The paper solves fractional scalar curvature problems on conformal infinities.
problem Prescribed fractional scalar curvature on conformal infinities.
method Introduced and solved the fractional scalar curvature problem on conformal infinities.
result Existence of smooth solutions to the fractional Yamabe problem in the endpoint case.
Study on fractional curvature flow on unit sphere, extending previous work.
problem Fractional Nirenberg problem on unit sphere.
method Fractional conformal curvature flow on unit sphere.
result Perturbation result for fractional Nirenberg problem with σ∈(1/2,1). Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…
We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient s…
We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli--Silvestre extension for (−Δ)γ when γ∈(0,1), and both…
New connected sum method for zero scalar curvature with constant mean curvature boundary.
problem Prescribing zero scalar curvature with constant mean curvature boundary on connected sums of manifolds.
method Boundary connected sum construction, exploiting nonlocal aspects and recent tools.
result Construction of a connected sum with zero scalar curvature and constant mean curvature boundary.
Second paper in series solves Einstein vacuum equations for three impulsive waves.
problem Solving local Cauchy problem for impulsive gravitational waves.
method Geometric commutators for energy estimates, fractional-derivative regularity, anisotropic Sobolev embedding.
result Scalar field becomes everywhere Lipschitz and C1,θ away from singular region. Compactness theorem for fractional Yamabe problem on asymptotically hyperbolic manifolds.
problem Compactness of solutions to the fractional Yamabe problem on conformal infinity.
method Analyzing convergence of scalar curvature and second fundamental form properties.
result Solution set is compact in C2(M) under specific conditions. Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each …
Introduces fractional length and nonlocal curvature for smooth curves.
problem Defining curvature for curves of fractional length.
method Introduces fractional length and derives nonlocal curvature using fractional perimeter analogy.
result Fractional length converges to traditional length with a multiplicative constant.
Study proves smooth solutions for fractional mean curvature flow within short time.
problem Short-time existence of smooth solutions for fractional mean curvature flow.
method Established using short-time existence theorem for bounded, C^{1,1}-regular initial sets.
result Smooth solutions exist for both fractional mean curvature flow and volume preserving flow.
Existence of unstable shrinking solutions in fractional mean curvature flow.
problem Existence and stability of self-shrinkers in fractional mean curvature flow.
method Existence proof of homothetically shrinking solutions with prescribed boundary conditions.
result Unstable shrinking solutions, except the ball, for fractional mean curvature flow.
First variation of fractional k-dimensional measure for submanifolds
problem Computing the first variation of a fractional k-dimensional measure for submanifolds method First variation computation
result Definition of a nonlocal mean-curvature vector for embedded submanifolds
We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity (Mn,[h]) of a Poincaré-Einstein manifold (Xn+1,g+) with either n=2 or n≥3 and (Mn,[h]) is locally flat - namely (M,h) is locally conformally flat. However, as for the classic…
Local fractional derivatives affect Riemann curvature tensor to zero.
problem Investigating how local fractional derivatives influence the Riemann curvature tensor.
method Introduced a general local fractional derivative operator and defined a specific Riemannian metric tensor field.
result The Riemann curvature tensor of the new metric is identically zero, indicating local isometry to Euclidean space.
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain qu…
Study finds non-uniqueness in sphere metrics with constant fractional curvature.
problem Non-uniqueness of metrics with constant positive fractional curvature on spheres.
method Bifurcation techniques applied to non-local equations with critical non-linearity.
result Non-uniqueness results for complete metrics on Sn∖Sk. The paper extends Heintze-Karcher inequalities to fractional Q-curvature.
problem Extending Heintze-Karcher inequalities to fractional Q-curvature.
method Generalization of Heintze-Karcher inequalities to fractional Q-curvature on conformally compact Einstein manifolds.
result Rigidity theorems for specific values of γ.
In this paper, we mainly study the scattering operators for the Poincaré-Einstein manifolds. Those operators give the fractional GJMS operators P2γ for the conformal infinity. If a Poincaré-Einstein manifolds (Xn+1,g+) is locally conformally flat and there exists an representative g for the conformal infi…
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
Study optimal partitions on spheres using fractional Q-curvature and variational methods.
problem Optimal partition problem on the sphere with fractional Q-curvature.
method Variational approach, symmetry analysis, Hölder regularity results.
result Existence of a symmetric minimal partition.
The paper examines complete Yamabe solitons with finite total scalar curvature.
problem Characterizing complete Yamabe solitons with specific curvature properties.
method Analyzing steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature.
result Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the Q-curvature on noncompact complete manifolds.
Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h-convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
The paper studies special Finsler spaces with Hp-scalar curvature.
problem Characterizing and investigating Finsler spaces with specific scalar curvatures.
method Intrinsic investigation and various conditions for transformations between Finsler spaces.
result Conditions for transforming Finsler spaces of scalar curvature to those of Hp-scalar curvature. Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.
problem Understanding surfaces with bounded fractional mean curvature.
method Investigates bounded L^p-norm of fractional mean curvature, proving control over local parametrization.
result Proves control over local parametrization, leading to lower Ahlfors-regularity, weak Michael-Simon type inequality, and stability application.
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fract…
The scattering operators associated to an ACHE metric of Bergman type on a strictly pseudovonvex domain are a one-parameter family of CR-conformally invariant pseudodifferntial operators of Heisenberg class with respect to the induced CR structure on the boundary. In this paper, we mainly show that if the boundary Webs…
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. Develops fractional de Rham theory for Maxwell equations.
problem Formulating fractional calculus for Maxwell equations.
method Fractional tangent functionals, Riemann-Liouville integral, polynomial algebra, exterior algebra.
result Fractional de Rham complex for Maxwell equations.
The study sets limits on scalar curvature in positive curvature manifolds.
problem Bounding scalar curvature in positive curvature manifolds.
method Establishing inequalities for manifolds with positive scalar curvature and scalar curvature bounded from below.
result Established inequalities for manifolds with positive scalar curvature.
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
Research on unique continuation principles in medical and seismic imaging.
problem Understanding unique continuation principles for inverse problems.
method Integral geometry and fractional calculus methods applied to various imaging problems.
result Developed new techniques for solving inverse problems with partial data.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
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.
Study finds curves with explicit formulas for curvature and torsion.
problem Finding curves with specific geometric properties.
method Developed a family of curves parametrized by arc length, dependent on angular and intrinsic fraction functions.
result Explicit formulas for curvature, torsion, and geodetic curvature found in terms of angular and intrinsic fraction functions.
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
The study explores scalar curvatures on manifolds with boundary properties.
problem Understanding scalar curvatures on manifolds with boundary constraints.
method Presentation of problems and results related to scalar curvatures and mean curvatures of boundaries.
result Exploration of natural and artificial constructions in scalar curvature studies.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
Minimal splitting factors help study scalar curvature constraints.
problem Scalar curvature constraints in geometry.
method Introducing minimal splitting factors with positive scalar curvature.
result Minimal splitting factors have properties similar to area minimizing hypersurfaces.