Bibliographic record
Abstract
In this work, first we will obtain some local curvature estimates for Kähler-Ricci flow on Kähler manifolds with initial metrics of nonnegative bisectional curvature. As a corollary, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete solution of the Kähler Ricci flow which satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue Rm left-parenthesis g left-parenthesis t right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to a t Superscript negative theta"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mtext>Rm</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi>a</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi> θ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|\text {Rm}(g(t))|\leq at^{-\theta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than theta greater-than 2"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi> θ </mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0>\theta >2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has nonnegative bisectional curvature, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> also has nonnegative bisectional curvature. This generalizes results in [Amer. J. Math. 140 (2018), pp. 189–220] and [J. Differential Geom. 45 (1997), pp. 94–220]. Using the local curvature estimate, we prove that for a complete solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Kähler-Ricci flow with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to have nonnegative bisectional curvature, to be noncollapsing, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript upper M times left-bracket tau comma upper T right-bracket Endscripts StartAbsoluteValue Rm left-parenthesis g left-parenthesis t right-parenthesis right-parenthesis EndAbsoluteValue greater-than plus normal infinity"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo> × </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi> τ </mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mtext>Rm</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mm
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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.000 | 0.004 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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; a candidate call from one teacher head, not a consensus.
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".