相关论文: The Fermi Flow and its Application to Geometry

Several results from differential geometry of hypersurfaces in R^n are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the…

A phenomenological theory is presented for two-dimensional quantum liquids in terms of the Fermi surface geometry. It is shown that there is a one-to-one correspondence between the properties of an interacting electron system and its…

We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the…

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity,…

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…

We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization…

The Ricci flow is one of the most important topics in differential geometry, and a central focus of modern geometric analysis. In this paper, we give an illustrated introduction to the subject which is intended for a general audience. The…

This paper gives some examples of hypersurfaces $\phi_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean…

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for…

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently…

Ricci flow on two dimensional surfaces is far simpler than in the higher dimensional cases. This presents an opportunity to obtain much more detailed and comprehensive results. We review the basic facts about this flow, including the…

We give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian metrics on complex manifolds. We emphasize open problems and new directions.

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,...,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among…

Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by…

We are interested in the impact of entropies on the geometry of a hypersurface of a Riemannian manifold. In fact, we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold, provided some geometric…

Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher…

Diffusion and flow models have become the dominant paradigm for generative modeling on Riemannian manifolds, with successful applications in protein backbone generation and DNA sequence design. However, these methods require tens to…

Modeling the mechanics of fluid in complex scenes is vital to applications in design, graphics, and robotics. Learning-based methods provide fast and differentiable fluid simulators, however most prior work is unable to accurately model how…

This is a survey paper focusing on the interplay between the curvature and topology of a Riemannian manifold. The first part of the paper provides a background discussion, aimed at non-experts, of Hopf's pinching problem and the Sphere…