 + (-∫ K dV) |ū| ≤ J(u) + C
for every C¹ function u. In particular J is bounded below, and a sequence on which J is
bounded has bounded Dirichlet energy and bounded means, hence is bounded in H¹.
Main results #
TauCeti.Manifold.liouvilleEnergy: the Liouville energyJ.TauCeti.Manifold.hasDerivAt_liouvilleEnergy_add_smul: the first variation ofJ, andTauCeti.Manifold.hasDerivAt_liouvilleEnergy_add_smul_eq_integral: its form∫ v (Δ⁺u + K + e^(2u)) dVon a closed manifold.TauCeti.Manifold.forall_hasDerivAt_liouvilleEnergy_add_smul_iff: the critical points ofJare the solutions ofΔ⁺u + K + e^(2u) = 0.TauCeti.Manifold.liouvilleEnergy_add_integral_mul_le: the convexity inequality forJ.TauCeti.Manifold.liouvilleEnergy_le_of_neg_laplaceBeltrami_add_add_exp_eq_zero: a solution minimizesJ, andTauCeti.Manifold.forall_liouvilleEnergy_le_iff: theC²minimizers ofJare exactly the solutions.TauCeti.Manifold.exists_le_liouvilleEnergy_add: the coercivity estimate, andTauCeti.Manifold.bddBelow_liouvilleEnergy_image:Jis bounded below.
References #
- M. S. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geom. 5 (1971), 325–332.
- J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14–47, §3.
- T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer (1998), Chapter 6.
The Liouville energy J(u) = ∫ (|∇u|² / 2 + K u + e^(2u) / 2) dV of a function u, for a
function K on a Riemannian manifold. Its Euler–Lagrange equation is Δ⁺u + K + e^(2u) = 0, with
Δ⁺ = -div grad; when K is the Gaussian curvature of a surface, this says that e^(2u) g has
curvature -1. Its classical meaning is for differentiable u.
Instances For
Unfolding lemma for TauCeti.Manifold.liouvilleEnergy.
On a compact manifold, the Liouville energy of a C¹ function u is
(∫ |∇u|² dV) / 2 + ∫ K u dV + (∫ e^(2u) dV) / 2, for an integrable K.
The first variation of the Liouville energy. On a compact manifold with a continuous metric,
let K be integrable and u, v be C¹. Then t ↦ J(u + t v) has derivative
∫ (⟪∇v, ∇u⟫ + K v + e^(2u) v) dV at t = 0.
Coercivity of the Liouville energy. On a compact connected manifold without boundary with a
continuous metric, let K be square integrable with ∫ K dV < 0. There is a constant C such that
every C¹ function u, with mean ū = ⨍ u, satisfies
(∫ |∇u|² dV) / 4 + (-∫ K dV) |ū| ≤ J(u) + C.
The Liouville energy is bounded below. On a compact connected manifold without boundary
with a continuous metric, if K is square integrable with ∫ K dV < 0, then the Liouville energy
is bounded below on C¹ functions.
The Euler–Lagrange equation on a closed manifold #
The first variation of the Liouville energy on a closed manifold. For integrable K, a
C² function u and a C¹ function v, t ↦ J(u + t v) has derivative
∫ v (Δ⁺u + K + e^(2u)) dV at t = 0, where Δ⁺ = -div grad.
The Euler–Lagrange equation of the Liouville energy. On a closed manifold, let K be
continuous. A C² function u is a critical point of the Liouville energy, in the sense that
t ↦ J(u + t v) has derivative 0 at t = 0 for every smooth v, exactly when it solves the
curvature equation Δ⁺u + K + e^(2u) = 0.
The convexity inequality for the Liouville energy. On a closed manifold, for integrable
K, a C² function u and a C¹ function v,
J(u) + ∫ v (Δ⁺u + K + e^(2u)) dV ≤ J(u + v).
Solutions of the curvature equation minimize the Liouville energy. On a closed manifold, if
a C² function u solves Δ⁺u + K + e^(2u) = 0, then J(u) ≤ J(w) for every C¹ function
w.
The minimizers of the Liouville energy. On a closed manifold, let K be continuous. A C²
function u minimizes the Liouville energy among C² functions exactly when it solves the
curvature equation Δ⁺u + K + e^(2u) = 0.