Exponential Euclidean Action Differential Geometry, Volume II: Explicit Calculations, Model Geometries, and Applications to Riemannian Variational Problems
Year: 2026
Type: preprint
Abstract: This volume applies the exponential realization, complete-action, and orbit-weak structures of Volume I to concrete geometric variational problems. Positive metrics are represented relative to a carrier, while connections, maps, complex structures, constraints, and configuration-space inertia retain their own equations and admissible variations. Classical curvature identities, elliptic estimates, and spectral decompositions supply the model-dependent calculations; their use is distinguished from the structural role of the realization and weak-limit interfaces. The initial space-form calculations separate stability of an original action from positivity of its Euler-defect action, with explicit reduced spectra and local weak rigidity under the stated regularity hypotheses. Subsequent applications treat coupled fields, singular geometries, and normalized spectral collapse. The later non-Kähler and homogeneous models include primary Kodaira surfaces, Heisenberg compact quotients, a standard Hopf surface, the Iwasawa manifold, and Sol geometry. Structure equations are used to distinguish Levi–Civita curvature from Hermitian curvature and torsion, and exact invariant constructions are checked against the complete equations before being interpreted on compact quotients. The final applications determine the full small-amplitude index and kernel of periodic Hopf connection solutions, classify the global weak minimizers of a quartic curvature functional on a Heisenberg quotient, and compute closed-axis Jacobi spectra and decorated collapse on Sol. The weak theories have three distinct outputs: static critical configurations modulo symmetry, weak trajectories for a declared inertial action, and phase-space measure solutions tested against observables. Passing between these outputs requires additional hypotheses. In particular, inertial stationarity is not static criticality, and a pushforward construction presupposes a flow on the relevant domain. General Riemannian applicability means that the interfaces do not require constant curvature; existence and regularity remain attached to the compactness, test-recovery, and nonlinear-closure estimates proved for each problem. The concluding synthesis fixes these distinctions and the scope of the results established in this volume. Keywords Exponential metric; Einstein sphere; Riemannian variational problem; normalized scalar curvature; Lichnerowicz Laplacian; Jacobi operator; spherical harmonics; harmonic map; minimal immersion; orbital weak solution; Yang–Mills connection; Hodge spectrum; configuration-space inertia; metric cone; Mosco convergence; spectral collapse; quadratic curvature; homogeneous Einstein metric; bifurcation analysis; geometric flow; minimizing movement; Ricci flow; mean-curvature flow; coupled geometric fields; measure-valued stress; non-Kähler geometry; Hermitian torsion; nilmanifold; solvmanifold.
Author: Kianming, Wang
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212016587
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