132 papers
This paper establishes necessary and sufficient conditions for reparametrizing nonholonomic mechanical trajectories as geodesics of a conformally modified Riemannian metric, while clarifying the relationships between these projective geodesic extensions and concepts like -simplicity, invariant measures, and Hamiltonization in Chaplygin systems.
Motivated by the Schoen–Marques–Neves conjecture, this paper verifies a sufficient pointwise Ambrozio–Carlotto–Sharp inequality for minimal isoparametric hypersurfaces with positive Ricci curvature in unit spheres, thereby establishing a lower bound on the Morse index proportional to the first Betti number for closed embedded minimal hypersurfaces in these ambient manifolds.
This paper defines analogs of -volume, Epstein surfaces, and Liouville action within the correspondence between anti-de Sitter 3-space and -conformal metrics, applying these constructions to derive finite invariants for positive curves in flag manifolds.
This paper establishes a pointwise geometric description of nonlinear Lebesgue spaces, including their length structure, curvature bounds, and speed definitions for absolutely continuous curves, by first proving a nonlinear Fubini–Lebesgue theorem that identifies L^p curves with mappings into the space of L^p curves.
This paper proves the stable degeneration conjecture for log Fano fibration germs by introducing the -invariant to identify a unique minimizing valuation that induces a special degeneration to a K-semistable germ, which further admits a unique K-polystable special degeneration.
This paper presents a vector generalization of the curvature-corrected Cramér-Rao bound for multivariate parameters in the nonasymptotic regime, utilizing extrinsic geometry and sum-of-squares relaxations to derive directional and matrix-valued refinements that offer more faithful estimation limits than classical second-order corrections, as demonstrated through curved Gaussian and spherical multinomial models.
This paper presents a geometric refinement of the classical Cramér-Rao bound in the non-asymptotic regime by leveraging the extrinsic curvature of the statistical model manifold's square root embedding to derive tighter, curvature-aware lower bounds on estimator variance.
This paper classifies all ancient smooth embedded finite-entropy curve shortening flows into five specific categories: static lines, shrinking circles, paper clips, translating grim reapers, and graphical ancient trombones, thereby establishing that all such compact flows are convex and characterizing the structure of non-compact ones.
This paper generalizes Simpson's main estimates for cyclic harmonic -bundles induced by split automorphisms and applies these results to classify Toda type -harmonic bundles.
This paper resolves the Einstein-scalar field conformal constraint equations under spherically symmetric and harmonic assumptions, revealing that while solutions on compact manifolds like the sphere exhibit nonexistence and instability in near-CMC regimes, the equations are always solvable on Euclidean and hyperbolic manifolds, thereby supporting the conformal method's utility for asymptotically flat and hyperbolic initial data.
This paper refines and extends recent results by Nagy and Semmelmann by demonstrating that the second-order Taylor expansion of Einstein deformations for negative Kähler-Einstein metrics is fully determined by the square of the initial deformation and the divergence of the Kodaira-Spencer bracket, thereby establishing a precise link between second-order Einstein deformation theory and the complex geometry of the underlying manifold.
This paper establishes a definition of quasi-local energy for initial data sets with an umbilic second fundamental form in an expanding de Sitter background and proves its positivity for certain bounded values of the cosmological constant by adapting the Liu-Yau energy framework.
This paper presents a proof of the reverse isoperimetric inequality for black holes in Einstein gravity with using a geometric-analytic approach, demonstrating that the reversal of the usual inequality stems from the curved background structure dictated by Einstein's equations.
This paper classifies left-invariant Ricci solitons on the Lie group as expanding and non-gradient, while also investigating harmonic maps into this geometry and characterizing a specific class of harmonic vector fields.
This paper proves that for a product manifold where has constant positive scalar curvature, the subcritical Yamabe equation admits -peak positive solutions for any when is sufficiently small, provided the scalar curvature of is constant or a specific dimensional constant vanishes and is a stable critical point of a curvature-dependent function.
This paper introduces a Bianchi-Calo type construction method for generating Bryant-type linear Weingarten surfaces within hyperbolic space.
This paper proposes a novel machine learning framework that combines gradient descent on the Grassmannian manifold with Donaldson's algorithm to efficiently compute Ricci-flat approximations of Calabi-Yau metrics, demonstrating its effectiveness and the emergence of nontrivial local minima on the Dwork family of threefolds.
This paper investigates cosmological spacetimes where the spatial curvature is promoted to a time-dependent function, demonstrating that such sign-changing transitions—which allow for topological changes from closed to open universes while avoiding infinite initial energy densities—are consistent with global hyperbolicity under specific conditions and constructing three distinct geometries that exhibit these properties.
This paper generalizes the spacetime positive mass theorem to multiple time dimensions, proving that energy remains nonnegative and bounded by the trace norm of linear momenta, with equality implying a foliation by flat submanifolds and, under an umbilicity assumption, an isometric embedding into a generalized pp-wave.
This paper clarifies the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces used in Cartan Neural Networks by proving that only Kähler spaces support Gibbs distributions, explicitly characterizing their generalized temperature spaces via adjoint orbits, and demonstrating the equivalence between various information and thermodynamical geometries while establishing the covariance of these distributions under the full symmetry group.