Scalar Curvature Map in Geometric Analysis

• Scalar Curvature Map is a tool defining how scalar curvature varies over Riemannian and Kähler geometries, providing a basis for local surjectivity results.
• It plays a central role in rigidity phenomena and moment map formulations, connecting geometric analysis with symplectic and quantum deformation theories.
• Applications span explicit constructions in Hopf surfaces and compact Lie groups, offering insights into curvature control, topological constraints, and deformation quantization.

The scalar curvature map is a fundamental object in differential geometry, capturing how the scalar curvature functional varies over spaces of Riemannian or Kähler metrics, almost complex structures, or more general geometric structures. It plays a central role in rigidity phenomena, deformation theory, geometric analysis, and moment map frameworks relating geometric analysis to symplectic geometry and quantum deformation theory.

1. Definition and Linearization of the Scalar Curvature Map

Let $M$ be a smooth $n$-dimensional manifold, possibly with boundary $\partial M$. The scalar curvature map

$F: \Met(M) \to C^\infty(M) \times C^\infty(\partial M), \quad F(g) = (R(g), H(g))$

assigns to a metric $g$ its scalar curvature $R(g)$ in the interior and the mean curvature $H(g)$ on the boundary with respect to the outward unit normal. The natural setting for $F$ is the space of metrics of class at least $C^4$ to ensure regularity of the curvature invariants (Sheng, 2024).

The linearization of $F$ at a background metric $g_0$ is given by

$D F|_{g_0}(h) = (L_{g_0}(h),\, B_{g_0}(h)),$

where, in local coordinates,

$L(h) = -\Delta(\tr h) + \div\div h - \langle h, \Ric \rangle,$

and at the boundary $\partial M$, setting $v$ the outward unit normal and $H_0 = H(g_0)$,

$B(h) = v(\tr h) - \div_{\partial M}(h(v,\cdot)) - h(v,v)H_0.$

The adjoint is

$D F|_{g_0}^*(u) = \big(-(\Delta u) g_0 + \Hess(u) - u\, \Ric_{g_0},\, u v^\flat - u H_0 \big|_{\partial M}\big).$

A crucial property is the generic surjectivity result: if $\ker D F|_{g_0}^* = \{0\}$, then local surjectivity holds for $F$ near $g_0$ onto prescribed pairs $(S, H')$ supported in compact subdomains (Sheng, 2024).

2. The Scalar Curvature Map in Kähler and Generalized Kähler Geometry

In Kähler geometry, the scalar curvature map emerges in an infinite-dimensional Hamiltonian framework. The space of $\omega$-compatible almost complex structures,

$\J(M,\omega) = \{ J \mid J^2 = -\mathrm{Id},\, \omega(J\cdot, J\cdot) = \omega(\cdot, \cdot),\, \omega(\cdot, J\cdot) > 0 \},$

admits a Kähler structure, with the Hamiltonian diffeomorphism group acting by isometries. The moment map for this action is

$\mu(J) = S(J) - \overline{S},$

where $S(J)$ is the scalar curvature of $g_J(\cdot,\cdot) = \omega(\cdot, J\cdot)$ and $\overline{S}$ its average. This characterizes constant scalar curvature Kähler (cscK) metrics as zeros of the moment map (Scarpa et al., 2018, Goto, 2016).

In generalized Kähler geometry, twisted by a closed 3-form $H$, the analogous structure is built from commuting generalized complex structures $(\J_1, \J_2)$. The scalar curvature is defined via spinor formalism, and Goto proved that it too gives a moment map on the space of $\J_2$-compatible almost generalized complex structures, with suitably twisted Hamiltonian action. In this context, the scalar curvature has a pure-spinor expression and underlies the existence of constant scalar curvature generalized Kähler structures even on manifolds lacking ordinary Kähler metrics, such as compact Lie groups and Hopf surfaces (Goto, 2021).

3. Surjectivity, Deformation, and Localized Scalar Curvature Mapping

Modern techniques, generalizing classical Corvino-Schoen theory, demonstrate the local surjectivity of the scalar curvature map under suitable conditions, both on closed and manifolds-with-boundary settings. On compact manifolds with boundary, localized deformations of the metric can be constructed to realize prescribed variations in interior scalar curvature and boundary mean curvature, provided generic absence of nontrivial static potentials: $\ker D F|_{g_0}^* = \{0\}.$ This ensures the map $F$ is locally surjective near $g_0$ in weighted Sobolev/Hölder topologies, allowing compactly supported deformations (Sheng, 2024). The method extends, with appropriate function spaces and boundary gauge conditions, to asymptotically locally hyperbolic (ALH) manifolds with compact boundary (Huang et al., 2021).

