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…
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48 results for “Bishop's theorem”
Enhanced Bishop-Gromov theorem for homogeneous and inhomogeneous spaces.
problem Bounding the volume growth of geodesic balls in spaces.
method Introducing coefficient shuffling and using the Raychaudhuri equation, geodesic flow conservation, and the full spectrum of Ricci curvature.
result Upper bounds on the average rate of growth of geodesics for finite-volume inhomogeneous spaces.
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.
We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.
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.
Enhances the Bishop-Gromov theorem for curved spaces, especially at late times.
problem The Bishop-Gromov theorem's volume growth upperbound is often too loose, especially at late times.
method Identified and quantified the effect of shear, using higher curvature invariants to improve the upperbound.
result Tighter upper bounds on late-time growth rates of geodesic balls in homogeneous spaces with non-positive sectional curvature.
Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.
problem Understanding the geometry and topology of manifolds with nonnegative Ricci curvature.
method New spectral inequalities and isoperimetric problems involving unequal weights and warped bubbles.
result Sharp spectral and isoperimetric bounds for manifolds with nonnegative Ricci curvature.
Sharp gradient estimates for positive Ricci curvature manifolds.
problem Understanding geometric properties of manifolds with positive Ricci curvature.
method Proving sharp gradient estimates and monotonicity formulae.
result Sharp gradient estimates and monotonicity formulae for positive Ricci curvature manifolds.
Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…
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.
Study of generalized Bishop frames on curves in 4D space.
problem Understanding frames on curves in 4D space.
method Introducing and studying four types of generalized Bishop frames on curves in .
result Every regular curve in admits all four types of generalized Bishop frames.
On Kahler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov's relative volume comparison, Bonnet-Meyers theorem, and Yau's gradient estimate for positive harmonic functions. The tool is a Boc…
We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant is of infinite…
In this paper, we study spinor Bishop equations of curves in E^3. We research the spinor formulations of curves according to Bishop frames in E^3. Also, the relation between spinor formulations of Bishop frames and Frenet frame are expressed.
In this study, we consider AW(k)-type curves according to the Bishop Frame in Euclidean space E^3. We give the relations between the Bishop curvatures k_1, k_2 of a curve in E^3.
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…
BiSHop tackles tabular data challenges with sparse Hopfield layers.
problem Non-rotationally invariant data structure and feature sparsity in tabular data.
method Sequential column-wise and row-wise processing through interconnected directional learning modules with generalized sparse modern Hopfield layers.
result BiSHop surpasses current SOTA methods with significantly less hyperparameter tuning.
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.
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 abstract develops weighted Ricci curvature in Lorentz-Finsler geometry and extends singularity theorems.
problem Extending singularity theorems in weighted Lorentz-Finsler geometry.
method Generalizing Jacobi, Riccati, and Raychaudhuri equations; applying generalized Bishop inequality.
result Weighted Lorentz-Finsler singularity theorems extended.
Study of generalized Bishop frames on time-like curves in 4D Lorentz space.
problem Characterize frames for time-like curves in 4D Lorentz space.
method Introduced and studied generalized Bishop frames for regular time-like curves in 4D Lorentz space.
result Hierarchy of frames exists for time-like curves in 4D Lorentz space, similar to Euclidean case.
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.
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.
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).
In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, ) to geometrically infinite discrete subgroups of isometries of negatively pinched Hadamard manifolds . We then generalize a theorem of Bishop to prove that every discrete geome…
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…
In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…
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…
Comparison and Rigidity Theorems for geodesic curvatures in two dimensional Alexandrov spacesmath.DG
Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.
problem Geodesic curvature in Alexandrov spaces with curvature below.
method Comparison and rigidity theorems for geodesic curvatures.
result Generalized known results for geodesic curvature in spaces with curvature above.
We extend Gaussian Differential Privacy to curved Riemannian manifolds.
problem Extending Gaussian Differential Privacy to curved spaces.
method Developed a Riemannian Gaussian distribution using the Bishop-Gromov theorem and a MCMC-based algorithm.
result Achieved Gaussian Differential Privacy on general Riemannian manifolds with bounded Ricci curvature.
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
The study compares spectral volumes of manifolds with weakly convex boundaries.
problem Establishing volume comparison theorems for manifolds with weakly convex boundaries.
method Using spectral methods and Ricci tensor eigenvalues, the study compares volumes and diameters of manifolds.
result Sharp upper bounds for the volume and diameter of manifolds with weakly convex boundaries.
Quantum complexity lowerbound proved using differential geometry.
problem Proving lower bounds on quantum complexity.
method Applied the Bishop-Gromov bound to Nielsen's complexity geometry.
result Lower bounds on quantum complexity are exponentially large.
The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further- more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane. The main drawback of the Bishop frame is that the principle normal vector N is not in …
In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regul…
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.
This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions;…
Sullivan showed that there exists such that if is a simply connected hyperbolic domain, then there exists a conformally natural -quasiconformal map from to the boundary of the convex hull of its complement which extends to the identity on . Explicit …
Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
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.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.
problem Investigating inequalities on Finsler manifolds with weighted Ricci curvature.
method Volume comparison, Bonnet-Myers theorem, Poincaré-Lichnerowicz inequality.
result Sharp lower bound for the first eigenvalue on Finsler manifolds.
Holomorphic discs converge to maximal surfaces under specific flows.
problem Understanding the evolution of holomorphic discs under mean curvature flow.
method Mean curvature flow with boundary conditions in the space of oriented lines.
result Holomorphic discs converge to Bishop filling by holomorphic discs under certain conditions.
The paper explores curvature-free effects in manifolds with volume growth and ends-counting.
problem Investigating curvature-free effects in manifolds with volume growth and ends-counting.
method Establishing two main theorems about volume growth and ends-counting.
result Proves the existence of smooth bounded mean-concave exhaustion and escaping geodesic lines.
A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemanni…
We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy on the -Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable -form in the sense of \cite{giglinonsmooth}. This ex…
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.