Geodesics in the space of Kähler metrics

Abstract

Let (X, ω) be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set H 0 of Kähler forms cohomologous to ω has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H 0 can be connected by a smooth geodesic, and show that the answer, in general, is "no". The Monge-Ampère foliation.Let Y be an m + 1 dimensional complex manifold, ω a real (1, 1) form on it, of class C 1 , dω = 0. If rk ω ≡ m, the kernels of ω form an integrable subbundle of T Y , and so Y is foliated by Riemann surfaces, whose tangent spaces are the kernels of ω. The foliation is of class C 1 . If w is a locally defined potential of

The Mabuchi geometry of finite energy classes

“…Chen proved that d 2 (u 0 , u 1 ) = 0 if and only if u 0 = u 1 , thus (H, d 2 ) is a metric space [17]. Unfortunately, smooth geodesics don't run between the points of H [35], however Chen proved (with complements by Błocki [9]) that for u 0 , u 1 ∈ H there exists a curve…”

“…Unfortunately the above problem does not usually have smooth solutions (see [35,24]), but a unique solution in the sense of Bedford-Taylor does exist and with bounded Laplacian (see [17]) and this regularity is essentially optimal (see [26]). This…”

Weak geodesics in the space of Kähler metrics

“…144], once one checks that for ω-plurisubharmonic u ∈ C ∂∂ (S × X) the Monge-Ampère measure (ω + i∂∂u) m+1 , as defined, e.g., in [BT], agrees with what is obtained by taking the exterior power of the continuous form ω + i∂∂u. Alternatively, the more elementary arguments for [D,Lemma 6] and the first paragraph of the proof of [LV,Proposition 2.3] also give uniqueness, provided one first checks the following: if Z is a complex manifold and w ∈ C ∂∂ (Z) is real valued, then i∂∂w ≥ 0 at any local minimum point of w. Because of Proposition 2.1, it suffices to verify this latter when dim Z = 1, and then it is straightforward: if i∂∂w < 0 at a point, then i∂∂w < 0 in a neighborhood, whence w is strongly superharmonic there, and has no local minimum.…”

“…Conversely, if a g-invariant v ∈ H does not satisfy (3.3), then (1.1) will have no ω-plurisubharmonic solution u ∈ C ∂∂ (S × X). Such v certainly exist (and form an open set among g-invariant potentials in H), because the matrices (v z jzk (x 0 )) = (p jk ) and (v z j z k (x 0 )) = (q jk ) can be arbitrarily prescribed for g-invariant v ∈ H, as long as (ω jk (x 0 ) + p jk ) is positive definite; see [LV,Lemma 3.3].…”

“…Theorem 1.1 corresponds to [LV,Theorem 1.2], but the C 3 regularity from [LV] has been lowered. The proofs here and in [LV] are similar in that, denoting by x 0 ∈ X an isolated fixed point of g, in both proofs we analyze the behavior of a regular solution u in a neighborhood of S × {x 0 }.…”