Study on charged parallel spinors and mass-charge inequalities.
problem Equality case of the spin positive mass theorem with charge.
method Investigation of charged parallel spinors and application to extremal charged manifolds.
result Characterization of the equality case of the mass-charge inequality.
The X-ADM mass is shown to be equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.
problem Proving the X-positive mass theorem for all dimensions.
method Conformal reduction argument.
result The X-ADM mass is equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.
The most general formulation of Penrose's inequality yields a lower bound for ADM mass in terms of the area, charge, and angular momentum of black holes. This inequality is in turn equivalent to an upper and lower bound for the area in terms of the remaining quantities. In this note, we establish the lower bound for a …
The paper proves positive energy-momentum theorems for charged AdS initial data sets.
problem Proving positive energy-momentum theorems for charged asymptotically AdS initial data sets.
method Introducing a charged energy-momentum functional and establishing positive theorems under a dominant energy condition.
result The charged energy-momentum functional is non-negative on a natural real cone.
Establish generalized Chen inequalities for Riemannian submersions and Riemannian maps with applications.
problem Generalized Chen inequalities for Riemannian submersions and Riemannian maps.
method Employing generalized δ-invariants introduced by Chen.
result Optimal inequalities involving δ-invariants and extrinsic invariants.
Establishes Calderón-Zygmund inequalities on evolving Riemannian manifolds.
problem Calderón-Zygmund inequalities on evolving Riemannian manifolds.
method Establishes various Calderón-Zygmund inequalities on evolving Riemannian manifolds with bounded curvature.
result Provides concrete applications of established inequalities.
The study establishes inequalities on path space for sub-Riemannian manifolds.
problem Understanding functional inequalities on path space for sub-Riemannian manifolds.
method Derivative and integration by parts formulae on path space with respect to a natural gradient operator, showing bounds of horizontal Ricci curvature.
result Established functional inequalities on path space analogous to Riemannian geometry.
The paper establishes new Casorati inequalities for various Riemannian maps and submersions.
problem Developing new inequalities for Riemannian maps and submersions.
method Using general forms of Casorati inequalities, the paper derives inequalities for specific Riemannian spaces.
result The paper provides new Casorati inequalities for Riemannian maps and submersions.
The paper studies bi-slant Riemannian maps to Kenmotsu manifolds and derives inequalities.
problem Investigating bi-slant Riemannian maps and their properties.
method Introducing and studying bi-slant Riemannian maps, deriving curvature relations and inequalities.
result Construction of Chen-Ricci inequalities, DDVV inequalities, and optimal inequalities involving Casorati curvatures.
Paper studies quaternionic space forms and Riemannian maps inequalities.
problem Investigate DDVV-type inequality for quaternionic space forms.
method Analyze Riemannian maps from quaternionic space forms to manifolds.
result Derived inequality with equality conditions discussed.
Extends Riemannian geometry inequalities with sharper estimates.
problem Deriving new inequalities on Riemannian manifolds.
method Investigates advanced Hardy and Rellich-type inequalities on complete noncompact manifolds with weight functions.
result Provides sharper estimates conforming to the geometry and structure of the manifold.
We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in t…
Sharp Hardy inequalities on Riemannian submanifolds with non-negative curvature.
problem Establishing Hardy inequalities for submanifolds in Riemannian geometry.
method Analyzing distance functions and using Riemannian submanifolds with non-negative curvature.
result Sharp weighted Hardy inequalities valid for compact and non-compact submanifolds, even in compact ambient manifolds.
The paper proves inequalities for submanifolds in Riemannian manifolds.
problem Proving geometric inequalities for submanifolds in Riemannian manifolds.
method Using Rauch's comparison theorem and first variation formula.
result General Li-Yau inequality applicable in bounded sectional curvature manifolds.
New inequalities for spectral zeta kernels on spheres and manifolds.
problem Establishing new inequalities for spectral zeta functions.
method Applying Kato's inequalities and majorisation techniques.
result Generalized Kato's comparison inequalities to higher dimensions.
The paper derives curvature inequalities for submersions from quaternionic space forms.
problem Deriving curvature inequalities for submersions from quaternionic space forms.
method Analyzing Ricci and scalar curvatures of horizontal and vertical distributions in anti-invariant submersions.
result Established Ricci curvature inequality for anti-invariant submersions.
The paper derives Chen-Ricci inequalities for Riemannian submersions and maps.
problem Chen-Ricci inequalities for Riemannian submersions and maps.
method General forms of Chen-Ricci inequalities for Riemannian submersions and maps are derived, involving curvatures of subspaces.
result New, easy, and elegant techniques for Chen-Ricci inequalities are established.
The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.
problem Investigating dilation type inequalities on weighted Riemannian manifolds.
method Introducing dilation profile and comparing it with model space under lower weighted Ricci curvature bounds.
result Showed several functional inequalities related to various entropies.
The paper derives inequalities for Riemannian submersions and applies them to specific space forms.
problem Deriving inequalities for Riemannian submersions.
method Introducing and deriving inequalities for vertical, horizontal, and mixed distributions of Riemannian submersions.
result Established relationships between intrinsic and extrinsic invariants of Riemannian submersions.
