arXiv research

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.

Estimates symplectic Calabi-Yau equation using Cheng-Yau method.

Paper proves new inequalities for hyperbolic space Laplacian eigenvalues.

Study Kähler-Ricci solitons on bounded domains, proving they are Kähler-Einstein.

The paper establishes eigenvalue inequalities for a specific operator on curved spaces.

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl's asymptotic formula. As an application, we find a lower bound for th…

The paper proves a conjecture about the Bergman metric of real analytic domains.

New proof confirms noncompact locally conformally flat manifolds are compact.

Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

Sharp gradient estimates extended to surfaces with lower Ricci curvature.

Paper improves volume gap between minimal submanifolds and unit spheres.

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions…

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

The study proves properties of self-shrinkers with bounded curvature.

This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negati…

We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjectu…

We improve the well known local gradient estimate of Cheng and Yau in the case when Ricci curvature has a negative lower bound.

Study shows quantum behavior near infinity in metric asymptotics.

Paper refines Chen-Cheng's estimates for Kähler metrics.

In this note we show how results in \cite{BaudoinBonnefont2016, BaudoinGarofalo2013, CoulhonJiangKoskelaSikora2017} yield the Cheng-Yau estimate on two classes of sub-Riemannian manifolds: Carnot groups and sub-Riemannian manifolds satisfying a generalized curvature-dimension inequality.

Commentary on Cheng's fairness comparison between tests and AI.

In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various …

In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an n-dimensional Euclidean space and obtain a lower bound for eigenvalues, which generalizes the results due to Cheng-Wei [5] and gives an improvement of results due to Cheng- Qi-Wei [3].

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

Let $M$ be a complex $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional curvature $1$, $σ$ its second fundamental form, and $\underline{|σ|}^2$ the mean value of the squared length of $σ$ on $M$. We derive a formula for $\underline{|σ|}^2$ an…

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…

The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.

Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.

Maximal diameter theorem for graphs with positive Ricci curvature.

We prove Cheng's eigenvalue comparison theorems for geodesic balls within the cut locus under weaker geometric hypothesis, and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci cur…

We study the Gauss map $G$ of surfaces of revolution in the 3-dimensional Euclidean space ${\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\square$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map …

The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.

New proof of Kähler-Einstein Fano manifold $L^\infty$ estimates.

We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng's lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean…

Study on Kähler metrics on ruled surfaces, proving existence and non-existence.

For each invariant polynomial $Φ$, we construct a global CR invariant via the renormalized characteristic form of the Cheng--Yau metric on a strictly pseudoconvex domain. When the degree of $Φ$ is 0, the invariant agrees with the total $Q'$-curvature. When the degree is equal to the CR dimension, we construct a primed …

Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.

Alternative proof for 4D shrinking Ricci solitons with constant scalar curvature.

New proof of shrinking gradient Ricci soliton rigidity.

The paper uses Nash-Moser iteration to prove gradient estimates for nonlinear equations on Riemannian manifolds.

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recas…

Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.

The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.

The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.

Proves rigidity of stable free boundary hypersurfaces in 5-manifolds.

Study critical quasilinear equations on Riemannian manifolds with curvature constraints.

Why the results of that article [arXiv:1304.3212] are immediate consequences of known ones.