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.
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.
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to (1+n)-dimensional Lorentz--Finsler manifolds. result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
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.
Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.
problem Volume comparison and monotonicity for Bakry-Émery Ricci curvature.
method Relative volume comparison theorem for LP-bound of Bakry-Émery Ricci curvature and gradient of potential function. result Modified proof for volume comparison and monotonicity of Kähler-Ricci flow.
New proof of Bishop's theorem using soap bubbles with singularities.
problem Proving Bishop's volume comparison theorem for manifolds with Ricci curvature.
method Using isoperimetric hypersurfaces (soap bubbles) with singularities.
result Successfully overcame the challenge of singularities to prove the theorem.
New volume comparison theorem for gradient Ricci almost solitons.
problem Volume comparison and rigidity of gradient Ricci almost solitons.
method Established a new volume comparison theorem with Bakry-Emery Ricci curvature.
result New volume rigidity result for gradient Ricci almost solitons.
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for f-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such 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 extends a theorem about scalar curvature and volume in higher dimensions.
problem Finding sharp volume bounds for manifolds with specific curvature conditions.
method Axis symmetry or upper bound on Ricci curvature used to extend the theorem.
result The extension of Bray's football theorem to higher dimensions.
We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.
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.
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 …
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.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…
Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local v…
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.
We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…
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.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
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…
In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.
Bounding characteristic numbers of Riemannian manifolds via volume.
problem Bounding characteristic numbers of Riemannian manifolds.
method Using Chern-Weil theory and connections constructed from harmonic metric tensors with bounded Hölder norms.
result Characteristic numbers are bounded proportionally to the volume of Riemannian manifolds.
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 ε-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 consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.
The study proves a gap theorem for shrinking gradient Ricci solitons with specific curvature and volume conditions.
problem Characterizing shrinking gradient Ricci solitons with given curvature and volume constraints.
method Combining Günther's volume comparison theorem and Yokota's gap theorem, the study proves a gap theorem.
result Complete shrinking gradient Ricci solitons with specific curvature and volume constraints are isometric to the Gaussian soliton.
This note explores comparison geometry concepts and theorems.
problem Exploring various comparison theorems in geometry.
method Analyzes Rauch and Toponogov theorems and their applications.
result Introduction of Gromov-Hausdorff convergence and Alexandrov Spaces.
The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
problem Proving finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
method The approach removes constraints of sectional curvature or conjugate radius and extends to previous related studies.
result Theorems are proven for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume, without the need for triangle comparison of Toponogov type.
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.
In this paper, we study the complete bounded λ-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with ∣A∣≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…
The study solves a problem in conformal geometry with applications to Q-curvature.
problem Existence of solutions to conformally invariant equations.
method Volume comparison theorems and volume rigidity theorems with respect to Q-curvature.
result Sufficient and necessary conditions for the existence of solutions to conformally invariant equations.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
Gradient estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. 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.
We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…
We establish volume comparison results for balls in Riemannian manifolds with C1,1-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with C1,1-metrics with a lower bound on the tim…
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we o…
The paper extends comparison theorems using a new tensor field and operator.
problem Extending comparison theorems in Riemannian geometry.
method Using a divergence type operator and an extended Ricci tensor.
result Upper bounds for the end of manifolds and new theorems.
Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.
problem Understanding conformal metrics with finite total Q-curvature.
method Introduces conformal mass and provides necessary and sufficient conditions for normality.
result Derives volume comparison theorems and proves a positive mass type theorem related to Q-curvature.
The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.
problem Analyzing diffusion processes on Riemannian manifolds with weighted metrics.
method Proving a Laplacian comparison theorem and applying it to various geometric and analytic properties of diffusion processes.
result Optimal conditions on m-Bakry-Émery Ricci tensor for various geometric and analytic properties to hold on weighted complete Riemannian manifolds. 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.
Second part of Q-curvature research focusing on volume comparison.
problem Volume control and rigidity of Q-curvature.
method Volume comparison and local rigidity analysis.
result Volume comparison theorem for metrics close to strictly stable positive Einstein metrics.
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…
In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the reversible case, a lower bound of injective radius is also obtained.