132 papers
This paper confirms the conjecture that every critical point metric is Einstein under the condition of constant norm of the traceless Ricci operator, and additionally proves it for the 3-dimensional case when the traceless Ricci operator satisfies a specific cubic trace inequality.
This paper establishes a uniform upper bound for the eigenvalues of the Hodge Laplacian on closed Riemannian manifolds with lower bounds on Ricci curvature and injectivity radius, as well as an upper bound on diameter, thereby extending Cheng-type eigenvalue comparison theorems to differential forms without requiring sectional curvature bounds.
This paper solves the prescribed Hermitian-Yang-Mills tensor problem on compact Kähler manifolds by proving the existence and uniqueness of a smooth Hermitian metric for any given positive definite tensor under specific positivity conditions, utilizing a new comparison theorem to derive quantitative Chern number inequalities for holomorphic vector bundles and Fano manifolds.
This paper establishes the existence, uniqueness, and exponential convergence of a Ricci flow with prescribed curvature on finite graphs, proving that constant curvature weights exist under specific density conditions and thereby providing an affirmative answer to Chow and Luo's Question 2 regarding combinatorial Ricci flows.
This paper establishes a complete proof of the Kobayashi-Hitchin correspondence for nef and big classes by introducing the concepts of adapted closed positive -currents and -adapted Hermitian-Yang-Mills metrics, thereby proving that a holomorphic vector bundle is slope polystable if and only if it admits such a metric, a result that extends to singular settings and yields new insights into projective flatness and the Bogomolov-Gieseker inequality.
This paper introduces a set of Python tools designed to bridge the gap between the theoretical framework of Kendall's 3D shape space and practical computational workflows, addressing specific limitations in existing libraries like Geomstats to facilitate advanced 3D shape analysis.
This paper establishes a conformal lower bound for weighted Dirac eigenvalues on compact spin manifolds with chiral boundary conditions in terms of the relative Yamabe constant, proving that equality is achieved if and only if the manifold is conformally equivalent to a hemisphere supporting a Killing spinor.
This paper establishes a framework of Courant algebroid relations to connect Dirac structures, spinors, and generalised complex/Kähler geometries, demonstrating how these relations induce T-duality in supersymmetric sigma-models and remain compatible with Type II supergravity equations.
This paper provides a complete local description of the moduli space of dynamical spherically symmetric black hole spacetimes near the Reissner-Nordström family, characterizing the black hole threshold as the extremal leaf of a foliation and establishing universal scaling laws with a critical exponent of $1/2$ alongside the activation of Aretakis instability for threshold solutions.
This paper derives the Hamiltonian formulation for axisymmetric magnetohydrodynamics on the three-sphere and extends Zeitlin's matrix model to three dimensions, creating the first discrete 3D MHD model that preserves the underlying Lie–Poisson geometric structure.
This paper extends the framework of invariant reduction for partial differential equations to handle geometric structures that are rescaled rather than strictly invariant by symmetries, establishing a shift rule that explains the emergence or loss of invariance in reduced systems and enabling the geometric construction of exact solutions without relying on integrability structures like Lax pairs.
This paper formulates a ten-dimensional holomorphic supergravity on a Calabi-Yau five-fold that reproduces heterotic moduli equations, exhibits factorized anomalies reconstructing a double-extension complex for moduli counting, and is conjectured to be an -twisted version of supergravity related to the Costello-Li theory.
This paper extends the non-abelian Hodge correspondence to compact Kähler spaces with klt singularities by establishing an equivalence between polystable Higgs bundles and semi-simple flat bundles on regular loci and proving a descent theorem for Higgs bundles along resolutions, ultimately yielding a quasi-uniformization theorem for projective klt varieties satisfying the orbifold Miyaoka-Yau equality.
This paper establishes a universal upper bound for the smallest area of a 2-dimensional stationary integral varifold in closed Einstein 4-manifolds, demonstrating that this bound depends solely on the manifold's volume and diameter under specific curvature and topological constraints.
This paper extends the probabilistic construction of Kähler-Einstein metrics to Fano manifolds with non-discrete automorphism groups by introducing Gibbs polystability and symmetry-breaking via moment map constraints, conjecturing its equivalence to metric existence and the emergence of unique metrics in the large-N limit, while proving these results for log Fano curves and deriving a strengthened logarithmic Hardy-Littlewood-Sobolev inequality with optimal stability constants.
This paper introduces an intrinsic deformation of smooth function algebras on compact Riemannian manifolds via spectral channel twisting, establishes conditions for its extension to Sobolev algebras and associativity, and demonstrates that classical strict deformations arising from abelian group actions are unified as specific instances of this framework.
This paper investigates the geometry of compact Kähler manifolds with partially semi-positive curvature by proving that such manifolds are rationally connected under specific positivity conditions and establishing structure theorems that classify them as either having high rational dimension or admitting a locally constant fibration with rationally connected fibers and a Ricci-flat base.
This paper investigates a geometric algorithm for constructing biharmonic and conformal-biharmonic maps into spheres, demonstrating that while the approach faces strong restrictions for biharmonic maps on closed domains due to the maximum principle, it offers greater flexibility on non-compact domains and successfully generates explicit critical points for conformal-biharmonic maps between spheres.
This paper establishes Allard-Michael-Simon-type -Sobolev inequalities with explicit, asymptotically sharp, and codimension-free constants for Euclidean minimal submanifolds of arbitrary codimension by utilizing optimal mass transport theory, while also providing a unified proof of recent isoperimetric inequalities by Brendle and Eichmair.
This paper presents new deformation and involutivity theorems for Poisson quasi-Nijenhuis manifolds under factorization hypotheses on their defining forms, supported by examples of involutive structures, to advance the application of this geometry to classical completely integrable systems.