Brownian motion on Perelman’s almost Ricci-flat manifold
Bibliographic record
Abstract
Abstract We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℳ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>M</m:mi> <m:mo>×</m:mo> <m:msup> <m:mi>𝕊</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>×</m:mo> <m:mi>I</m:mi> </m:mrow> </m:mrow> </m:math> {\mathcal{M}=M\times\mathbb{S}^{N}\times I} , whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>→</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> {N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℳ</m:mi> </m:math> {\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝒪</m:mi> <m:mo></m:mo> <m:mi>ℳ</m:mi> </m:mrow> </m:math> {\mathcal{OM}} . As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.003 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.006 | 0.006 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.003 | 0.000 |
| Scholarly communication | 0.005 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.002 | 0.013 |
| Insufficient payload (model declined to judge) | 0.007 | 0.006 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".