132 papers
This paper establishes a necessary and sufficient condition for embedding homogeneous quandles associated with a triplet into conjugation quandles, thereby generalizing existing results for Alexander quandles and providing new insights into core, Grassmann, and rotation quandles.
This paper provides an affirmative solution to an open problem posed by Brezis and Mironescu in their book *Sobolev Maps to the Circle*, demonstrating that the least mass of area-minimizing integral rectifiable currents with a smooth boundary equals the infimum of areas among smoothly immersed submanifolds sharing that boundary.
This paper establishes the existence and asymptotic behavior of foliations by area-minimizing hypersurfaces in arbitrary-dimensional asymptotically flat manifolds with multiple ends, while also proving a global regularity result for free-boundary area-minimizing hypersurfaces in dimensions up to eight.
This paper establishes the existence of sign-changing solutions to a critical elliptic equation involving a Yamabe type operator on a compact manifold with boundary, contingent upon specific geometric conditions.
This paper establishes sufficient conditions for the existence of longest arcs in left-invariant three-dimensional contact sub-Lorentzian structures on solvable Lie groups and the universal cover of SL(2, R), addressing the nontrivial existence question inherent in such optimal control problems with unbounded control sets.
This paper investigates a one-parametric family of left-invariant, SO(1,1)-symmetric Lorentzian structures on the universal covering of SL(2,R), analyzing their global geodesic optimality and examining how these properties deform into those of a limiting sub-Lorentzian structure.
This paper extends Kajiura's construction of noncommutative deformations of holomorphic line bundles on complex tori by viewing them through the lens of twisted trivial bundles and investigates their mirror duals within the SYZ framework.
This paper establishes a logarithmic Bott localization formula for global holomorphic sections of on a compact complex manifold with a simple normal crossings divisor, extending the theory to non-isolated zero schemes that are local complete intersections and providing a current-theoretic formulation that identifies the local residue with a Coleff-Herrera current.
This paper establishes a sharp rate of convergence for a free-boundary curve shortening flow within a convex domain in , demonstrating that the flow evolves to a round half-point in finite time.
This paper establishes the scattering rigidity of positively homogeneous Hamiltonian systems on manifolds with boundary by proving that the Hamiltonian is uniquely determined up to boundary-fixing canonical transformations via the inversion of X-ray and light ray transforms on Hamiltonian curves, a result applied to demonstrate semiglobal lens rigidity for non-trapping Finsler manifolds.
This paper classifies central extensions for the loop group of area-preserving diffeomorphisms of the 2-sphere and demonstrates that their associated Lie algebra cocycles emerge as fuzzy sphere limits of Kac-Moody cocycles for large in (twisted) loop algebras of , provided a specific rescaling by $6/k^3$.
This paper resolves the third gap problem for closed minimal surfaces in the unit sphere across the entire interval by establishing refined integral identities and new curvature bounds, thereby proving rigidity at the endpoints and providing improved quantitative estimates for the squared norm of the second fundamental form.
This paper resolves a question posed by Steinerberger by proving that the factor in the two-dimensional Green–Wasserstein inequality is necessary and cannot be removed while maintaining a uniform remainder for point sets on compact connected surfaces.
This paper resolves a specific open problem by proving that any complete Riemannian manifold with uniform lower bounds on both Ricci curvature and injectivity radius admits an -close bi-Lipschitz smooth metric that preserves two-sided Ricci curvature bounds and a uniform positive lower bound on injectivity radius.
This paper establishes Liouville theorems, nonexistence results, and gradient estimates for solutions to the quasi-linear equation on complete Riemannian manifolds satisfying a -type Sobolev inequality with integral-bounded negative Ricci curvature, leading to new geometric applications such as proving that manifolds with non-negative Ricci curvature outside a compact set and sufficiently small -norm of negative Ricci curvature possess exactly one end.
This paper establishes a 2-systolic inequality for compact Kähler surfaces with positive scalar curvature that admit a nonconstant holomorphic map to a positive-genus Riemann surface, a class of manifolds identified as ruled surfaces fibred over such curves.
This paper generalizes a 2011 result by Wang and Ou to arbitrary dimensions, proving that any Riemannian submersion from an -dimensional manifold with constant sectional curvature to an -dimensional manifold is biharmonic if and only if it is harmonic.
This essay introduces conformal symmetry by examining the Yamabe operator and its applications in conformal differential geometry and representation theory.
This paper establishes that for generic smooth compact initial surfaces, the mean curvature flow in develops only isolated spherical or nondegenerate neck pinch singularities at the first singular time.
This paper establishes sharper polyhomogeneous mapping properties for the Radon transform and backprojection operator on the unit ball by constructing a double -fibration to desingularize the point-hyperplane relation and utilizing associated -fibration operations to improve upon classic Mellin-based estimates.