arXiv research

Develops interpretable low-dimensional kernels with conic discriminant functions.

Study heat flow on collapsing K3 surfaces, handling conic singularities.

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the coni…

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of th…

Defines Perelman's functionals on manifolds with non-isolated conical singularities.

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

Study curve shortening flow on Riemann surfaces with conic singularities.

Following Donaldson's oppenness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use t…

Paper classifies conic submanifolds in control systems.

In this paper, we develop the theory of Perelman's $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvatur…

The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.

Study conic Laplacian on \(\mb P^1\) with explicit model and boundary data.

New approach to prescribing Gaussian curvature on spheres with conical singularities.

Study on spherical conical metrics and their reducibility on compact Riemann surfaces.

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 paper, we apply the method developed in [Ti97] and [TZ00] to proving the properness of log $F$-functional on any conic Kähler-Einstein manifolds. As an application, we give an alternative proof for the openness of the continuity method through conic Kähler-Einstein metrics.

We prove functional identities for conic webs on del Pezzo surfaces.

In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the a…

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singul…

Krein's formula for conic Laplacians on compact Riemann surfaces

The paper proves geodesics and conic sections are length-minimizing under specific metrics.

Study shortest geodesics on flat cone spheres with conical singularities.

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's $λ$ and $ν$ functionals of cones and characterize their finiteness in terms of the $λ$-functional of…

Extends existence results for scalar curvature on conical manifolds.

The purpose of this paper is to prove the uniqueness of conical Kähler-Einstein metrics, under the condition that the twisted $Ding$-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted Kähler-Einstein metrics, and then use these smooth p…

In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, …

Regularized zeta function for polyhedra calculated from Riemann surface invariants.

The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle,…

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prov…

Paper develops compact formulations for optimization problems with rank-one convex functions and indicator variables.

The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.

We describe the range of the Radon transform on the space $M$ of irreducible conics in $\CP^2$ in terms of natural differential operators associated to the $SO(3)$-structure on $M=SL(3, \R)/SO(3)$ and its complexification. Following \cite{moraru} we show that for any function $F$ in this range, the zero locus of $F$ is…

In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^\infty$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^\infty (M)$ we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of $M$, and the …

Paper proposes a method to find approximate SOSP for nonconvex conic optimization problems.

We prove a regularity result for Monge-Ampère equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $β$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical singularities along divisors and give a geometric sufficient condition on the divisor …

Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $\mathsf m$ be the standard round metric of curvature $1$ on the Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ is…

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on t…

We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry …

Study spectral properties on manifolds with conical singularities, proving new inequalities.

New conical metrics found on toric varieties with convex cones.

SOC-ICNN expands neural network representational capacity by using conic optimization.

The paper studies metrics on hyperkähler manifolds using sub-twistor constraints.

Two unique conic-line arrangements with degree 9 are found.

Study conic singular manifolds, proving Lipschitz normal embedding.

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case a…

Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.

We view a conic optimization problem that has a unique solution as a map from its data to its solution. If sufficient regularity conditions hold at a solution point, namely that the implicit function theorem applies to the normalized residual function of [Busseti et al., 2018], the problem solution map is differentiabl…

Paper shows invertibility of tensor X-ray transform on certain manifolds.