SYSTEMATIC MATHEMATICS

Ricci Flow & Geometric Evolution

Use eighty-five visible bilingual chapters in twelve content-sized units to build Ricci flow from time-dependent metric bookkeeping through DeTurck gauge, short-time existence, evolution and maximum principles, exact and self-similar solutions, entropy and noncollapsing, singularity rescaling, bounded surface and three-dimensional routes, normalized long-time behavior, Kähler specialization, weak-flow frameworks, and verified numerical evidence. Every chapter includes plain-language orientation, symbol-level definitions, a stated theorem or bounded claim, an inspectable proof route, a failure boundary, a worked example, matched practice, a complete solution, and a diagnostic check. Read each displayed expression from left to right: identify the object and its type, say what each index, subscript, superscript, derivative, norm, integral, scale, and time parameter means, list the hypotheses, derive the next line, then test dimensions, signs, normalization, equality, and failure cases. A named theorem is never a substitute for its usable statement, and a proof route explicitly marks any deep theorem it invokes. Exact models serve as sign and scale oracles; numerical models retain raw values, refinement, residuals, uncertainty, and the first unproved claim. The course does not claim a line-by-line proof of the full three-dimensional surgery and geometrization program, a complete pluripotential or minimal-model treatment of Kähler–Ricci flow, a complete modern theory of singular metric flows, RCD evolution, or discrete Ricci flows, or mastery of every postdoctoral frontier.

Before this course: Completed Riemannian Geometry II: Comparison & Geometric Analysis; Nonlinear Elliptic & Parabolic Regularity; Geometric Measure Theory & Minimal Surfaces; Differential Forms, de Rham Cohomology & Hodge Theory; Sobolev Spaces, Distributions & Weak PDE; and Calculus of Variations. No prior complete course in Ricci flow, Kähler–Ricci flow, three-dimensional surgery, singular metric flows, or numerical geometric evolution is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A Ricci flow changes the metric, not the underlying point set

Objective: Does a zero of a(t) describe a smooth metric at that time?

A geometric flow is a path of measuring rules on one fixed smooth manifold. Points do not move until a time-dependent diffeomorphism is deliberately introduced. Before calculating, write the fixed manifold, time interval, metric convention, curvature sign, normalization, regularity class, compactness or completeness hypothesis, and whether the conclusion is pointwise, integral, local in spacetime, global, or valid only before a singular time.

Read every object in words: Let M be fixed and let g(t) be a smooth positive-definite symmetric 2-tensor for t in I. Write h=∂_t g. A Ricci flow satisfies h=−2Ric(g(t)). The exact result used here is: At each time, g(t) defines its own distance, volume, connection, and curvature. Differentiating any derived object must include its dependence on g(t). Separate an identity valid for every metric variation from a theorem that needs the Ricci-flow equation, a maximum principle, compactness, or a noncollapsing hypothesis.

The complete proof or honestly bounded proof route, at the length this argument actually requires, is: 1. Fix the manifold and time-independent coordinate labels before differentiating. 2. Define h as the metric velocity and expand each derived object as a function of g. 3. Substitute h=−2Ric only after the general variation identity is established. Record every pullback, scale factor, time orientation, boundary term, cutoff, curvature bound, basepoint, subsequence, and equality case rather than hiding it in notation.

Reconstruct the model “For g(t)=a(t)g₀ on a fixed n-manifold, compute h and state when g(t) remains Riemannian.” from the stated metric. The checked result is h=a′g₀=(a′/a)g, and the allowed time set is exactly {t:a(t)>0}. The nearest failure boundary is: If coordinates themselves depend on time, an additional Lie-derivative term appears. Ignoring it confuses geometric evolution with relabeling. A rendered surface, finite mesh, fitted curvature, short simulation, or small residual is evidence only for the recorded discretization and tolerance; it is never a proof of smooth continuation, uniqueness through singularities, topology change, or a global classification.

At each time, g(t) defines its own distance, volume, connection, and curvature. Differentiating any derived object must include its dependence on g(t).

Fix the manifold and time-independent coordinate labels before differentiating.

Define h as the metric velocity and expand each derived object as a function of g.

Substitute h=−2Ric only after the general variation identity is established.

For g(t)=a(t)g₀ on a fixed n-manifold, compute h and state when g(t) remains Riemannian.

  1. Differentiate to obtain h=a′(t)g₀.
  2. Rewrite h=[a′(t)/a(t)]g(t) wherever a(t)>0.
  3. Positive definiteness is equivalent to a(t)>0 because g₀ is positive definite.

Result: h=a′g₀=(a′/a)g, and the allowed time set is exactly {t:a(t)>0}.