(Monograph) Surgery on Lorentzian Manifolds: A full account of singularity resolution in gravitational collapse, conservation laws for angular momentum, Geometric Vorticity and the final-state problem for rotating black holes

DISCLAIMER: The results presented herein are not final and undergoing refinements to meet some technical issues and the overall presentation, This is a conceptual step towards refinement which are by measure a collection of study preprints that are yet to be classified as final technical verifications [see version V3 for immediate results].This monograph presents a comprehensive mathematical framework for resolving spacetime singularities through constrained geometric evolution with surgical intervention. Building on the analogy between Ricci flow in Riemannian geometry and gravitational dynamics in general relativity, the work introduces a constrained Lorentzian flow coupled to a Clebsch potential structure that preserves crucial topological and physical invariants. The central innovation is a surgical protocol that excises regions of extreme curvature, analogous to singularities forming in gravitational collapse, and replaces them with regular geometric data, enabling global evolution. Within this framework, this account develops the complete Komar-Clebsch theory, a mathematical formalism for angular momentum that extends classical Komar integrals to incorporate matter contributions through Clebsch potentials. Key results include: Global existence theorems for the constrained surgical flow on finite and infinite time intervals Exact conservation laws for mass and angular momentum both between and across surgical interventions Convergence proofs showing that under appropriate symmetry (spherical/axisymmetric), solutions asymptotically approach stationary solitons: Schwarzschild for non-rotating collapse and Kerr for rotating collapse Pointwise and integral inequalities binding geometric vorticity to angular momentum, providing a priori estimates for collapse scenarios Explicit verification that the Kerr metric naturally embeds as a stationary Clebsch–Killing soliton within the framework, with its angular momentum correctly given by the Komar–Clebsch formula Edge calculus and microlocal analysis at null corners, providing Sobolev estimates, spectral flow calculations, and constraint preservation proofs for the surgical boundaries, with explicit derivation of the Hölder regularity exponent from geometric monodromy. The work is entirely self-contained, with all results derived deductively from the flow equations and constraint structure without additional physical assumptions. It bridges geometric analysis, partial differential equations, and classical general relativity, offering both theoretical insights into the nature of singularities and rotation, as well as practical foundations for numerical implementations that track angular momentum without horizon finding. Suitable for researchers and graduate students in geometric analysis, mathematical general relativity, and theoretical physics, this monograph provides a complete account of an avenue toward a singularity-free classical theory of gravitational collapse and black hole formation. Figures and briefing available after the last chapter on edge calculus analysis. Keywords: Lorentzian geometry, geometric surgery, spacetime singularities, Komar integrals, Clebsch potentials, angular momentum conservation, gravitational collapse, Kerr metric, black hole final state, constrained geometric evolution.