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Longteng Chen

Publications and source records attributed to Longteng Chen.

6 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

Stability of asymptotically conical gradient K\"ahler-Ricci expanders

In this work, we consider a perturbation of an asymptotically conical gradient expanding K\"ahler-Ricci soliton metric $g$ in the same K\"ahler class. We demonstrate that, under suitable assumptions, the normalized K\"ahler-Ricci flow starting from the initial perturbed metric exists for all time and converges uniformly to an asymptotically conical gradient expanding K\"ahler-Ricci soliton metric $g_\infty$. Moreover, if the perturbed initial metric is asymptotic to $g$ at spatial infinity, then the limiting metric coincides with the original soliton, that is, $g_\infty = g$.

math.DG↗
arXiv2025

Uniqueness of asymptotically conical K\"ahler-Ricci flow

We study the uniqueness problem for the K\"ahler-Ricci flow with a conical initial condition. Given a complete gradient expanding K\"ahler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity--appearing as a K\"ahler cone--must coincide with the forward self-similar K\"ahler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its K\"ahler form remains in the same cohomology class as that of the soliton's self-similar K\"ahler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.

math.DG↗
arXiv2023

Semantic-Enhanced Image Clustering

Image clustering is an important and open-challenging task in computer vision. Although many methods have been proposed to solve the image clustering task, they only explore images and uncover clusters according to the image features, thus being unable to distinguish visually similar but semantically different images. In this paper, we propose to investigate the task of image clustering with the help of a visual-language pre-training model. Different from the zero-shot setting, in which the class names are known, we only know the number of clusters in this setting. Therefore, how to map images to a proper semantic space and how to cluster images from both image and semantic spaces are two key problems. To solve the above problems, we propose a novel image clustering method guided by the visual-language pre-training model CLIP, named \textbf{Semantic-Enhanced Image Clustering (SIC)}. In this new method, we propose a method to map the given images to a proper semantic space first and efficient methods to generate pseudo-labels according to the relationships between images and semantics. Finally, we propose performing clustering with consistency learning in both image space and semantic space, in a self-supervised learning fashion. The theoretical result of convergence analysis shows that our proposed method can converge at a sublinear speed. Theoretical analysis of expectation risk also shows that we can reduce the expected risk by improving neighborhood consistency, increasing prediction confidence, or reducing neighborhood imbalance. Experimental results on five benchmark datasets clearly show the superiority of our new method.

cs.CV↗