In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scala…
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In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain …
The study establishes comparison theorems for weighted Finsler manifolds and spacetimes.
problem Analyzing weighted Finsler manifolds and spacetimes with curvature conditions.
method Using weight function and -range, the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem are formulated.
result New comparison theorems for weighted Finsler manifolds and spacetimes are derived, including those for weighted Riemannian manifolds.
We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to …
Paper proves a new volume comparison theorem for Riemannian manifolds.
problem Comparing volumes of boundaries in Riemannian manifolds.
method Inspired by Schur's theorem, applies to Riemannian manifolds with Ricci curvature.
result Provides a new Schur's type volume comparison theorem.
Comparison geometry of holomorphic bisectional curvature for Kaehler manifolds and limit spacesmath.DG
Triangle comparison for Kaehler manifolds with curvature bounds.
problem Understanding curvature bounds in Kaehler manifolds and their limits.
method Analog of triangle comparison for Kaehler manifolds with holomorphic bisectional curvature.
result Curvature bounds pass to noncollapsed Gromov-Hausdorff limits.
New theorems compare Laplacian on Kähler manifolds.
problem Comparing Laplacian on Kähler manifolds.
method New curvature notions between Ricci and holomorphic bisectional curvatures.
result Established Laplacian comparison theorems and rigidity theorems.
Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler manifolds.
We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted -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 -Laplacian on geodesic balls.
result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted -Laplacian.
We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.
Equivalences among parabolicity, comparison principle and capacity on complete Riemannian manifoldsmath.DG
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between -parabolicity and the comparison principle for the -Laplace equation.
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
The paper develops heat kernel comparison theorems and applies them to spectral geometry.
problem Developing mathematical tools for spectral geometry.
method Established weighted heat kernel comparison theorems for manifolds with bounded radial curvatures.
result Two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian.
Comparison theorems in centro-affine differential geometry
problem rigidity phenomena of comparison theorems
method study of centro-affine differential geometry
result examples of rigidity phenomena
The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
problem Investigating volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
method Applying similar techniques to derive local rigidity theorems for strictly stable Ricci flat manifolds.
result Derives local rigidity theorems for strictly stable Ricci flat manifolds with respect to σ2-curvature.
The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.
problem Proving Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.
method Analyzing complete noncompact Riemannian manifolds with nonnegative Ricci curvature, applying Talenti's comparison theorem to Poisson equations.
result Obtained the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, - and -moment spectrum, and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.
On Kahler manifolds with Ricci curvature lower bound, assuming the real analyticity of the metric, we establish a sharp relative volume comparison theorem for small balls. The model spaces being compared to are complex space forms, i.e, Kahler manifolds with constant holomorphic sectional curvature. Moreover, we give a…
Proves curvature comparison theorem for manifolds with conical singularities.
problem Comparing scalar mean curvature of manifolds with conical singularities.
method Uses Dirac operator and index theory to prove curvature comparison theorem.
result Proves curvature comparison theorem without knowing the index of the twisted Dirac operator.
Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.
problem Comparing isoperimetric profiles on manifolds with integral Ricci curvature bounds.
method Extending previous work, the study uses integral bounds on Ricci curvature to prove comparison results for isoperimetric profile functions.
result Comparison results for the Isoperimetric profile function in manifolds with integral bounds on Ricci curvature.
Sharp estimates for parabolic equations on manifolds using symmetrization.
problem Estimating solutions to parabolic equations on manifolds.
method Symmetrization techniques and isoperimetric inequalities.
result Generalization of Bandle's comparison to Riemannian setting.
Study compares H-type sub-Riemannian manifolds using uniform metrics.
problem Comparing H-type sub-Riemannian manifolds with Riemannian metrics.
method Establishes sub-Hessian and sub-Laplacian comparison theorems for a family of approximating Riemannian metrics.
result Proves a sharp sub-Riemannian Bonnet-Myers theorem.
