Mabuchi–Semmes Connection in Kähler Geometry

• Mabuchi–Semmes connection is a canonical infinite-dimensional Levi–Civita connection on the space of Kähler potentials that provides a formal Riemannian structure with nonpositive curvature.
• It induces geodesics characterized by a homogeneous complex Monge–Ampère equation, linking Kähler geometry with least action principles and the spectral analysis of Toeplitz operators.
• Its nonpositive curvature and convexity properties underpin uniqueness and stability in energy minimization and metric geometry, offering critical insights for complex differential geometry.

The Mabuchi–Semmes connection is a canonical infinite-dimensional Levi–Civita connection defined on the space of Kähler potentials of a compact Kähler manifold. Introduced independently by Mabuchi and Semmes, this connection arises naturally from a formal Riemannian metric—now called the Mabuchi metric—endowing the space of Kähler metrics with a formal structure of nonpositively curved symmetric space. Geodesics for this connection play a key role not only in Kähler geometry, complex Monge–Ampère theory, and canonical metric problems, but also in the spectral analysis of Toeplitz operators through their asymptotic behavior. The connection is fundamental for the geometric framework of energy functionals, least action principles, and the study of metric convexity in spaces of Kähler potentials, as established in recent research (Finski, 3 Feb 2025, Lempert, 2020).

1. The Space of Kähler Potentials and the Mabuchi Metric

Let $(X,\omega_0)$ be a compact Kähler manifold with fixed Kähler form $\omega_0$, often realized as $2\pi\,c_1(L,h_0^L)$ for a positive Hermitian metric $h_0^L$ on an ample line bundle $L\to X$. The space of Kähler potentials is

$\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$

an open convex subset of $C^\infty(X,\R)$ modulo constants. At $u \in \Hscr$, the tangent space is identified with $C^\infty(X,\R)/\R$. The Mabuchi metric, a formal $L^2$-type Riemannian metric, takes the form: $\omega_0$0 where $\omega_0$1 and $\omega_0$2. The space $\omega_0$3 is an infinite-dimensional weak Riemannian manifold of nonpositive sectional curvature.

2. The Mabuchi–Semmes Connection: Construction and Key Properties

Associated to $\omega_0$4 is a unique metric, torsion-free connection $\omega_0$5, the Mabuchi–Semmes connection, on $\omega_0$6. It admits two equivalent descriptions:

• Parallel Transport: Along a curve $\omega_0$7, parallel transport is induced by pull-back via time-dependent symplectomorphisms $\omega_0$8.
• Christoffel Formula: For vector fields $\omega_0$9 on $2\pi\,c_1(L,h_0^L)$0,

$2\pi\,c_1(L,h_0^L)$1

where the pointwise gradient and metric refer to $2\pi\,c_1(L,h_0^L)$2. This structure ensures compatibility with $2\pi\,c_1(L,h_0^L)$3 and vanishing torsion.

The sectional curvature of this connection is nonpositive, reflecting the underlying structure of a nonpositively curved symmetric space (Finski, 3 Feb 2025, Lempert, 2020).

3. Geodesics and the Complex Monge–Ampère Equation

A smooth path $2\pi\,c_1(L,h_0^L)$4 is a geodesic with respect to the Mabuchi–Semmes connection if

$2\pi\,c_1(L,h_0^L)$5

which locally becomes

$2\pi\,c_1(L,h_0^L)$6

where derivatives and gradients are taken with respect to $2\pi\,c_1(L,h_0^L)$7 and the metric $2\pi\,c_1(L,h_0^L)$8. Semmes and Calabi observed the equivalence between this real geodesic equation and the homogeneous complex Monge–Ampère equation: consider

$2\pi\,c_1(L,h_0^L)$9

then

$h_0^L$0

on $h_0^L$1, with Dirichlet boundary conditions corresponding to endpoints in $h_0^L$2. This links Kähler geometry directly to complex Monge–Ampère theory (Finski, 3 Feb 2025, Lempert, 2020).

4. The Principle of Least Action and Energy Minimization

For curves $h_0^L$3 in $h_0^L$4, define the kinetic energy (least-action) functional as

$h_0^L$5

and the action on a path as

$h_0^L$6

The Euler–Lagrange equation for this action is precisely the Mabuchi geodesic equation $h_0^L$7. Consequently, Mabuchi geodesics are stationary points of the action and minimize $h_0^L$8 among all sufficiently smooth paths with fixed endpoints, establishing the "principle of least action" in this infinite-dimensional context (Lempert, 2020).

5. Convexity of the Action and Metric Geometry

A central feature of the Mabuchi–Semmes geometry is the convexity of the least-action function with respect to geodesics. For endpoints $h_0^L$9 and the least action

$L\to X$0

there holds convexity along geodesics: $L\to X$1 is convex for any pair of Mabuchi geodesics $L\to X$2. This generalizes properties of nonpositive curvature and triangle inequalities in finite dimensions, and has significant implications for uniqueness, regularity, and stability within Kähler geometry. The nonpositive curvature of the metric is also reflected in the convexity of this geodesic distance function (Lempert, 2020).

6. Interaction with Toeplitz Operators and Spectral Asymptotics

The Mabuchi–Semmes connection has deep applications in the semiclassical spectral analysis of Toeplitz operators. For a nonnegative symbol $L\to X$3 with $L\to X$4, the logarithmic spectral distribution of the Toeplitz operator $L\to X$5 is governed asymptotically by the initial speed function of the geodesic connecting the reference metric $L\to X$6 to the Lebesgue envelope metric $L\to X$7: $L\to X$8 where the principal symbol $L\to X$9 collects the initial velocities along the Mabuchi geodesic. The smallest eigenvalue of $\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$0 decays exponentially: $\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$1 with $\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$2 determined by the geodesic envelope. More generally, for any continuous test function $\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$3, the weak convergence result

$\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$4

shows that the limiting spectral measure is pushed forward by $\Hscr = \{ u\in C^\infty(X,\R) : \omega_u := \omega_0 + \tfrac{i}{2\pi}\,\partial\bar\partial u > 0 \},$5, encoding precise geometric information in the spectrum (Finski, 3 Feb 2025). These links are explicit in settings such as projective space and classical Toeplitz matrices, where the Mabuchi–Semmes geometry emerges concretely in the spectral asymptotics.