arXiv research

Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.

Study behavior of curvatures near singular points of frontals.

Study precise asymptotic behavior of functions in singular metric spaces.

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

Differential forms and symmetric tensors show contrasting singular behaviors in a specific geometric setting.

Study on manifolds with kinks and Gaussian kernel behavior.

Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.

We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to natura…

Study on solutions near isolated singularities in 6D Yamabe equation.

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points o…

We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S^3 with varying amounts of S^2 neck pinching, we find critical behavior at the threshold of singularity formation.

Paper compares two equivariant η-forms, revealing their singular behavior.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

Study shows how solutions to Yamabe flow can develop Type II singularities.

Study finds solutions to Yamabe equation with specific behavior near singular points.

Study shows asymptotic behavior of metric near singular points of a Monge-Ampère equation.

Study of 17 surface behaviors and singularities for elliptic Weingarten equations.

This is a continuation of paper \cite{Li}. On any toric Fano manifold, we discuss the behavior of limit metric of a sequence of metrics, which are solutions to a continuity family of complex Monge-Ampere equations in Kahler-Einstein problem. We show that the limit metric satisfies a singular complex Monge-Ampere equati…

Study of symplectic manifolds degenerating into singular spaces.

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…

Study singularity formation in Ricci flow solutions.

In earlier work, we provided a general description of the forces of attraction and repulsion, encountered by two parallel vertical plates of infinite extent and of possibly differing materials, when partially immersed in an infinite liquid bath and subject to surface tension forces. In the present study, we examine som…

This paper tackles the dynamics of singularities in geometric flows.

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

The main result is a computation of the Nahm transform of a SU(2)-instanton over RxT^3, called spatially-periodic instanton. It is a singular monopole over T^3, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is described.

Investigates optimal strategies for behavioral control problems with finite variation controls.

Studying the behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler-Ricci flow on mildly singula…

In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.

Study on singularities of Gauss maps of wave fronts with specific properties.

The paper shows singularities in geometric flows can wander off.

Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ri…

The study connects minimal and maximal surfaces in 3D and 3-L space.

Study of singular solutions to a fourth order system in a ball with a singularity.

We introduce thermodynamic response functions for singular Bayesian models.

Study shows twist tori equidistribute in moduli space, with other families having singular distributions.

Study on singularity behavior of mean curvature flow with bounded curvature and index.

In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schrödinger operator $-4Δ+R$ consists of discrete eigenvalues with finite multiplicities, if the scalar curvature $R$ satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior fo…

In this short paper, we show that Kähler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some mild assumption.

New method for analyzing learning dynamics in singular models.

Study focal surfaces of wave fronts with unbounded curvatures.

Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, us…

We study a behavior of the conformal Laplacian operator $Ł_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator $Ł_g$ on such manifolds. We describe the asymptotic of a general solution …

In \cite{CM5}, Colding and Minicozzi describe a type of compactness property possessed by sequences of embedded minimal surfaces in $\Real^3$ with finite genus and with boundaries going to $\infty$. They show that any such sequence either contains a sub-sequence with uniformly bounded curvature or the sub-sequence has …

In this paper, we study the boundary behavior of the negatively curved Kähler-Einstein metric attached to a log canonical pair $(X,D)$ such that $K_X+D$ is ample. In the case where $X$ is smooth and $D$ has simple normal crossings support (but possibly negative coefficients), we provide a very precise estimate on the p…

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.