Study eigenvalues of Pucci operator on geodesic balls, extending Cheng's theorem.
problem Eigenvalue problem for the Riemannian Pucci operator on geodesic balls.
method Established upper and lower bounds for Pucci eigenvalues based on curvature.
result Cheng's bounds extended to Pucci eigenvalues on geodesic balls.
Maximal diameter theorem for graphs with positive Ricci curvature.
problem Diameter comparison in directed graphs with positive Ricci curvature.
method Introduced a Lin-Lu-Yau type Ricci curvature for directed graphs and investigated rigidity properties for the equality case.
result Concluded a maximal diameter theorem of Cheng type.
Study vanishes L2 cohomology groups on Hessian manifolds and cones.
problem Vanishing L2 cohomology groups on Hessian manifolds and cones. method Analyzes L2 cohomology groups of Kodaira-Nakano type on complete Hessian manifolds and a regular convex cone with the Cheng-Yau metric. result Obtains vanishing theorems for L2 cohomology groups on Hessian manifolds and cones. The paper establishes eigenvalue inequalities for a specific operator on curved spaces.
problem Eigenvalue estimation for a specific operator on curved domains.
method Bochner type formula and Rauch comparison theorem.
result Universal inequalities for eigenvalues of the drifted Cheng-Yau operator.
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…
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…
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
New proof confirms noncompact locally conformally flat manifolds are compact.
problem Rigidity of Schouten tensor under conformal transformations.
method Proof of Cheng's theorem using modified Schouten tensor.
result Noncompact locally conformally flat manifolds are compact.
We study the Gauss map G of surfaces of revolution in the 3-dimensional Euclidean space E3 with respect to the so called Cheng-Yau operator □ 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 …
Study on constant mean curvature hypersurfaces in Anti-de Sitter space.
problem Characterizing and classifying hypersurfaces in Anti-de Sitter space.
method Analytic foliation and classification of hypersurfaces based on their asymptotic boundary.
result Every admissible sphere is the boundary of a unique hypersurface for any given mean curvature.
This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.
problem Proving a more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.
method Using a combination of Ricci curvature bounds and Riemannian universal cover properties to establish a quantitative rigidity result.
result If a manifold has positive Ricci curvature and a diameter close to the maximal possible, it is diffeomorphic and bi-Hölder close to the sphere.
Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.
problem Establishing uniform bounds for eigenvalues of the Hodge Laplacian on manifolds with specific geometric constraints.
method Using uniform bounds on Ricci curvature, injectivity radius, and diameter, we derive eigenvalue estimates for the Hodge Laplacian.
result Uniform upper bounds for eigenvalues of the Hodge Laplacian on differential forms on manifolds with given geometric constraints.
Sharp estimates lead to new comparison theorems in Riemannian and Kähler geometry.
problem Developing precise geometric inequalities for curvature assumptions.
method Quantitative Laplacian estimates and integral curvature assumptions.
result Derive quantitative comparison theorems for Riemannian and Kähler manifolds.
Sharp gradient estimates extended to surfaces with lower Ricci curvature.
problem Rigidity of Cheng-Yau gradient estimates on surfaces with lower Ricci curvature.
method Extending Cheng-Yau gradient estimates to surfaces with lower Ricci curvature bound and higher-dimensional Riemannian manifolds.
result Pointwise Cheng-Yau gradient estimates for higher-dimensional Riemannian manifolds and monotonicity formulas for positive harmonic functions.
The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.
problem Existence and convergence of Hesse-Einstein metrics on compact Hessian manifolds.
method Study of the Hesse-Koszul flow and its convergence properties.
result The flow converges to the unique Hesse-Einstein metric under certain conditions.
Estimates symplectic Calabi-Yau equation using Cheng-Yau method.
problem Estimating solutions to the Calabi-Yau equation on symplectic 4-manifolds.
method Applies a Cheng-Yau type estimate in the symplectic setting.
result Proves an a priori estimate for the symplectic Calabi-Yau equation.
The paper proves a conjecture about the Bergman metric of real analytic domains.
problem Proving the Cheng-Yau conjecture for real analytic pseudoconvex domains.
method Localization of Bergman kernels, extension theorem, and Einstein metrics.
result The Bergman metric of a bounded pseudoconvex domain with real-analytic boundary is Einstein if and only if the domain is biholomorphic to the unit ball.
Note shows Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
problem Proving Cheng-Yau gradient estimate for specific geometric structures.
method Utilizing previous results on sub-Riemannian manifolds and Carnot groups.
result Established Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
Paper improves volume gap between minimal submanifolds and unit spheres.
problem Volume gap between minimal submanifolds and unit spheres.
method Modified Cheng-Li-Yau coefficients and applied Cheng-Yang eigenvalue estimate for Laplacian.
result Enhanced volume gap between minimal submanifolds and unit spheres.
Develops a Barta theorem for p-Laplacian on manifolds.
problem Sharp lower bounds for p-fundamental tone on Riemannian manifolds.
method Extends Barta-type formulation to nonlinear setting on Riemannian manifolds.
result Sharp lower bounds for p-fundamental tone without boundary regularity assumptions.
Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.
problem Extending Cheng's eigenvalue comparison theorem to non-smooth spaces.
method Localization technique, synthetic Ricci curvature bounds via optimal transport.
result Sharp upper bounds on eigenvalues in metric measure spaces.
