Article
Mathematics
Xiaochun Rong, Yusheng Wang
Summary: For an n-dimensional Alexandrov space X with curvature >= 1, X can be isometric to a finite quotient of join if it contains two compact convex subsets X, without boundary, such that their dimensions sum up to n-1 and they are at least pi/2 apart.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Elia Brue, Andrea Mondino, Daniele Semola
Summary: We solve a conjecture raised by Kapovitch, Lytchak, and Petrunin in [KLP21] by proving that the metric measure boundary vanishes on any RCD(K, N) space (X, d, H-N) without boundary. Our result, in conjunction with [KLP21], settles an open question about the existence of infinite geodesics on Alexandrov spaces without boundary raised by Perelman and Petrunin in 1996.
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2023)
Article
Astronomy & Astrophysics
Alex S. Arvanitakis, David Tennyson
Summary: This study introduces a new technique to achieve brane wrapping and double dimensional reduction in the AKSZ topological sigma models, and discovers a new relation between two manifolds.
Article
Mathematics, Applied
Shicheng Xu, Xuchao Yao
Summary: This article proves the generalized Margulis lemma on an Alexandrov n-space X with curvature bounded below, providing a bound on the index of nilpotent subgroups in the fundamental group. It also discusses regular almost Lipschitz submersions constructed by Yamaguchi and gives a detailed proof on gradient push.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2022)
Article
Mathematics
Jian Ge, Nan Li
Summary: Perel'man's Doubling Theorem and Petrunin's Gluing Theorem have many applications in the study of spaces with lower curvature bounds. This study explores the gluing of multiple Alexandrov spaces and advances the research on the Gluing Conjecture through specific conditions and proofs.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Curtis Pro, Frederick Wilhelm
Summary: This paper discusses the convergent sequence of Riemannian n-manifolds under certain conditions and answers the Diffeomorphism Stability Question. The results show that in special cases, all but finitely many of the M(alpha)s are diffeomorphic.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Shengqi Hu, Xiaole Su, Yusheng Wang
Summary: This note presents an elementary proof for Toponogov's theorem in Alexandrov geometry with a lower curvature bound. The proof is based on the idea that in Riemannian geometry, sectional curvature can be manifested in geodesic variations.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Dorin Bucur, Ilaria Fragala
Summary: In this study, we investigate the rigidity problem of measurable sets with constant nonlocal h-mean curvature. It is shown that under certain conditions, these sets can be characterized as finite unions of equal balls, and the radius and mutual distance of these balls can be controlled by suitable parameters.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2023)
Article
Mathematics
Josef Schicho
Summary: This paper investigates the configurations of mechanical linkages and provides mathematical explanations for unexpected mobility in some cases.
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Xiaochun Rong, Yusheng Wang
Summary: This paper proves the Soul Conjecture in Alexandrov geometry in dimension 4, which states that in a complete non-compact 4-dimensional Alexandrov space with non-negative and positive curvature, the soul of the space is a point.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Christian Ketterer, Yu Kitabeppu, Sajjad Lakzian
Summary: We extend the rigidity property of the first spectral gap to compact infinitesimally Hilbertian spaces with non-negative Ricci curvature and bounded dimension. This category of metric measure spaces includes various types of spaces such as non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry-Emery manifolds, and measured Gromov-Hausdorff limits. Our results show that in these spaces, the first eigenvalue is equal to π^2/diam^2 if and only if the space is one-dimensional with a constant density function. We use new techniques combining Sobolev theory and singular 1D-localization.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics
Nina Lebedeva, Shin-ichi Ohta, Vladimir Zolotov
Summary: In this article, we show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound that includes finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. Our condition, regarded as a strengthened doubling condition, also holds for certain metric spaces with upper curvature bound. Additionally, we demonstrate the non-embeddability of large snowflakes into metric spaces in the same class, following a strategy previously applied to spaces with upper curvature bound.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Interdisciplinary Applications
Yao Pengfei
Summary: The optimal exponents of thickness in the geometry rigidity inequality of shells indicate the geometry rigidity of the shells. Lower bounds of the optimal exponents for hyperbolic, parabolic, and elliptic shells are found to be 4/3, 3/2, and 1 respectively through the construction of Ansatze.
JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
(2021)
Article
Mathematics, Applied
Ben Andrews, Xuzhong Chen, Yong Wei
Summary: This paper focuses on studying flows of hypersurfaces in hyperbolic space and their applications in proving geometric inequalities. By analyzing curvature functions and convergence results, conclusions are drawn regarding the validity of geometric inequalities under different conditions.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Ara Basmajian, Hrant Hakobyan, Dragomir Saric
Summary: In this work, the parabolicity of Riemann surfaces is studied, with a focus on the influence of the Fenchel-Nielsen parameters of hyperbolic pants decompositions. The study reveals that the modulus of nonstandard half-collars depends on the twist parameter and can provide a complete characterization of parabolicity in certain cases.