Sharp comparison theorems for 3D manifolds with scalar curvature bound.
problem Understanding the geometry and topology of 3D manifolds with scalar curvature constraints.
method Sharp comparison results for Green's function and spectrum, derived from scalar curvature bounds.
result Sharp upper and lower bounds for the Green's function and spectrum of 3D manifolds.
The paper compares heat kernels on manifolds with Robin boundary conditions.
problem Comparing heat kernels on manifolds with different boundary conditions.
method Proving comparison theorems for heat kernels on geodesic balls and minimal submanifolds.
result Eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds.
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.
Uniform Laplace comparison for Kähler Ricci flow on Fano manifolds.
problem Proving convergence of Kähler-Ricci flows on Fano manifolds.
method Uniform integral Laplace comparison, Cheeger-Colding theory, and previous results.
result Direct proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower -weighted Ricci curvature bounds with -range to study comparison geometry.
result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
Study improves Poisson equation solutions on various manifolds.
problem Improving solutions to Poisson equation on different types of manifolds.
method Established estimates for mixed boundary conditions on manifolds with specific curvature properties.
result Generalized existing theorems to broader Riemannian settings.
In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient mani…
CR Sub-Laplacian Comparison and Liouville-type Theorem in a Complete Noncompact Sasakian Manifoldmath.AP
In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…
Gauge theory for familiesmath.GT
Gauge theory for families helps compare 4-manifold groups.
problem Comparing diffeomorphism and homeomorphism groups of 4-manifolds.
method Gauge theory applied to families of 4-manifolds.
result Improved understanding of group comparisons in 4-manifolds.
The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.
problem Volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.
method Gradient integral estimates and level set analysis.
result Sharp volume and area comparisons derived from a gradient integral estimate.
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to -dimensional Lorentz--Finsler manifolds.
result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.
problem Comparing solutions of Poisson equations on Riemannian manifolds with Robin boundary.
method Using Schwarz rearrangement and isoperimetric inequalities.
result Extends results on Poisson equations with .
Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
The paper extends volume comparison results to total σ_l-curvature.
problem Comparing total σ_l-curvature with σ_k-curvature.
method Volume comparison theorem extension to σ-curvature comparison.
result Comparison holds for metrics close to strictly stable positive Einstein metrics.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower boundmath.DG
Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.
problem Analyzing positive solutions of semilinear Dirichlet problems on Riemannian manifolds with Ricci lower bound.
method Sharp pointwise gradient comparison method, derived from admissibility and structural conditions on f.
result Explicit isoperimetric-type inequality and quantitative hot-spot localization estimate.
We show that if a Riemannian manifold satisfies (3,3)-bipolar comparisons and has an open flat subset then it is flat. The same holds for a version of MTW where the perpendicularity is dropped. In particular we get that the (3,3)-bipolar comparison is strictly stronger than the Alexandrov comparison.
For a complete Riemannian manifold with an (1,1)-elliptic Codazzi self-adjoint tensor field on it, we use the divergence type operator and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems like mean curvature…
The paper proves gradient and comparison inequalities for RCD spaces.
problem Gradient and comparison inequalities for RCD spaces.
method Elliptic Dirichlet problems and Talenti-type comparison.
result Sharp, rigid, and stable Talenti-type comparison results.
The paper explores dualities in differential equations and their applications in Riemannian geometry.
problem Developing comparison theorems for mixed type differential equations.
method Utilizing dualities in differential equations and inequalities, and applying them to Riemannian geometry.
result Proves Hessian and Laplacian comparison theorems under various curvature assumptions.
Examines different approaches to Poisson structures in Banach spaces.
problem Exploring various definitions of Poisson structures in Banach spaces.
method Presenting and comparing different definitions of Poisson structures.
result Illustrates differences between existing definitions with examples.
Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.
problem Investigate the first Robin eigenvalue of the p-Laplacian on Riemannian manifolds.
method Prove eigenvalue comparison theorems and establish sharp bounds.
result Sharp lower bounds for the first Robin eigenvalue of the p-Laplacian.