For ALH manifolds, the scalar curvature map on the space of metrics asymptotic (in appropriate weighted Hölder spaces) to a reference ALH background satisfies surjectivity in two contexts:

• Exponentially decaying boundary-gauge
• Prescribed Bartnik data (induced metric and mean curvature on the boundary)

These surjectivity results permit precise control of curvature invariants under interior metric deformations even in global, noncompact, or singularity-admitting settings (Huang et al., 2021).

4. Moment Map Structures, Quantization, and Generalizations

The scalar curvature's identification as a moment map links geometric analysis, symplectic geometry, and geometric invariant theory. In the formal deformation quantization framework (Fedosov quantization), the moment map structure is deformed: the infinite-dimensional symplectic manifold of compatible almost complex structures is endowed with a formal 2-form and a moment map valued in $C^\infty(\J)[[v]]$. In the classical (Kähler) limit, this recovers the Fujiki–Donaldson picture; in the quantum-deformed regime, the scalar curvature becomes part of an infinite hierarchy of geometric invariants reflecting quantum corrections (Fuente-Gravy, 2022).

This structure further generalizes to hyperkähler reductions, where coupled moment map equations deform the cscK condition, producing coupled scalar curvature-Higgs field PDEs with solutions on spaces lacking cscK metrics, e.g., certain ruled surfaces (Scarpa et al., 2018).

5. Applications: Rigidity, Mass, and Topological Constraints

The scalar curvature map is pivotal in rigidity phenomena and global geometric analysis. In the ALH setting, local surjectivity of the scalar curvature map enables the analysis of Bartnik-type mass functionals and their minimizers. The Chruściel–Herzlich mass, defined for ALH manifolds via a flux integral at infinity, is nonnegative under suitable scalar curvature and boundary mean curvature bounds, with mass-zero rigidity forcing the interior metric to be the hyperbolic or Birmingham–Kottler model (Huang et al., 2021).

Scalar curvature also underpins topological rigidity. For instance, via Bochner-type identities and harmonic map techniques, one establishes inequalities relating scalar curvature to Euler characteristics of level-set surfaces, Thurston norms, and systolic invariants, producing rigidity and nonexistence theorems for positive scalar curvature metrics on certain 3-manifolds (Stern, 2019).

6. Explicit Constructions and Examples

• Hopf Surface: On $X = S^3 \times S^1$, explicit twisted generalized Kähler structures with constant scalar curvature are constructed using explicit pure spinors and 3-form $H$, yielding $S(\J_1,\J_2) = 1$ (Goto, 2021).
• Compact Lie Groups: Any compact even-dimensional Lie group admits a twisted generalized Kähler structure with constant scalar curvature. The construction proceeds through Lie-theoretic data (Cartan subalgebras, maximal isotropic subalgebras, and pure spinors) and yields scalar curvature $S = 2|P|^2$, which is nonzero and constant (Goto, 2021).
• Deformation Quantization: Formal moment map constructions unify the scalar curvature picture with Fedosov star-product quantization, introducing higher order "quantum" scalar curvature invariants that generalize cscK equations and relate to existence of closed star-products (Fuente-Gravy, 2022).

7. Broader Significance and Contemporary Research Directions

The scalar curvature map, via its geometric, analytic, and algebraic incarnations, mediates between local deformation theory (control of curvature via metric perturbations), global rigidity (uniqueness of mass-minimizing or symmetric models), moment map formalism (linking symplectic actions and stability conditions), and geometric quantization (formal and "quantum" generalizations). Its surjectivity results provide structural tools for constructing metrics of controlled curvature, establishing uniqueness and stability of geometric models, and revealing subtle topological constraints.

Current research continues to explore:

• New rigidity and gluing results in ALH and asymptotically flat settings (Huang et al., 2021, Sheng, 2024).
• Quantum deformations of scalar curvature and their implications in geometric quantization and stability (Fuente-Gravy, 2022).
• Generalized Kähler theories and their Lie-theoretic, topological, and analytic classifications (Goto, 2021, Goto, 2016).
• Hyperkähler reductions and systems coupling scalar curvature with auxiliary geometric data (Higgs fields, Poisson structures) (Scarpa et al., 2018).

These advances provide both a deepening of foundational understanding and a broadening of the scalar curvature map’s role in geometric analysis.