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Nan Li
JOURNAL OF TOPOLOGY AND ANALYSIS
(2015)
Article
Mathematics
Vitali Kapovitch, Nan Li
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2018)
Article
Mathematics, Applied
Nan Li, Feng Wang
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS
(2014)
Article
Mathematics
Nan Li, Xiaochun Rong
PACIFIC JOURNAL OF MATHEMATICS
(2012)
Article
Mathematics
Shu-Cheng Chang, Yingbo Han, Nan Li, Chien Lin
Summary: This paper shows the existence of a nonconstant CR-holomorphic function of polynomial growth in a complete noncompact Sasakian manifold with nonnegative pseudohermitian bisectional curvature and the CR maximal volume growth property. This is the first step towards the CR analogue of the Yau uniformization conjecture, which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group. The paper constructs CR-holomorphic functions with controlled growth in a sequence of exhaustion domains in Sasakian manifolds using the Cheeger-Colding theory, and then obtains a nonconstant CR-holomorphic function of polynomial growth by considering the CR analogue of a tangent cone at infinity and the three-circle theorem.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2023)
Article
Mathematics
Nan Li, Xiaochun Rong
JOURNAL OF DIFFERENTIAL GEOMETRY
(2012)
Article
Mathematics
Matija Bucic, Richard Montgomery
Summary: This article improves upon previous research by showing that any n-vertex graph can be decomposed into O(n log* n) cycles and edges.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Summary: The paper investigates the relationship between linear optimization degree and geometric structure. By analyzing the geometric structure of the conormal variety of an affine variety, the Chern-Mather classes of the given variety can be completely determined. Additionally, the paper shows that these bidegrees coincide with the linear optimization degrees of generic affine sections.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
William Chan, Stephen Jackson, Nam Trang
Summary: Under the determinacy hypothesis, this paper completely characterizes the existence of nontrivial maximal almost disjoint families for specific cardinals kappa, considering the ideals of bounded subsets and subsets of cardinality less than kappa.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhenguo Liang, Zhiyan Zhao, Qi Zhou
Summary: This paper investigates the reducibility of the one-dimensional quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form. It provides a description and upper bound for the growth of the Sobolev norms of the solution, and demonstrates the optimality of the upper bound.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhao Yu Ma, Yair Shenfeld
Summary: This study provides a new approach to understanding the extremal cases of Stanley's inequalities by establishing a connection between the combinatorics of partially ordered sets and the geometry of convex polytopes.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurent Laurent, Rosa M. Miro-Roig
Summary: This paper discusses the problem of constructing matrices of linear forms of constant rank by focusing on vector bundles on projective spaces. It introduces important examples of classical Steiner bundles and Drezet bundles, and uses the classification of globally generated vector bundles to describe completely the indecomposable matrices of constant rank up to six.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Nicoletta Cantarini, Fabrizio Caselli, Victor Kac
Summary: In this paper, we construct a duality functor in the category of continuous representations to study the Lie superalgebra E(4, 4). By constructing a specific type of Lie conformal superalgebra, we obtain that E(4, 4) is its annihilation algebra. Furthermore, we also obtain an explicit realization of E(4, 4) on a supermanifold in the process of studying.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Rotem Assouline, Bo'az Klartag
Summary: This article studies the horocyclic Minkowski sum of two subsets in the hyperbolic plane and its properties. It proves an inequality relating the area of the subsets when they are Borel-measurable, and provides a connection to other inequalities.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Alessio Porretta
Summary: This article discusses Fokker-Planck equations driven by Levy processes in the entire Euclidean space, under the influence of confining drifts, similar to the classical Ornstein-Ulhenbeck model. A new PDE method is introduced to obtain exponential or sub-exponential decay rates of zero average solutions as time goes to infinity, under certain diffusivity conditions on the Levy process, including the fractional Laplace operator as a model example. The approach relies on long-time oscillation estimates of the adjoint problem and applies to both local and nonlocal diffusions, as well as strongly or weakly confining drifts.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Weichao Qian, Yong Li, Xue Yang
Summary: In this paper, we investigate the persistence of resonant invariant tori in Hamiltonian systems with high-order degenerate perturbation, and prove a quasiperiodic Poincare theorem under high degeneracy, answering a long-standing conjecture on the persistence of resonant invariant tori in general situations.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Julius Ross, David Witt Nystroem
Summary: This article extends Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity, and applies it to the interpolation problem of convex functions and convex sets, introducing a new notion of harmonic interpolation.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Airi Takeuchi, Lei Zhao
Summary: In this article, we explore the connection between several integrable mechanical billiards in the plane through conformal transformations. We discuss the equivalence of free billiards and central force problems, as well as the correspondence between integrable Hooke-Kepler billiards. We also investigate the integrability of Kepler billiards and Stark billiards, and the relationship between billiard systems and Euler's two-center problems.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Damiano Rossi
Summary: In this study, we prove new results in generalised Harish-Chandra theory by providing a description of the Brauer-Lusztig blocks using the p-adic cohomology of Deligne-Lusztig varieties. We then propose new conjectures for finite reductive groups by considering geometric analogues of the p-local structures. Our conjectures coincide with the counting conjectures for large primes, thanks to a connection established between p-structures and their geometric counterparts. Finally, we simplify our conjectures by reducing them to the verification of Clifford theoretic properties.
ADVANCES IN MATHEMATICS
(2024)
