arXiv research

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives…

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

The paper studies Ricci curvature on Kähler-Ricci flow.

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2…

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curva…

Study conic singular manifolds, proving Lipschitz normal embedding.

Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.

Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may…

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

Proves existence of Yamabe metrics on conical manifolds with conical points and links.

Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.

The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.

Study constant Q-curvature metrics on conic 4-manifolds.

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…

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

Unique circle patterns on spheres found for spherical conical metrics.

Proves conic version of YTD conjecture on log Fano manifolds.

Classifies 4D toric Hermitian ALF metrics with conical singularities.

New connection between complex analysis and PDE for spherical conical metrics.

Study describes ALF instantons with conical singularities.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

Study X-ray transform on conic spaces, proving injectivity under certain conditions.

We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a gene…

Characterizes conical angles for metrics with dihedral symmetry.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

Survey on metrics with conic singularities on Riemann surfaces.

Constructs scalar-flat Kähler metrics with varying conical singularities.

Analytic torsion studied for fibred boundary metrics, with applications to conic degeneration.

In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2π)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…

The paper proves stability of Kähler-Ricci flows on Fano manifolds.

The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.

The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.

The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

The study proves convergence of conic 4-spheres' geometry to boundary cases.

In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than $2π$; in particular, we define and study the Teichmüller space $\mathcal{T}^{\mathrm{conic}}_{γ,k}$ of conic constant curvature metrics on a surface of genus $γ$ with $k$ …

The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.

Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.

Study examines heart and football-shaped metrics, verifying geometric structure.

We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic singularities here can be along a simple normal crossing divisor.

Recently it was shown by H. Guenancia and M. Paun that a singular metric satisfying the conical Kahler-Einstein equation with a simple normal crossing divisor is equivalent to a conical metric along that divisor. In this note, we present an alternative proof of their theorem.

New Calabi-Yau metrics with conical singularities are created near complex lines.

Finite number of metrics found for specific spherical angles.

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

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.