The paper extends inequalities to closed Riemannian manifolds.
problem Extending inequalities from Euclidean domains to Riemannian manifolds.
method Proving explicit bounds using geometric quantities.
result Explicit bounds on Riemannian manifolds are derived.
Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
problem Proving Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
method Using the Alexandrov-Bakelman-Pucci method to prove Michael-Simon type inequalities.
result Extends existing inequalities to the k-Ricci curvature setting and provides isoperimetric inequalities. The paper derives inequalities for Riemannian submersions and their applications.
problem Characterizing Casorati inequalities for Riemannian submersions.
method Algebraic and geometric analysis of Casorati inequalities for normalised scalar and Casorati curvatures.
result Characterization of equality cases for Casorati inequalities in Riemannian submersions.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
Throughout the literature on the charged Riemannian Penrose inequality, it is generally assumed that there is no charged matter present; that is, the electric field is divergence-free. The aim of this article is to clarify when the charged Riemannian Penrose inequality holds in the presence of charged matter, and when …
The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.
problem Proving nonexistence results for parabolic inequalities on Riemannian manifolds.
method Using a test function argument and weighted volume growth assumptions.
result Established Liouville-type theorems for (p,q)-Laplacian operator inequalities. This review explores Ricci soliton inequalities in Riemannian geometry.
problem Understanding geometric and analytic characteristics of Riemannian manifolds.
method Comprehensive study of Ricci soliton inequalities, summarizing historical evolution and current developments.
result Complex interactions between curvature conditions and geometric inequalities.
Introduces systolic inequalities in Riemannian and symplectic geometry.
problem Exploring systolic inequalities in different geometric settings.
method Comparing classical Riemannian metrics to recent symplectic measurements.
result Illustrates connections between Riemannian and symplectic geometry.
Paper proves inequality for p-Laplacian eigenvalues on curved spaces.
problem Eigenvalue inequalities for p-Laplacian on curved manifolds.
method Robin boundary conditions, lower Ricci bounds, positive asymptotic volume ratio.
result Bossel-Daners inequality extends to compact submanifolds.
New inequality for special forms on manifolds.
problem Understanding coexact 1-forms on manifolds.
method Proved a Cheeger-like inequality.
result New inequality for coexact 1-forms.
Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.
problem Linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.
method Deriving inequalities linking these measures on Riemannian manifolds.
result Strengthening and extending existing inequalities to Riemannian manifolds.
Study curvature invariants in sub-Riemannian manifolds.
problem Understand curvature invariants in sub-Riemannian geometry.
method Prove geometrical inequalities for submanifolds with orthogonal distributions.
result Inequalities for submanifolds with orthogonal distributions are derived.
Sharp inequalities for manifolds with nonnegative curvature.
problem Establishing inequalities for manifolds with nonnegative curvature.
method Using the ABP-method, generalizing previous work by Brendle.
result Sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds.
Sharp inequalities on curved spaces with bounded curvature.
problem Establishing inequalities on curved spaces with curvature constraints.
method Using Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below.
result Best constants for inequalities on curved spaces with curvature constraints.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.
problem Proving sharp isoperimetric and Sobolev inequalities in nonnegative Ricci curvature spaces.
method Optimal mass transport theory, symmetrization techniques, and volume non-collapsing properties.
result Sharp isoperimetric and Sobolev inequalities established in Riemannian manifolds with nonnegative Ricci curvature.
The paper explores inequalities on weighted Riemannian manifolds with boundary.
problem Developing inequalities on weighted Riemannian manifolds with boundary.
method Using a Reilly type integral formula associated with the φ-Laplacian.
result Provided inequalities of Brascamp-Lieb type and Colesanti type.
Paper establishes new inequality for Riemannian maps and applies it to various space forms.
problem Developing a new inequality for Riemannian maps and its applications.
method Proposed and utilized a general Chen's first inequality for Riemannian maps and applied it to various space forms.
result Validated the new inequality and compared results with existing approaches.
The paper proves inequalities on Riemannian manifolds using a test function method.
problem Proving differential inequalities with (p,q)-Laplacian on Riemannian manifolds. method Using a test function argument.
result Established Liouville-type theorems under manifold's geometry and potential behavior.
Sharp inequality found for hypersurfaces in curved spaces.
problem Establishing geometric inequalities for hypersurfaces in curved spaces.
method Standard comparison methods in Riemannian Geometry.
result Sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.
Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del …
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 Ric≥(n−1)κ. Sharp inequalities and symmetries on Riemannian surfaces quantified.
problem Understanding symmetries and asymmetries in Riemannian surfaces.
method Introducing scattering energy to measure asymmetry and proving isoperimetric inequalities.
result Sharp quantitative isoperimetric inequalities and domains with vanishing scattering energy characterized.
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
Smooth metrics satisfying Penrose inequality are necessarily smooth.
problem Rigidity of Penrose inequality with singular metrics.
method Showed suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality are smooth in specified coordinates.
result Smooth metrics satisfying Penrose inequality are necessarily smooth.
The paper extends Liouville theorems to sub-Riemannian manifolds.
problem Generalizing Liouville theorems to sub-Riemannian manifolds.
method Constructing 'good' cut-off functions and applying a nonnegative generalized curvature-dimension inequality.
result The Liouville theorems are extended to sub-Riemannian manifolds.
Sharp Sobolev inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived.