arXiv subjects

Lucas Lavoyer

Publications and source records attributed to Lucas Lavoyer.

4 recordsLinked to original sources

arXiv2026

K\"ahler-Ricci Tangent Flows in the Analytic Minimal Model Program

We describe certain finite-time singularities of the K\"ahler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical K\"ahler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of K\"ahler potentials. Consequently, every noncollapsed K\"ahler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

math.DG↗

arXiv2026

Expanding Soliton Models for K\"ahler-Ricci Flow Near Conical Singularities

Let $(Y,g_0)$ be a compact analytic space with a finite number of singular points, where the metric at each singular point is modelled on a K\"ahler cone with smooth canonical model. We show that the K\"ahler-Ricci flow with such initial data satisfies a $C/t$ curvature bound, and that the flow near each singular point is modelled on the unique K\"ahler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the K\"ahler--Ricci flow emerging from singularities arising in the analytic minimal model program.

math.DG↗

arXiv2025

Steady Gradient Ricci Solitons with $O(p)\times O(q)$ Symmetry

We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p) \times O(q)$ symmetry in dimension $p+q$ for any pair of integers $p,q \geq 2$ and discuss their asymptotic geometry.

math.DG↗

arXiv2023

Ricci flow from spaces with edge type conical singularities

We study the Ricci flow out of spaces with edge type conical singularities along a closed, embedded curve. Under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively curved cone and a line, we show existence of a solution to Ricci flow $(M,g(t))$ for $t\in (0,T],$ which converges back to the singular space as $t\searrow 0$ in the pointed Gromov-Hausdorff topology. We also prove curvature estimates for the solution and, for edge points, we show that the tangent flow at these points is a positively curved expanding Ricci soliton solution crossed with a line.

math.DG↗