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted p-Laplacian on manifolds with modified Ricci curvature. method Established Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted p-Laplacian on geodesic balls. result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted p-Laplacian. The paper uses Nash-Moser iteration to prove gradient estimates for nonlinear equations on Riemannian manifolds.
problem Proving gradient estimates for solutions of nonlinear equations on Riemannian manifolds.
method Employing Nash-Moser iteration technique to establish gradient estimates.
result Gradient estimates for solutions of nonlinear equations on Riemannian manifolds are proven.
We improve the well known local gradient estimate of Cheng and Yau in the case when Ricci curvature has a negative lower bound.
Rigidity theorem for Bergman metric on Hartogs domains over bounded homogeneous domains.
problem Proving rigidity of Bergman metric on Hartogs domains.
method Combining explicit formula for Bergman kernel and structural invariants of bounded homogeneous base.
result Conditions for Bergman metric to be Kähler-Einstein, homogeneous, and biholomorphic to Bn+m are equivalent. New theorem splits spaces with maximal variance of 1-Lipschitz functions.
problem Understanding the structure of metric measure spaces.
method Analyzing isoperimetric profiles and variance of 1-Lipschitz functions.
result Spaces with maximal variance are foliated by minimal geodesics.
The paper proves Liouville theorems for V-harmonic maps under specific curvature conditions.
problem Proving Liouville theorems for V-harmonic maps in Riemannian manifolds with non-negative (m,V)-Ricci curvature. method Probabilistic proof extending previous results by Cheng, Hildebrandt-Jost-Wideman, and Stafford.
result Extends Liouville theorems to a broader class of manifolds and curvature conditions.
Study shows quantum behavior near infinity in metric asymptotics.
problem Quantum behavior of metrics near infinity on quasi-projective manifolds.
method Analysis of Bergman kernel function near smooth divisor at infinity of Cheng-Yau metric.
result Quantum phenomenon observed for points very close to the divisor at infinity.
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.
Paper refines Chen-Cheng's estimates for Kähler metrics.
problem Uniform boundedness of scalar curvature assumption.
method Replacing uniform boundedness with Lp-boundedness. result Improved estimates for Kähler metrics under Lp-boundedness. Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.
problem Proving entire zero mean curvature graphs are hyperplanes in Lorentz-Minkowski space.
method Using line theorems at degenerate light-like points to generalize Bernstein theorem.
result Entire zero mean curvature graphs in Lorentz-Minkowski space are hyperplanes if they only contain space-like or light-like points.
Commentary on Cheng's fairness comparison between tests and AI.
problem Distinction between equality and equity in fairness.
method Systematic comparison of test fairness and algorithmic fairness.
result Importance of causality in fairness research.
In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minim…
The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
problem Analyzing modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
method Proving Laplacian comparison theorems using modified m-Bakry-Emery Ricci tensors under m≤1.
result Optimal conditions for modified m-Bakry-Emery Ricci tensors under m≤1 are derived.
The paper proves rigidity theorems on holomorphic isometries into homogeneous domains.
problem Characterizing and comparing holomorphic isometries into homogeneous bounded domains.
method Two rigidity theorems on holomorphic isometries into homogeneous bounded domains.
result The flat (definite or indefinite) complex Euclidean space is not a relative of a homogeneous bounded domain.
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].
The paper constructs global CR invariants from renormalized characteristic forms.
problem Global CR invariants on strictly pseudoconvex domains.
method Renormalized characteristic forms of the Cheng--Yau metric.
result Generalizations of I′-curvature on CR five-manifolds. Rigidity properties of hypercube graphs via curvature methods.
problem Rigidity of hypercube graphs under curvature constraints.
method Semigroup methods and new direct methods translating curvature to combinatorial properties.
result Sharp inequalities for diameter and eigenvalues only hold for hypercubes.
The paper proves gap results for self-shrinkers in r-mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in r-mean curvature flow. method Proving gap results using a modified second fundamental form and a differential operator.
result Proper self-shrinkers are parabolic for a certain second-order differential operator.
Study explores Liouville's theorem and SLP for harmonic functions on cones and surfaces.
problem Investigating Liouville's theorem and SLP for harmonic functions on Riemannian cones and surfaces.
method Reinterprets classical Liouville property in terms of radial eigenfunctions, providing explicit estimates and constructing examples.
result Explicit estimates for slowest-growing nonconstant harmonic functions and a unified geometric perspective on Liouville phenomena.
Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.
problem Understanding Brownian motions and heat kernel bounds on specific geometric manifolds.
method Sharp Laplacian comparison theorems and Cheeger-Yau type lower bounds for heat kernels.
result Sharp Cheeger-Yau type lower bounds for heat kernels and Dirichlet eigenvalues of metric balls.
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
The paper extends gradient estimates for harmonic maps into singular spaces.
problem Estimating gradients for harmonic maps into singular spaces.
method Using Alexandrov curvature bounds and extending Cheng and Choi's work.
result Yau's gradient estimates for harmonic maps into singular spaces.
Extends three circle theorem to almost Hermitian manifolds.
problem Three circle theorem for Kähler manifolds to almost Hermitian manifolds.
method General maximum principle established to prove three circle theorem.
result Sharp dimension estimates and Liouville theorems for holomorphic functions.
Solves a 1979 conjecture about Bergman metrics and constructs hyperbolic metrics.
problem Affirmative solution to a 1979 conjecture about Bergman metrics.
method Establishes Kähler-Einstein metrics and constructs hyperbolic metrics.
result Affirmative solution to Cheng's conjecture about Bergman metrics.